Dissipative Solitons: The Structural Chaos And The Chaos Of Destruction Vladimir L. Kalashnikov Institute for Photonics, Technical University of Vienna, Vienna, Austria (E-mail: [email protected]) Abstract. Dissipative soliton, that is a localized and self-preserving structure, de- velops as a result of two types of balances: self-phase modulation vs. dispersion and dissipation vs. nonlinear gain. The contribution of dissipative, i.e. environ- mental, effects causes the complex “far from equilibrium” dynamics of a soliton: it can develop in a localized structure, which behaves chaotically. In this work, the chaotic laser solitons are considered in the framework of the generalized complex nonlinear Ginzburg-Landau model. For the first time to our knowledge, the model of a femtosecond pulse laser taking into account the dynamic gain saturation cov- ering a whole resonator period is analyzed. Two main scenarios of chaotization are revealed: i) multipusing with both short- and long-range forces between the solitons, and ii) noiselike pulse generation resulting from a parametrical interaction of the dissipative soliton with the linear dispersive waves. The noiselike pulse is characterized by an extremely fine temporal and spectral structure, which is similar to that of optical supercontinuum. Keywords: Dissipative soliton, Complex nonlinear Ginzburg-Landau equation, Chaotic soliton dynamics. 1 Introduction The nonlinear complex Ginzburg-Landau equation (NCGLE) has a lot of applications in quantum optics, modeling of Bose-Einstein condensation, condensate-matter physics, study of non-equilibrium phenomena, and non- linear dynamics, quantum mechanics of self-organizing dissipative systems, and quantum field theory [1]. In particular, this equation being a general- ized form of the so-called master equation provides an adequate description of pulses generated by a mode-locked laser [2]. Such pulses can be treated as the dissipative solitons (DSs), that are the localized solutions of the NC- GLE [3]. It was found, that the DS can demonstrate a highly non-trivial dynamics including formation of multi-soliton complexes [4], soliton explo- sions [5], noise-like solitons [6], etc. The resulting structures can be very complicated and consist of strongly or weakly interacting solitons (so-called soliton molecules and gas) [7] as well as the short-range noise-like oscillations inside a larger wave-packet [8]. The nonlinear dynamics of these structures can cause both regular and chaotic-like behavior. In this article, the different scenarios of the soliton structural chaos will be considered for the chirped DSs formed in the all-normal group-delay dis- Proceedings, 4 th Chaotic Modeling and Simulation International Conference 31 May – 3 June 2011, Agios Nikolaos, Crete Greece
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Dissipative Solitons: The Structural ChaosAnd The Chaos Of Destruction
Vladimir L. Kalashnikov
Institute for Photonics, Technical University of Vienna, Vienna, Austria(E-mail: [email protected])
Abstract. Dissipative soliton, that is a localized and self-preserving structure, de-velops as a result of two types of balances: self-phase modulation vs. dispersionand dissipation vs. nonlinear gain. The contribution of dissipative, i.e. environ-mental, effects causes the complex “far from equilibrium” dynamics of a soliton: itcan develop in a localized structure, which behaves chaotically. In this work, thechaotic laser solitons are considered in the framework of the generalized complexnonlinear Ginzburg-Landau model. For the first time to our knowledge, the modelof a femtosecond pulse laser taking into account the dynamic gain saturation cov-ering a whole resonator period is analyzed. Two main scenarios of chaotizationare revealed: i) multipusing with both short- and long-range forces between thesolitons, and ii) noiselike pulse generation resulting from a parametrical interactionof the dissipative soliton with the linear dispersive waves. The noiselike pulse ischaracterized by an extremely fine temporal and spectral structure, which is similarto that of optical supercontinuum.Keywords: Dissipative soliton, Complex nonlinear Ginzburg-Landau equation,Chaotic soliton dynamics.
1 Introduction
The nonlinear complex Ginzburg-Landau equation (NCGLE) has a lot ofapplications in quantum optics, modeling of Bose-Einstein condensation,condensate-matter physics, study of non-equilibrium phenomena, and non-linear dynamics, quantum mechanics of self-organizing dissipative systems,and quantum field theory [1]. In particular, this equation being a general-ized form of the so-called master equation provides an adequate descriptionof pulses generated by a mode-locked laser [2]. Such pulses can be treatedas the dissipative solitons (DSs), that are the localized solutions of the NC-GLE [3]. It was found, that the DS can demonstrate a highly non-trivialdynamics including formation of multi-soliton complexes [4], soliton explo-sions [5], noise-like solitons [6], etc. The resulting structures can be verycomplicated and consist of strongly or weakly interacting solitons (so-calledsoliton molecules and gas) [7] as well as the short-range noise-like oscillationsinside a larger wave-packet [8]. The nonlinear dynamics of these structurescan cause both regular and chaotic-like behavior.
In this article, the different scenarios of the soliton structural chaos willbe considered for the chirped DSs formed in the all-normal group-delay dis-
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persion region [9]. The first scenario is an appearance of the chaotic finegraining of DS. For such a structure, the mechanism of formation is identi-fied with the parametric instability caused by the resonant interaction of DSwith the continuum. The second scenario is formation of the multi-solitoncomplexes governed by both short-range forces (due to solitons overlapping)and long-range forces (due to gain dynamics). The underlying mechanism offormation is the continuum amplification, which results in the soliton pro-duction or/and the dynamical coexistence of DSs with the continuum.
2 Chirped dissipative solitons of the NCGLE
Formally, the NCGLE consists of the nondissipative (hamiltonian) and dissi-pative parts. The nondissipative part can be obtained from variation of theLagrangian [10]:
L =i
2
[A∗ (x, t)
∂A (x, t)∂t
−A (x, t)∂A∗ (x, t)
∂t
]+
+β
2∂A (x, t)
∂t
∂A∗ (x, t)∂t
− γ
2|A (x, t)|2 , (1)
where A(x, t) is the field envelope depending on the propagation distance xand the “transverse” coordinate t (that is the local time in our case), β is thegroup-delay dispersion (GDD) coefficient (negative for the so-called normaldispersion case), and γ is the self-phase modulation (SPM) coefficient [11].The dissipative part is described by the driving force:
Q = −iΓA (x, t) + iρ
1 + σ∫∞−∞ |A|
2dt′
[A (x, t) + τ
∂2
∂t2A (x, t)
]+
+iκ[|A (x, t)|2 − ζ |A (x, t)|4
]A (x, t) , (2)
where Γ is the net-dissipation (loss) coefficient, ρ is the saturable gain (σis the inverse gain saturation energy if the energy E is defined as E ≡∫∞−∞ |A|
2dt′ ), τ is the parameter of spectral dissipation (so-called squared
inverse gainband width), and κ is the parameter of self-amplitude modula-tion (SAM). The SAM is assumed to be saturable with the correspondingparameter ζ.
Then, the desired CNGLE can be written as
i∂A (x, t)
∂x− β
2∂2
∂t2A (x, t)− γ |A (x, t)|2 A (x, t) =
= −iΓA (x, t) + iρ
1 + σ∫∞−∞ |A|
2dt′
[A (x, t) + τ
∂2
∂t2A (x, t)
]+
+iκ[|A (x, t)|2 − ζ |A (x, t)|4
]A (x, t) . (3)
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Eq. (3) is not integrable and only sole exact soliton-like solution is knownfor it [11,12]. Nevertheless, the so-called variational method [10] allows ex-ploring the solitonic sector of (3). The force-driven Lagrange-Euler equations
∂∫∞−∞L dt
∂f− ∂
∂x
∂∫∞−∞L dt
∂f= 2<
∫ ∞
−∞Q
∂A
∂fdt (4)
allow obtaining a set of the ordinary first-order differential equations for aset f of the soliton parameters if one assumes the soliton shape in the formof some trial function A(x, t) ≈ F (t, f). One may chose [13]
F = a (x) sech(
t
T (x)
)exp
[i
(φ(x) + ψ(x) ln
(sech(
t
T (x))))]
, (5)
with f = a(x), T (x), φ(x), ψ(x) describing amplitude, width, phase, andchirp of the soliton, respectively.
Substitution of (5) into (4) results in four equations for the soliton parame-ters. These equations are completely solvable for a steady-state propagation(i.e. when ∂xa = ∂xT = ∂xψ = 0, ∂xφ 6= 0 ). The analysis demonstratesthat the solitonic sector can be completely characterized by two-dimensionalmaster diagram, that is the DS is two-parametrical and the correspondingdimensionless parameters are c ≡ τγ/|β|κ and the “energy” E ≡ Eb
√κζ/τ
(here b ≡ γ/κ).The master diagram is shown in Fig. 1. The curves correspond to the
stability threshold defined as Γ − ρ/(
1 + σ∫∞−∞ |A|
2dt
)= 0. Positivity of
this value provides the vacuum stability. As will be shown, the vacuum desta-bilization is main source of the soliton instability causing, in particular, thechaotic dynamics. The master diagram in Fig. 1 reveals a very simple asymp-totic for the maximum energy of DS: E ≈ 17 |β|/√κζτ . The continuum risesabove this energy.
3 Resonant excitation of continuum
The stability threshold shown in Fig. 1 corresponds to an unperturbed DSof (3). The physically meaningful perturbation results from a higher-orderdispersion correction to the Lagrangian: L = L0 + iδ
2∂2A∂t2
∂A∗∂t , where L0 is
the unperturbed Lagrangian (1) and δ is the third-order dispersion (TOD)parameter.
The DS of unperturbed Eq. (3) does not interact with the continuum andthe collapse-like instability is suppressed by ζ >0. Nevertheless, the DS peakpower on the asymptotic stability threshold of Fig. 1 is ≈ 1.1/ζ > 1/ζ that,in accordance with [8] has to result in the chaotic behavior. However, such achaotic layer in the vicinity of stability threshold has not been revealed by ournumerical simulations. A possible explanation is that the solitonic sector (5)
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-6 -5 -4 -3 -2 -1 1
2
4
6
Fig. 1. Master diagram for the chirped DS. The vertical axis is log10 E and thehorizontal axis is log10 c. The stability thresholds are shown for b =100 (brown), 5(green), 2 (cyan) and the asymptotic E = 17/c (magenta). The DS is stable belowthe corresponding curves.
is not isolated and there exists another stable solitonic sector correspondingto the DS with the so-called “finger-like” spectra [9,14]. Such a spectrumhas a main part of the power in the vicinity of the spectrum center. As aresult, a spectral loss decreases that corresponds to energy growth close to theboundary of the DS stability. In turn, a concentration of power close to thespectrum center corresponds to a similar power concentration around a pulsepeak in the time domain, that allows the stable DSs with the peak powers> 1/ζ and, thus, a new solitonic sector appears. Such a sector correspondsto
F =a (x)√
θ (x) + cosh(
tT (x)
) exp[i
(φ (x) + ψ (x) ln
(θ (x) + cosh
(t
T (x)
)))]
(6)with θ(x) >1 and requires a further exploration.
Nevertheless for any solitonic sector, δ 6=0 can result in an appearance ofinteraction with the continuum at some frequency [15]. Hence, the stabilitythreshold becomes lower than that shown in Fig. 1. As the resonance occursin the spectral domain, an exploration of the DS spectrum is most informativein this case. The numerical results corresponding to a mode-locked Cr:ZnSeoscillator [16] are shown in Fig. 2. Non-zero δ can be treated as a frequency-dependence of net-GDD (dashed curves). As a result of such dependence, thezero GDD shifts towards the DS spectrum with the growing |δ| (black solidcurve and black dashed line correspond to δ =0). The vertex of truncated DSspectrum (solid curves) traces the GDD line (dashed lines) logarithmically.When the zero GDD (see orange dashed line) reaches the DS spectrum, theresonant generation of continuum (dispersive wave) begins (longer wavelength
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side of the spectrum demonstrates such a wave; orange and violet curves inFig. 2).
2.35 2.40 2.45 2.50 2.55 2.60 2.65
10-12
10-11
10-10
10-9
spec
tral p
ower
, arb
. un.
, m
-900
-600
-300
0
300
600
900
GD
D, fs
2
Fig. 2. Spectra of the chirped DSs (solid curves) corresponding to the differentnet-GDD (dashed curves). The GDD slope depends on the TOD value.
As a result of resonant interaction with the dispersive wave, when thezero GDD approaches the central wavelength of an unperturbed DS spectrum(≈2.5 µm in our case), the irregular beatings of the DS peak power develop(violet curve in Fig. 3, left). In the time domain, the resonant interactionforms a fine (femtosecond) structure in the vicinity of pulse peak (Fig. 3,right). Such a structure enhances with the shift of zero GDD towards thecentral wavelength of an unperturbed DS spectrum and spreads to a wholespectrum. As a result, the DS envelope becomes to be strongly distorted(gray curve Fig. 3, right) and disintegrates.
4000 4500 50000.04
0.06
0.08
peak
pow
er, a
rb. u
n.
transit number
-15000 -10000 -5000 0 50000.00
0.02
0.04
0.06
0.08
pow
er, a
rb. u
n.
t, fs
Fig. 3. Peak power evolution (left) and DS envelopes (right) for the GDD curvesof Fig. 2.
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4 Nonresonant excitation of continuum
Another correction to (3) is defined by the energy-dependent gain/loss termsin (2). As a result, a large-scale solitonic (multi-soliton) structure appears(Fig. 4) and, the picosend satelites appear both nearby (few picosecond)the main pulse and far (nanoseconds) from it. Strong interaction betweenthe pulses with the contribution from a gain dynamics results in a chaoticbehavior. For the chirped DS, the dynamic loss/gain saturation causes aparametric resonance, as well. Hence, the DS becomes finely structured [17].
0 50 100 150 2000.00
0.01
0.02
0.03
0.04
0.05
-10 -5 0 5 100.0
0.5
1.0
1.5
2.0
pow
er, a
rb. u
n.
time, ns
peak
pow
er, a
rb. u
n.
time, s
Fig. 4. Multiple DS evolution in the presence of the dynamic gain saturation.
5 Conclusion
Unlike a classical soliton, a chirped DS posses a nontrivial internal structure.As a result, the dynamics of such DS can be very complicated. In particular,a chaotic interaction with an excited vacuum (continuum) develops. Suchan interaction can be nonresonant (as it takes a place for the Schrodingersoliton) and resonant. The last results in the chaotic behavior with a strictlocalization of the DS spectrum and power envelope even for a “far fromequilibrium” regime. The strong localization of a chaotic structure resultsin the chirped DS, which remains to be traceable in an even chaotic regime.Such a traceability promises a lot of applications in the spectroscopy, forinstance.
Acknowledgements
This work was supported by the Austrian Science Foundation (FWF projectP20293).
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References
1.I.S. Aranson, L. Kramer. The world of the complex Ginzburg-Landau equation.Rev. Mod. Phys., 74:99–143, 2002.
2.F.X. Kartner, U. Morgner, Th. Schibli, R. Ell, H.A. Haus, J.G. Fujimoto, E.P.Ippen. Few-cycle pulses directly from a laser. In F.X. Kartner, editor, Few-cycle Laser Pulse Generation and its Applications, pages 73-178, Berlin, 2004.Springer.
3.N. Akhmediev, A. Ankiewicz. Dissipative solitons in the complex Ginzburg-Landau and Swift-Hohenberg equations. In N. Akhmediev, A. Ankiewicz, edi-tors, Dissipative Solitons, pages 1–18, Berlin, 2005. Springer.
4.J.M. Soto-Crespo, Ph. Grelu. Temporal multi-soliton complexes generated by pas-sively mode-locked lasers. In N. Akhmediev, A. Ankiewicz, editors, DissipativeSolitons, pages 207–240, Berlin, 2005. Springer.
5.S.T. Cundiff. Soliton dynamics in mode-locked lasers. In N. Akhmediev, A.Ankiewicz, editors, Dissipative Solitons, pages 183–206, Berlin, 2005. Springer.
6.M. Horowitz, Y. Barad, Y. Silberberg. Noiselike pulses with a broadband spec-trum generated from an erbium-doped fiber laser. Opt. Letters, 22:799–801,1997.
7.A. Zavyalov, R. Iliew, O. Egorov, F. Lederer. Dissipative soliton molecules withindependenly evolving or flipping phases in mode-locked fiber lasers. Phys. Rev.A, 80:043829, 2009.
8.S. Kobtsev, S. Kukarin, S. Smirnov, S. Turitsyn, A. Latkin. Generation of double-scale femto/pico-second optical lumps in mode-locked fiber lasers. Optics Ex-press, 17:20707–20713, 2009.
9.V.L. Kalashnikov. Chirped dissipative solitons. In L.F. Babichev, V.I.Kuvshinov,editors, Nonlinear Dynamics and Applications, pages 58–67, Minsk, 2010. Re-publican Institute of higher school.
10.D. Anderson, M. Lisak, and A. Berntson. A variational approach to nonlinearequations in optics. Pramana J. Phys. 57:917-936, 2001.
11.N. Akhmediev, A. Ankiewicz. Solitons: Nonlinear Pulses and Beams, London,1997. Chapman&Hall.
12.R. Conte, M. Musette. Solitary waves of nonlinear nonintegrable equations. In N.Akhmediev, A. Ankiewicz, editors, Dissipative Solitons, pages 373–406, Berlin,2005. Springer.
13.V.L. Kalashnikov, A. Apolonski. Energy scalability of mode-locked oscillators:a completely analytical approach to analysis. Optics Express, 18:25757–25770,2010.
14.E. Podivilov, V.L. Kalashnikov. Heavily-chirped solitary pulses in the normaldispersion region: new solutions of the cubic-quintic complex Ginzburg-Landauequation. JETP Letters, 82:467–471, 2005.
15.V.L. Kalashnikov. Dissipative solitons: perturbations and chaos formation. InProceedings of 3nd Chaotic Modeling and Simulation International Conference,pages 69-1–8. 1-4 June, 2010, Chania, Greece.
Abstract. This work considers the Hamilton equations of general relativity in asetting of arithmetic, algebra, and topology provided by Observer’s Mathematics(see [1], [2], [3]). Certain results and communications pertaining to solution of theseproblems are provided.Keywords: Hamilton, observer, Lagrange, probability.
1 Introduction
The Hamilton equations are generally written as follows:
p = −∂H∂q
q =∂H∂p
In the above equations, the dot denotes the ordinary derivative with respectto time of the functions p = p(t), called generalized momenta, and q = q(t),called generalized coordinates, taking values in some vector space, and H =H(p, q, t) is the so-called Hamiltonian, or (scalar valued) Hamiltonian function.Thus, more explicitly, one can equivalently write
d
dtp(t) = − ∂
∂qH(p(t), q(t), t)
d
dtq(t) = − ∂
∂qH(p(t), q(t), t)
and specify the domain of values in which the parameter t (time) varies.
2 Basic Physical Interpretation
The simplest interpretation of the Hamilton equations is as follows, applyingthem to a one-dimensional system consisting of one particle of mass m un-der time independent boundary conditions: the Hamiltonian H represents the
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energy of the system, which is the sum of kinetic and potential energy, tradi-tionally denoted T and V , respectively. Here q is the x−coordinate and p isthe momentum, mv. Then
H = T + V
T =p2
2m
V = V (q) = V (x)
Now, the time-derivative of the momentum p equals the Newtonian force,and so here the first Hamilton equation means that the force on the particleequals the rate at which it loses potential energy with respect to changes in x,its location.
The time-derivative of q here means the velocity: the second Hamiltonequation here means that the particle’s velocity equals the derivative of itskinetic energy with respect to its momentum. (Because the derivative withrespect to p of p2/2m equals p/m = mv/m = v.)
3 Using Hamilton’s Equations
In terms of the generalized coordinates q and generalized velocities q, we canperform the following steps:
1. Write out the Lagrangian L = T − V . Express T and V as though La-grange’s equation were to be used.
2. Calculate the momenta by differentiating the Lagrangian with respect tovelocity:
p(q, q, t) =∂L∂q
3. Express the velocities in terms of the momenta by inverting the expressionsin step 2.
4. Calculate the Hamiltonian using the usual definition of H as the Legendretransformation of L via
H = q∂L∂q− L = qp− L
Substitute for the velocities using the results in step (3).5. Apply Hamilton’s equations.
4 Deriving Hamilton’s Equations
We can derive Hamilton’s equations by looking at how the total differentialof the Lagrangian depends on time, generalized positions q and generalizedvelocities q.
dL =
(∂L∂q
dq +∂L∂q
dq
)+
∂L∂t
dt
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Now the generalized momenta were defined as p = ∂L∂q and Lagrange’s equa-
tions tell us that
d
dt
∂L∂q− ∂L
∂q= 0
We can rearrange this to get
∂L∂q
= p
and substitute the result into the total differential of the Lagrangian
dL = (pdq + pdq) +∂L∂t
dt
dL = (pdq + d(pq)− qdp) +∂L∂t
dt
and rearrange again to get
d (pq − L) = (−pdq + qdp)− ∂L∂t
dt
The term on the left-hand side is just the Hamiltonian that we have definedbefore, so we find that
dH = (−pdq + qdp)− ∂L∂t
dt =
[∂H∂q
dq +∂H∂p
dp
]+
∂H∂t
dt
where the second equality holds because of the definition of the total dif-ferential of H in terms of its partial derivatives. Associating terms from bothsides of the equation above yields Hamilton’s equations
∂H∂q
= −p, ∂H∂p
= q,∂H∂t
= −∂L∂t
5 As a Reformulation of Lagrangian Mechanics
Starting with Lagrangian mechanics, the equation of motion is based on gen-eralized coordinates q and matching generalized velocities q. We write theLagrangian as
L(q, q, t)
with the subscripted variables understood to represent these variables of thattype. Hamiltonian mechanics aims to replace the generalized velocity variableswith generalized momentum variables, also known as conjugate momenta. Bydoing so, it is possible to handle certain systems, such as aspects of quantummechanics, that would otherwise be even more complicated.
For each generalized velocity, there is one corresponding conjugate momen-tum, defined as:
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p =∂L∂q
The Hamiltonian is the Legendre transform of the Lagrangian:
H(q, p, t) = qp− L(q, q, t)
.If the transformation equations defining the generalized coordinates are in-
dependent of t, and the Lagrangian is a product of functions (in the generalizedcoordinates) which are homogeneous of order 0, 1 or 2, then it can be shownthat H is equal to the total energy E = T + V .
Each side in the definition of H produces differential:
dH =
[∂H∂q
dq +∂H∂p
dp
]+
∂H∂t
dt =
[qdp + pdq − ∂L
∂qdq − ∂L
∂qdq
]− ∂L
∂tdt
Substituting the previous definition of the conjugate momenta into thisequation and matching coefficients, we obtain the equations of motion of Hamil-tonian mechanics, known as the canonical equations of Hamilton:
∂H∂q
= −p, ∂H∂p
= q,∂H∂t
= −∂L∂t
6 Observer’s Mathematics Point of View
The main relation in classical case is
(p + ∂p)× (q + ∂q)− p× q = p× ∂q + q × ∂p
In Observer’s Mathematics in Wn (from m−observer point of view withm > 4n), the left hand side (LHS) becomes:
(p +n ∂p)×n (q +n ∂q)−n p×n q
while the right hand side (RHS) becomes
p×n ∂q +n q ×n ∂p
Crucial difference is that LHS is not always equal to RHS.Next, we prove the following four theorems.
Theorem 1. If p, q ∈W2, from m−observer point of view with m > 8, then
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Proof. If we put ∂p = 0.01 and ∂q = 0.01 and take p = xy.za and q = uv.wb,x, y, z, u, v, w ∈ 0, 1, . . . , 9 with a 6= 9, b 6= 9, then we have this identity. Butif a = 9 or b = 9, then this identity becomes wrong.
Theorem 2. If p, q ∈Wn, from m−observer point of view with m > 4n, then
Abstract We describe a sub microscopic mechanism which is responsible for the appearance of crop circles on the surface of the Earth. It is shown that the inner reason for the mechanism is associated with intra-terrestrial processes that occur in the outer core and the mantle of the terrestrial globe. We assume that magnetostriction phenomena should take place at the boundary between the liquid and the solid nickel-iron layers of the terrestrial globe. Our previous studies showed that at the magnetostriction a flow of inertons takes out of the striction material (inertons are carriers of the field of inertia, they represent a substructure of the matter waves, or the particle's psi-wave function; they transfer mass properties of elementary particles and are able to influence massive objects changing their inner state and behaviour). At the macroscopic striction in the interior of the Earth, pulses of inerton fields are irradiated, and through non-homogeneous channels of the globe's mantle and crust they reach the surface of the Earth. Due to the interaction with walls of these channels, fronts of inerton flows come to the surface as fringe images. These inerton flows affect local plants and bend them, which results in the formation of the so-called crop circles. It is argued that the appearance of crop circles under the radiation of inertons has something in common with the mechanism of formation of images in a kaleidoscope, which happens under the illumination of photons. Key words: crop circles, inertons, mantle and crust channel, magnetostriction of rocks 1. Introduction Crop circles attract attention of many researchers. Studies (see, e.g. Refs. 1-3) show that in these circles stalks are bent up to ninety degrees without being broken and something softened the plant tissue at the moment of flattening. Something stretches stalks from the inside; sometimes this effect is so powerful that the node looks as exploded from the inside out. In many places crop formation is accompanied with a high degree of magnetic susceptibility, which is caused by adherent coatings of stalks with the commingled iron oxides, hematite (Fe203) and magnetite (Fe304) fused into a heterogeneous mass [2]. Researchers [2-4] hypothesized that crop formations involve organised ion plasma vortices, which deliver lower atmosphere energy components of sufficient magnitude to produce bending of stalks, the formation of expulsion cavities in plant stems and significant changes in seedling development. It should be noted that an idea of the origin of crop circles associated with the atmosphere energy and/or UFO is wildly accepted.
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On the other hand, researchers who study geophysical processes and the earthquakes note about possible regional semi-global magnetic fields that might be generated by vortex-like cells of thermal-magmatic energy, rising and falling in the earth's mantle [5]. Another important factor is magnetostriction of the crust – the alteration of the direction of magnetization of rocks by directed stress [6,7]. Moreover, recent study [8] has suggested a possible mechanism of earthquake triggering due to magnetostriction of rocks in the crust. The phenomenon of magnetostriction in geophysics is stipulated by mechanical deformations of magnetic minerals accompanied by changes of their remanent or induced magnetization. These deformations are specified by magnetostriction constants, which are proportional coefficients between magnetization changes and mechanical deformations. A real value of the magnetostriction constant of the crust is estimated as about 10-5 ppm/nT, which is a little larger than for pure iron. Yamazaki’s calculation [8] shows that effects connected to the magnetostriction of rocks in the crust can produce forces nearly 100 Pa/year and even these comparatively small stress changes can trigger earthquakes. Of course, weaker deformations associated with magnetostriction of rocks also take place. These are the magnetostriction deformations that we put in the foundation of the present study of field circles. 2. Preliminary Our theoretical and experimental studies have shown that the phenomenon of magnetostriction is accompanied with the emission of inerton fields from the magnetostrictive material studied. What is the inerton field? Bounias and one of the authors [9-12] proposed a detailed mathematical theory of the constitution of the real physical space. In line with this theory, real space is constrained to be a mathematical lattice of closely packed topological balls with approximately the Planck size,
hG /c3 ~ 10−35 m. It was proven that such a lattice is a fractal lattice and that it also manifests tessellation properties. It has been called a tessel-lattice. In the tessel-lattice volumetric fractalities of cells are associated with the physical concept of mass. A particle represents a volumetrically deformed cell of the tessel-lattice. The motion of such a particle generates elementary excitations of the tessel-lattice around the particle. These excitations, which move as a cloud around the particle, represent the particle’s force of inertia. That is why they were called inertons [13,14]. The corresponding submicroscopic mechanics developed in the real space can easily be connected to conventional orthodox quantum mechanics constructed in an abstract phase space. Submicroscopic mechanics associates the particle’s cloud of inertons with the quantum mechanical wave ψ-function of this particle. Thus, the developing concept turns back a physical sense to the wave ψ-function: this function represents the field of inertia of the particle under consideration. Carriers of the field of inertia are inertons. A free inerton, which is released from the particle’s cloud of insertions, possess a velocity that exceeds the velocity of light c [15]. In condensed media entities vibrating at the equilibrium positions periodically irradiate and absorb their clouds of inertons back [16]; owing to such a behaviour the mass of entities varies. This means that under special conditions the matter may irradiate a portion of its inertons. Lost inertons then can be absorbed by the other system, which has to result in changes of physical properties of the system. One of such experiments was carried out in work [17]. Continuous-wave laser illumination of ferroelectric crystal of LiNbO3 resulted in the production of a long-living stable electron droplet with a size of about 100 µm, which freely moved with a velocity of
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about 0.5 cm/s in the air near the surface of the crystal experiencing the Earth's gravitational field. The role of the restraining force of electrons in the droplet was attributed to the inerton field, a substructure of the particles’ matter waves, which was expelled from the surface of crystal of LiNbO3 together with photoelectrons by a laser beam. Properties of electrons after absorption of inertons changed very remarkably – they became heavy electrons whose mass at least million of times exceeded the rest mass of free electrons. Only those heavy electrons could elastically withstand their Coulomb repulsion associated with the electrical charge, which, of course, is impossible in the case of free electrons. We have shown [16] that in the chemical industry inerton fields are able to play the role of a field catalyst or, in other words, inerton fields can serve to control the speed of chemical reactions. In the reactive chamber we generated inerton fields by using magnetostriction agents: owing to the striction the agents non-adiabatically contract, which is culminated in the irradiation of sub matter, i.e. inertons, from the agents. Then under the inerton radiation, the formation of a new chemical occurred in several seconds, though usually these chemical reactions last hours. Therefore, these results allow us to involve inerton fields, which originate from the ground, in a study of the formation of crop circles. The thickness of the crust is about 20 km. The mantle extends to a depth above 3000 km. The mantle is made of a thick solid rocky substance. Due to dynamical processes in the interior of the Earth, magnetostrictive rocks contract with a coefficient of about 10-5 [8], which is a trigger mechanism for the appearance of a flow of inerton radiation. This flow of inertons shoots up from a depth by coming through the mantle and crust channel. Such channels are usual terrestrial materials with some non-homogenous inclusions down to tens or hundreds of kilometres from the surface of the terrestrial globe (compare with bio-energy channels in our body: the crude morphological structure is the same, but the fine morphological structure is different, which allows these bio-energy channels to display a higher conductivity). A mantle-crust channel can be modelled as a cylindrical tube, which has a cross-section area equal to A , along which a flow of inertons travels out from the interior of the globe. The inner surface of the channel has to reflect inerton radiation, at least partly, so that the flow of inertons will continue to follow along the channel to its output, i.e. the surface of the Earth. 3. Elastic rod bending model Let us evaluate conditions under which the stalks of herbaceous plants will bend affected by mantle inertons. A stalk of a plant can be modelled for the first approximation by an elastic rod (Fig. 1). We suppose that it is deformed by an external force f distributed uniformly over the rod
length l . This external force is a force caused by a flow of inertons going from the ground due to a weak collision of the mantle and crust rocks as described above. The rod profile in the projections to the horizontal and vertical axes is described as follows [18].
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x
y
ϑ
θ
fy
x
y
ϑ
θ
fx
a) b)
Figure 1. Elastic rod model. I. Vertical force fy (Fig. 1a)
x =2I E
fy
1− cosϑ l − cosϑ − cosϑ l( ), (1)
y =I E
2 fy
cosϑ dϑcosϑ − cosϑ l0
ϑ
∫ . (2)
Here I = πR4 /4 is the rod’s moment of inertia, R is the rod’s radius, and E is the Young’s modulus of the rod’s material. The length of the rod is explicitly given as
l =I E
2 fy
dϑcosϑ − cosϑ l0
ϑ l
∫ . (3)
At the maximum bending we have ϑmax = ϑl = π /2, so that
l =I E
2 fy
dϑcosϑ0
π / 2
∫ =I E
fy
K(1/2), (4)
where K(1/2) ≈1.854 is the complete elliptic integral of the first kind. Hence, we come to an expression for the force required to bend the rod by a π /2 angle:
fy =I E
l2 K 2(1/2)≈ 3.44IE
l2 . (5)
II. Horizontal force fx (Fig. 1b)
x =IE
2 fx
sinϑ dϑsinϑ l − sinϑ0
ϑ
∫ , (6)
y =2I E
fx
sinϑ l − sinϑ l − sinϑ( ). (7)
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The length of the rod is explicitly given as
l =IE
2 fx
dϑsinϑ l − sinϑ0
ϑ l
∫ . (8)
In this case the maximum bending angle should be smaller than π /2 (no such a force exists that can bend the rod by this angle). So, we select the maximum bending angle at ϑl = π /3 and write the corresponding relationship between the rod’s length and the acting force:
l ≈I E
2 f x
2.61 or fx ≈ 3.41I E
l2 , (9)
which is nearly the same as in the previous case (5). Now let us evaluate the value of the breaking force f = fx ≅ f y. We have to substitute
numerical values l = 0.5 m, R =1.5 ×10−3 m for the rod and the value of elasticity (Young’s) modulus E to expressions (5) or (9). The value of E has been measured for many different grasses, see, e.g., Refs. 19-23. According to these data, E varies approximately from 0.8 to about 109 kg/(m⋅s2). For instance, in the case of wheat we can take E ≈ 3×109 kg/(m⋅s2), which gives f ≈ 0.16 N. Besides, the authors [19-23] emphasize that for grassy stalks in addition to the elasticity modulus one has to take into account the bending stress, the yield strength (tensile strength) and the shearing stress. These parameters range from 7 ×106 to about 50×106 kg/(m⋅s2) and, hence, significantly decrease the real value of f , which is capable to bend stalks. For example, putting for E the value of the maximal tensile stress 50×106 kg/(m⋅s2) we derive for the bending force f ≈ 0.0027 N. The gravity force acting on the rod is 033.02
g ≈=== glRVgmgf ρπρ N (10)
where ρ is the rod’s material density about ρ =103 kg/m3, m and V are its mass and volume, and g = 9.8 m/s2 is the acceleration due to gravity. Comparing the gravity force gf with the banding forcef we may conclude that the latter
is not enough to fracture a grassy stalk. Only the breaking forces (5) and (9) can exceeds the gravity force (10). 4. Motion in the rotating central field The inner surface of a mantle-crust channel can be described by a retaining potential U , which is holding a flow of inertons spreading along the channel from an underground source. Let µ be the mass of an effective batch of terrestrian inertons from this source, which interact with a grassy stalk. The planar motion of such a batch of inertons in the central field is described by the Lagrangian
( ) ),(2
222 ϕϕµ&&& rUrrL −+= , (11)
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which is here written in polar coordinates r and ϕ; dot standing for the derivative with respect to time. To model a spreading inerton field, the potential should include a dependence on the angular velocity, ),( ϕ&rU , which means that we involve the proper rotation of the Earth relative to the flow of inertons. For instance, the potential can be chosen in the form of the sum of two potentials: ϕβαϕ &&
22
22),( rrrU += . (12)
In the right hand side of expression (12) the first term is a typical central-force harmonic potential, which describes an elastic behaviour of the batch of inertons in the channel and the surrounding space; the second term includes a dependence on the azimuthal velocity, which means that it depicts the rotation-field potential. The introduction of this potential allows us to simulate more correctly the reflection of inertons from the walls of the mantle channel, which of course only conditionally can be considered round in cross-section. The equations of motion are then written as
0=∂∂−
∂∂
ii q
L
q
L
dt
d&
, i =1, 2, q1 ≡ ρ, q2 ≡ ϕ , (13)
or in the explicit form
02 =++− ϕµβ
µαϕ &&&& rrrr , (14)
02
2 =
−+µ
βϕϕ &&&& rr . (15)
These equations can be integrated explicitly or solved numerically at the given initial conditions r(0) , )0(r& , ϕ(0), )0(ϕ& , and the trajectory of motion can be plotted in rectangular coordinates rcosϕ, rsinϕ . The second equation represents the conservation of the angular momentum M :
02
2 =
−µ
βϕµ &rdt
d or const
22 =
−=µ
βϕµ &rM . (16)
Figures 2 and 5 show two possible trajectories at particular values of the parameters.
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Figure 2. Trajectory of the motion of intertons in the rotating central field with parameters α µ =1 s–2, β µ = 0.5 s–1; r(0) =10 m, 0)0( =r& , 0)0( =ϕ , 01.0)0( =ϕ& s–1.
Figure 3. Velocity 222|| ϕ&&&r rrr += of the batch of inertons versus time for the case of the
trajectory shown in Fig. 2. The maximal velocity is υmax =10 m/s.
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Figure 4. Acceleration 222 )2()(|| ϕϕϕ &&&&&&&&&r rrrrr ++−= of the batch of inertons versus time
for the case of the trajectory shown in Fig. 2. The maximal acceleration is amax ≈10 m/s2.
Figure 5. Trajectory of the motion of inertons in the rotating central field with parameters α µ =1 s–2, β µ = 0.5 s–1; r(0) =10 m, 0)0( =r& , ϕ(0) = 0, 1)0( =ϕ& s–1.
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Figure 6. Velocity 222|| ϕ&&&r rrr += of the batch of inertons versus time for the case of the
trajectory shown in Fig. 5. The maximal velocity is υmax ≈12 m/s.
Figure 7. Acceleration 222 )2()(|| ϕϕϕ &&&&&&&&&r rrrrr ++−= of the batch of inertons versus time
for the case of the trajectory shown in Fig. 5. The maximal acceleration is amax ≈15 m/s2.
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In the case of the Newton-type potential, expression (12) changes to
ϕβγϕ &&2
2),( r
rrU +−= . (17)
Then the equations of motion for the Lagrangian (11) become
02
2 =++− ϕµβ
µγϕ &&&& rr
rr , (18)
02
2 =
−+µ
βϕϕ &&&& rr (19)
The solution to these equations is shown in Fig. 8.
Figure 8. Trajectory of the motion of inertons in the rotating central field with parameters γ /µ =1 m3 s–2, β µ = 0.1 s–1; r(0) =10 m, 0)0( =r& , ϕ(0) = 0, 01.0)0( =ϕ& s–1. In Fig. 9 we show the solution to the equations of motion of a batch of inertons for the case of simplified potential (17), namely, when it is represented only by the Newton-type potential U(r) = −γ /r . Figures 4 and 7 give an estimate for the acceleration a of the batch of inertons: a =10 to 15 m/s2.
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Figure 9. Elliptic trajectory of the motion of inertons in the Newton-type potential with parameters γ /µ =1 m3⋅s–2, β µ = 0 s–1; r(0) =10 m, 0)0( =r& , ϕ(0) = 0, 01.0)0( =ϕ& s–1. Figures 2, 4, 8 and 9 depict possible patterns of crop circles generated by flows of the mantle-crust inertons. Let us estimate now the intensity of inerton radiation needed to form a crop circle of total area A ≈100 m2. Let M rocks be the mass of the mantle-crust rocks that generate inertons owing to their magnetosriction activity. We have to take into account the magnetostriction coefficient C , which describes an extension strain of rocks. In view of the fact of that low frequencies should accompany geophysical dynamical processes, we can assume that the striction activity of a local group of rocks occurs at a low frequency ν (i.e. rocks collide N times per a time ∆t of radiation of inertons). Having these parameters, we can evaluate a flow of mass µΣ that is shot in the form of inerton radiation at the striction of rocks:µΣ ≈ NCMrocks.
If we put 710~M kg, C ~10−5, and 5=N we obtain 500≈Σµ kg. This mass µΣ is distributed along the area of A in the form of a flow of the inerton field. Let each square metre be the ground for the growth of 1000 stalks. Then 105 stalks can grow in the area of A =100 m2. This means that each stalk is able to catch an additional mass µ = µΣ /105 5= g from the underground inerton flow; this value is of the order of the mass of a stalk itself. Knowing the mass 3105 −×=µ kg of the batch of inertons which interacts with a stalk and the acceleration of this inerton batch a =10 to 15 m/s2, we can rate the force of inertons that bends and breaks up stalks in the large area A : 05.0≈F to 075.0 N. This estimation exceeds not only the threshould bending forcef , but also the gravity force gf (10) evaluated
in section 3. Therefore, the model developed in this work is plausible.
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5. Kaleidoscope model This kaleidoscope model gives a static description of inerton structures. We assume that a bunch of inertons depicted in the centre of Fig. 10 is reflected from the walls, whose geometry was selected rectangular in this particular example. Multiple reflections from the walls produce the pattern shown in Fig. 10. This model can be assumed as an analogy of geometrical optics with light reflecting from the mirrors. Uniting the rotating central field model described in the previous section and the kaleidoscope model can generate yet more complex patterns.
Figure 10. Kaleidoscope model 6. Conclusion In this study we have shown a radically new approach to the conception and description of crop circles. The theory developed is multi-aspect and based on first submicroscopic principles of fundamental physics. The theory sheds light also on fine processes occurring in the crust and the mantle of the terrestrial globe. The investigation will allow following researchers to improve the mathematical model of the description of shapes of crop circles, to correctly concentrate on biological changes in plants taken from crop circles, to reach more progress in understanding a subtle dynamics of the earth crust, and to contemplate a more delicate approach to the development of new methods of earthquake prediction.
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References
[1] W. C. Levengood, Anatomical anomalies in crop formation plants, Physiologia Plantarum 92, 356-363 (1994).
[2] W. C. Levengood and J. Bruke, Semi-Molten Meteoric Iron Associated with a Crop Formation, J.
Scient. Exploration 9, No. 2, pp. 191-199 (1995).
[3] J. A. Bruke, The physics of crop formations, MUFON Journal, October, pp. 3-7 (1998).
[4] W. C. Levengood and N. P. Talbott, Dispersion of energies in worldwide crop formations, Physiologia Plantarum 105, 615-624 (1999).
[5] J.-C. Pratsch , Reative motion in geology: some pholosophical differences, J. Petroleum Geology 13,
No. 2, 229–234 (1990).
[6] W. H. Munk and G. J. F. Macdonald, The Rotation of the Earth. A Geophysical Discussion (Cambridge University Press, London, 1975).
[7] H. Jeffreys, The earth. Its origin, history and physical constitution (Cambridge University Press,
London, 1976).
[8] K. Yamazaki, Possible mechanism of earthquake triggering due to magnetostriction of rocks in the crust, American Geophysical Union, Fall Meeting 2007, abstract #S33B-1307, Dec. 2007.
[9] M. Bounias and V. Krasnoholovets, Scanning the structure of ill-known spaces: Part 1. Founding
principles about mathematical constitution of space, Kybernetes: The Int. J. Systems and Cybernetics 32, Nos. 7/8, 945-975 (2003); arXiv.org: physics/0211096.
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construction of physical space, ibid. 32, Nos. 7/8, 976-1004 (2003); arXiv: physics/0212004.
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[12] M. Bounias and V. Krasnoholovets, The universe from nothing: A mathematical lattice of empty sets.
Int. J. Anticipatory Computing Systems 16, 3-24 (2004); arXiv.org: physics/0309102
[13] V. Krasnoholovets, Submicroscopic deterministic quantum mechanics, Int. J. Computing Anticipatory Systems 11, 164-179 (2002); arXiv: quant-ph/0109012.
[14] V. Krasnoholovets, Inerton fields: Very new ideas on fundamental physics, American Inst. Phys. Conf.
Proc. – Dec. 22, 2010 - Volume 1316, pp. 244-268. Search for fundamental theory: The VII International Symposium Honoring French Mathematical Physicist Jean-Pierre Vigier (12-14 July 2010, Imperial College, London); doi:10.1063/1.3536437.
[15] V. Krasnoholovets and J.-L. Tane, An extended interpretation of the thermodynamic theory including
an additional energy associated with a decrease in mass, Int. J. Simulation and Process Modelling 2, Nos. 1/2, 67-79 (2006); also arXiv.org: physics/0605094.
[16] V. Krasnoholovets, Variation in mass of entities in condensed media, App. Phys. Research 2, No. 1, 46-
59 (2010).
[17] V. Krasnoholovets, N. Kukhtarev and T. Kukhtareva, Heavy electrons: Electron droplets generated by photogalvanic and pyroelectric effects. Int. J. Modern Phys. B 20, No. 16, 2323-2337 (2006); arXiv.org: 0911.2361[quant-ph].
[18] L. D. Landau and E. M. Lifshits, The theory of elasticity (Nauka, Moscow, 1987), pp. 106-107 (in
Russian).
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[19] G. Skubisz, The dependence of the Young's modulus of winter wheat stalk in various phenological
phases, Proceedings of the 2nd International Conference on physical properties of agricultural materials, Godollo, Hungary, 26-28 August 1980, Vol. 2. (1980), p. 9.
[20] G. H. Dunn and S. M. Dabney, Modulus of elasticity and moment of inertia of grass hedge stems,
Transactions of the ASAE 39, No. 3, 947-952 (1996).
[21] M. Nazari Galedar, A. Jafari, S. S. Mohtasebi, A. Tabatabaeefar, A. Sharifi, M. J. O'Dogherty, S. Rafiee and G. Richard, Effects of moisture content and level in the crop on the engineering properties of alfalfa stems, Biosystems Engineering 101, No. 2, 199-208 (2008).
[22] H. Tavakoli, S.S. Mohtasebi, A. Jafari, Effects of moisture content, internode position and loading rate
on the bending characteristics of barley straw, Research in Agricultural Engineering 55, No. 2, 45-51 (2009).
[23] A. Esehaghbeygi, B. Hoseinzadeh, M. Khazaei and A. Masoumi, Bending and shearing properties of
wheat stem of alvand variety, World Applied Sciences J. 6, No. 8, 1028-1032 (2009).
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Studying the Non-Linearity of Tumour Cell
Populations under Chemotherapeutic Drug
Influence1
George I. Lambrou, Apostolos Zaravinos, Maria Adamaki and Spiros
Vlahopoulos
1st Department of Pediatrics, University of Athens, Choremeio Research
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Chaos and Complexity Models and Sustainable
Building Simulation
Xiaoshu Lu1,3
, Charles Kibert2, Martti Viljanen
1
1Department of Civil and Structural Engineering, School of Engineering, Aalto
University, PO Box 12100, FIN-02150, Espoo, Finland
E-mail: [email protected] 2Powell Center for Construction & Environment, University of Florida, PO Box
115703, Gainesville, Florida 32611-5703 USA 3Finnish Institute of Occupational Health, Finland
Abstract: This paper intends to provide suggestions of how sustainable building
simulation might profit from mathematical models derived from chaos and complexity
approaches. It notes that with the increasing complexity of sustainable building systems
which are capable of intelligently adjusting buildings' performance from the environment
and occupant behavior and adapting to environmental extremes, building performance
simulation is becoming more crucial and heading towards new challenges, dimensions,
concepts, and theories beyond the traditional ones. The paper then goes on to describe
how chaos and complexity theory has been applied in modeling building systems and
behavior, and to identify the paucity of literature and the need for a suitable methodology
of linking chaos and complexity approaches to mathematical models in building
sustainable studies. Chaotic models are proposed thereafter for modelling energy
consumption, nonlinear moisture diffusion, and building material properties in building
simulation. This paper provides an update on the current simulation models for
sustainable buildings.
Keywords: Chaos theory, Sustainable uilding simulation, Energy consumption,
Moisture diffusion and Material properties.
1. Introduction Buildings represent a large share of world’s end-use energy consumption. Due
to rapid increase in energy consumption in the building sector, the climate
change driven by global warming, and rising energy shortage, there is no doubt
that renewable energy and sustainable buildings play an role in the future.
Today, sustainable buildings are seen as a vital element of a much larger
concept of sustainable development that aims to meet human needs while
preserving the environment so that the needs can be met not only in the present,
but in the indefinite future [1]. Moreover, the concept itself keeps on evolving
and resulting in iterations of sustainability [2]. Technically, sustainable
buildings require integration of a variety of computer-based complex systems
which are capable of intelligently adjusting their performance from the
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environment and occupant behavior and intelligently adapting to environmental
extremes [2].
With the increasing complexity of building systems, simulation based design
and predictive control of building performances are more and more important
for a sustainable energy future. Consequently, this makes building performance
models more complex and crucial which are heading towards new challenges,
dimensions, concepts, and theoretical framework beyond the traditional building
simulation theories. It has been suggested that as a basis chaos and complexity
theory is valid and can handle the increasingly complexity of building systems
that have dynamic interactions among the building systems on the one hand, and
the environment and occupant behavior on the other. Here we do not distinguish
chaos and complexity theories in this paper even though there has been a debate
about their differences [3].
The chaos models have already been applied to some problems in building
simulation applications. Chow et al. investigated chaos phenomena of the
dynamic behavior of mixed convection and air-conditional systems for buildings
with thermal control [4]. Weng et al. applied chaos theory to the study of
backdraft phenomenon in room fires [5]. Morimoto et al. studied an intelligent
control technique for keeping better quality of fruit during the storage process
[6]. For humidity control purpose, the sampled relative humidity data in storage
house were measured and analyzed. Chaos phenomenon was identified in such
measured relative humidity time series over daytime hours.
In spite of the studies discussed above, the application of chaos theory to
building performance simulation, especially to sustainable buildings, is still in
its infancy. Building performance simulation models can be roughly classified
into either the physical model or the black-box approach. Some may be difficult
to categorize in this way. As far as the physical model is concerned, there is a
voluminous literature on the models ranging from detailed to local thermal
analysis of energy demand, passive design, environmental comfort and the
response of control [7,8]. These physical models often require sufficient
information on systems, control and environmental parameters for buildings.
The output of the model is only as accurate as input data.
Presently many input data for buildings are poorly defined, which creates
ambiguity or uncertainty in interpreting the output. This is the general drawback
of these models. Therefore, for many practical applications, a black-box
approach, a model without internal mechanisms or physical structure, is often
adopted. For example, neutral networks and fuzzy logic models [9] and time
series models [10] which are generally better suited for prediction. However,
these models have several limitations. Take neural networks as an example.
Firstly, large experimental input and output data are needed in order to build
neural networks which can be difficult and expensive to obtain in practice.
Secondly, they are susceptible to over-training. Above all, the models have been
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criticized as 'black box' model with no explanation of the underlying dynamics
that drive the study systems [11].
More specifically, as for sustainable buildings, the current models often lack the
long-term economics factors, evolving factors, and flexibility necessary for
dynamic predictions. These weaknesses and the current status of sustainable
building simulation models have encouraged us to focus instead on a chaos-
based model incorporating physical model to enhance our understanding and
prediction of building physical behaviour. Chaos theory is characterized by the
so-called ‘butterfly effect’ described by Lorenz [12]. It is the propensity of a
system to be sensitive to initial conditions so that the system becomes
unpredictable over time. Yet, a chaotic process is not totally random and has
broadened existing deterministic patterns with some kind of structure and order
[12]. This paper extends the literature by proposing potential chaotic models in
sustainable building simulation. Below we describe three such models. The first
is building energy consumption model. The second deals with nonlinear
moisture diffusion model. The third is related to building material properties.
2. Building Energy Consumption Model Swan provided an up-to-date review of various simulation models used for
modeling residential sector energy consumption and sustainability [13]. Most
models rely on input data whose levels of details can vary dramatically. Li
presented an overview of literature regarding long-term energy demand and CO2
emission forecast scenarios [14]. These reviews reflect general modeling
approaches currently in existence for sustainable buildings. Two approaches are
generally adopted: top-down and bottom-up. The top-down approach utilizes
historic aggregate energy values and regresses the energy consumption of the
housing stock as a function of top-level variables such as macroeconomic
indicators. While the general employed techniques may account for future
technology penetration based on historic rates of change, they lack of evolving
factors. Hence an inherent drawback is that there is no guarantee that values
derived from the past will remain valid in the future, especially given the fact
that the levels of details of input data vary significantly [13].
The bottom-up approach extrapolates the estimated energy consumption of a
representative set of individual houses to regional and national levels, and
consists of two distinct methodologies: the statistical method and the
engineering method [13]. Methodologically, extrapolation has been questioned
for many good reasons. It is therefore noted that the statistical technique is
hampered by multicollinearity resulting in poor prediction of certain end-uses
while engineering technique requires many more inputs and has difficulty
estimating the unspecified loads [13, 15].
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The major disadvantage of these models is their lack of flexibility due to the fact
that there is no deterministic structure provided to characterize the input data. In
this context, chaos theory offers a solid theoretical and methodological
foundation for interpreting the fundamental deterministic structure of the data
which present the increasingly complexity of building systems. Karatasou
applied chaos theory in analyzing time series data on building energy
consumption [16]. The correlation dimension 3.47 and largest Lyapunov
exponent 0.047 were estimated for the data, which indicated that chaotic
characteristics exist in the energy consumption data set. Therefore, chaos theory
techniques, based on phase space dynamics for instance, can be used to model
and predict buildings energy consumption.
3. Strong Nonlinear Moisture Diffusion Model Building envelopes can be susceptible to moisture accumulation which may
cause growth of moulds and the deterioration of both occupant health and
building materials. A certain duration of exposure conditions, such as humidity,
temperature, and exposure time, is required for the growth of organisms and the
start of the deterioration process. Critical exposure duration depends on the
particular exposure and material. Take a critical moisture level as an example. If
the moisture level in the material exceeds the critical level, there is a risk of
damage [17] and mould growth [18]. Trechsel summarised that the critical
moisture level can be presented as the critical factors such as 'the critical
moisture content' and 'the critical accumulative exposure time' [19]. He
emphasized that with qualitative criteria it is not possible to assess the risk.
Qualitative criteria can be used only if performance limit states are known
which need statistical data. Evidence has shown the existence of inherent
randomness and nonlinearity in mould growth and the data [18]. Therefore,
moisture transfer process manifestly has chaos.
From a physical modelling point of view, heat and moisture transfer phenomena
in a medium are governed by heat or diffusion equations which are partial
differential equations. For a homogeneous and isotropic medium, the diffusivity
coefficient is often assumed to be constant in the entire domain under study. In
inhomogeneous media, it depends on the coordinates and even on the
temperature [20]. Until now, there is no model that considers time-dependent
diffusivity. However, time-dependent diffusivity, which might be due to the
time-dependent perturbation of environment such as sudden structural change, is
an optional explanation for the critical moisture level.
Yao studied one-dimensional Kuramoto–Sivashinsky (KS) equation, a nonlinear
partial differential equation, in the hope of clarify the role of nonlinear terms the
their consequences [21]:
0)(4 =+++ xxxxxxxt uuuuu λ (1)
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Nonlinear stability analysis was investigated with respect to time-dependent λ. After certain time (t =4), the chaotic behavior was observed.
It is not difficult to see that the KS equation and heat or diffusion equations do
not differ significant. Thus the KS equation example is expected to more easily
expose major points and hopefully identify open questions that are related to the
critical moisture level or mould phenomena in related to chaos phenomena.
4. Material Properties Model Porous materials have played a major role in building engineering applications.
They are important elements of heat and mass conservation for buildings and
have been extensively studied [22]. A porous material has a unique structure of
complex geometry which is characterized by the presence of a solid matrix and
void phases with porosity. The heat and mass transport behavior of porous
media is largely governed by the interactions among coexisting components.
These interactions occur through interfaces. Theoretically, transport processes in
a porous medium domain may be described by a continuum at the microscopic
level, based on the Navier-Stokes equations for example, as taking into account
the multi-phase nature of the domain. However, for most cases this is
impractical because of the inability to describe the complex geometry and trace
a large number of interfacial boundaries for the porous domain. Therefore, the
porous media models are often constructed through averaging the governing
equations, for example Navier-Stokes equations, in continua at the microscopic
level over a length scale such as representative elementary volume [23]. During
the averaging process some integrals are performed, introducing a weighted
average of the relevant variables, parameters and properties which can be
determined by laboratory and field measurements.
However, both laboratory and field measurements are often tedious, time
consuming and expensive. This has motivated researchers toward the
development of mathematical modeling approaches from routinely measured
properties. In general, three types of mathematical models are used to model
material transport properties: empirical, bundle of tubes, and network models
[24]. The empirical models provide a set of analytical functions to fit the
measurement data for material properties. The model has the advantage of
simplicity but the disadvantage of limited flexibility and adjustability and hence
low reliability.
Depending on how they represent the geometry of the material, both the bundle
of tubes and the network models rely on the pore structure, such as distribution,
connectivity and tortuosity, to derive the material’s transport properties. These
models are also called pore-distribution models and were pioneered by Fatt [25-
27]. The bundle of tubes model approximates the pore structure in a fairly
simple way, for example a set of parallel tubes [24]. Networks models
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approximate the pore structure by a lattice of tubes and throats of various
geometrical shapes on the microscopic scale. Creating a network model is
laborious and not straightforward especially for 3D models [28].
Most importantly, these models, or current state of property modeling
approaches, are case sensitive depending on the excited boundary or the
environment. Therefore, variations of material properties under different
conditions are large, which has been a challenge for modelers. On a longer time
scale, large quantity of data is often needed to build the model and this can be
difficult and expensive to accomplish in practice. In addition, in a wide
environment setting when different environmental phenomena overlap, material
properties become complicated and difficult to predict [29]. This is due to the
lack of a deterministic structure or a core mechanism characterizing the material
transport properties. Chaos theory provides a set of diagnostic tools to exploit
the underlying structures that appear random or unpredictable under traditional
analysis.
Stazi et al. applied chaos theory to investigate the hygrometric properties of
building materials, such as adsorption and suction curves [29]. The constitute
relationship of material’s water content and the environment humidity, the core
of this study, was modelled on the basis of fractal geometry using the material’s
pore radius as:
u = u(φ, D) (2)
where u is the hygroscopic content inside the material and φ the relative humidity of the material. Their relationship was determined through finding the
material 's fractal dimension of water inside the pores, D, which was 2.5265 for
mortar [29].
The novelty of the model lies in its ability to construct the relationship between
the water content inside the material and the relative humidity of the
environment based on the material's geometric property characterized by fractal
dimension. The knowledge of the fractal dimension of the pore spacing in a
porous medium is enough to work out the suction and adsorption curves of the
material. It is, therefore, natural for us to consider chaos theory as a source of
inspiration to envisage the importance of the concerns raised in research in
different fields of building material properties.
5. Conclusions This paper aims to provide a suggestion to update the current status of
simulation models for sustainable buildings. Three chaotic models are proposed.
The first is the building energy consumption model as chaotic characteristics has
been observed in the specific energy consumption data set. The second is
dealing with investigation of nonlinearity of the moisture diffusion model. The
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third model involves the investigation of material physical properties. The
conclusion to be drawn is that chaos theory may reflect real situations, deepen
our understanding, and make predictions more realistic in sustainable building
Abstract: A hollow vortex core in shallow liquid, produced inside a cylindrical reservoir using a rotating disk near the bottom of the container, exhibits interfacial polygonal patterns. These pattern formations are to some extent similar to those observed in various geophysical, astrophysical and industrial flows. In this study, the dynamics of rotating waves and polygonal patterns of symmetry-breaking generated in a laboratory model by rotating a flat disc near the bottom of a cylindrical tank is investigated experimentally. The goal of this paper is to describe in detail and to confirm previous conjecture on the generality of the transition process between polygonal patterns of the hollow vortex core under shallow water conditions. Based on the image processing and an analytical approach using power spectral analysis, we generalize in this work – using systematically different initial conditions of the working fluids – that the transition from any N-gon to (N+1)-gon pattern observed within a hollow core vortex of shallow rotating flows occurs in an universal two-step route: a quasi-periodic phase followed by frequency locking (synchronization). The present results also demonstrate, for the first time, that all possible experimentally observed transitions from N-gon into (N+1)-gon occur when the frequencies corresponding to N and N+1 waves lock at a ratio of (N-1)/N.
Keywords: Swirling flow, patterns, transition, quasi-periodic, synchronization. 1. Introduction Swirling flows produced in closed or open stationary cylindrical containers are of fundamental interest; they are considered as laboratory model for swirling flows encountered in nature and industries. These laboratory flows exhibit patterns which resemble to a large extent the ones observed in geophysical, astrophysical and industrial flows. In general, the dynamics and the stability of such class of fluid motion involve a solid body rotation and a shear layer flow. Because of the cylindrical confining wall, the shear layer flow forms the outer region while the inner region is a solid body rotation flow. The interface between the flow regimes can undergo Kelvin-Helmholtz instability because of the jump in velocity at the interface between the inner and outer regions, which manifests as azimuthal waves. These waves roll up into satellite vortices which impart the interface polygonal shape (e.g., see Hide & Titman 1967; Niño &
* Paper accepted for the 4th Chaotic Modeling and Simulation International Conference (CHAOS 2011), Crete, Greece 31 May – 3 June, 2011.
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Misawa 1984; Rabaud & Couder 1983; Chomaz et al. 1988; Poncet and Chauve 2007). The inner solid body rotation region can also be subjected to inertial instabilities which manifest as Kelvin’s waves and it is this type of waves that will be investigated in this paper. In our experiment a hollow core vortex, produced by a rotating disk near the bottom of a vertical stationary cylinder, is within the inner solid body rotation flow region and acts as a wave guide to azimuthal rotating Kelvin’s waves. The shape of the hollow core vortex was circular before it breaks into azimuthal rotating waves (polygonal patterns) when some critical condition was reached. A fundamental issue that many research studies were devoted to the study of rotating waves phenomena is the identification and characterization of the transition from symmetrical to non-symmetrical swirling flows within cylindrical containers. Whether confined or free surface flow, the general conclusion from all studies confirmed that, the Reynolds number and aspect ratio (water initial height H / cylinder container radius R) are generally the two dominant parameters influencing the symmetry breaking phenomenon’s behaviour. Vogel (1968) and Escudier (1984) studied the transitional process in confined flows and found that symmetry breaking occurs when a critical Reynolds number was reached for each different aspect ratio. Vogel used water as the working fluids in his study where he observed and defined a stability range, in terms of aspect ratio and Reynolds number, for the vortex breakdown phenomenon which occurred in the form of a moving bubble along the container’s axis of symmetry. Escudier (1984) later extended the study by using an aqueous glycerol mixture (3 to 6 times the viscosity of water) and found that varying the working fluid viscosity caused changes in the critical Reynolds number values. He also observed that for a certain range of aspect ratio and viscosity, the phenomenon of vorticity breakdown has changed in behaviour, revealing more vortices breakdown stability regions than the conventional experiments using water as the working fluid. Where in open free surface containers under shallow liquid conditions using water as the working fluid, Vatistas (1990) studied the transitional flow visually and found that the range of the disc’s RPM where the transitional process occurs shrinks as the mode shapes number increased. Jansson et al. (2006) concluded that the endwall shear layers as well as the minute wobbling of the rotating disc are the main two parameters influencing the symmetry breaking phenomenon and the appearance of the polygonal patterns. Vatistas et al. (2008) studied the transition between polygonal patterns from N to N+1, using image processing techniques, with water as the working fluid and found that the transition process from N to a higher mode shape of N+1 occurs when their frequencies ratio locks at (N-1)/N, therefore following a devil staircase scenario which also explains the fact that the transition process occurs within a shorter frequency range as the mode shapes increase. Speculating the transition process as being a bi-periodic state, the only way for such system to lose its stability is through frequency locking (Bergé et al. 1984). From nonlinear dynamics consideration, Ait Abderrahmane et al. (2009) proposed the transition between equilibrium states under similar configurations using classical nonlinear dynamic theory approach and found that
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the transition occurs in two steps being, a quasi-periodic and frequency locking stages, i.e., the transition occurs through synchronization of the quasi-periodic regime formed by the co-existence of two rotating waves with wave numbers N and N+1. Their studies however was built mainly on the observation of one transition, from 3-gon to 4-gon. In the present paper, we provide further details on the symmetry-breaking pattern transitions and confirm the generalized mechanism on the transition from N-gon into (N+1)-gon using power spectra analysis. This study systematically investigates different mode transitions, the effect of working fluid with varying viscosity, liquid initial height on the polygonal pattern instability observed within the hollow core. 2. Experimental Setup and Measurement Technique The experiments were conducted in a 284 mm diameter stationary cylindrical container with free surface (see Fig. 1). A disk, located at 20 mm from the bottom of the container, with radius Rd = 126 mm was used and experiments with three initial water heights above the disk, ho = 20, 30 and ho = 40 mm, were conducted. Similar experiment was conducted by Jansson et al. (2006) within a container of different size where the distance of the disk from the bottom of the container is also much higher than in the case of our experiment. In both experiments similar phenomenon − formation of a polygonal pattern at the surface of the disk − was observed. It appears therefore that the dimension of the container and the distance between the disk and the container bottom do not affect the mechanism leading to the formation of the polygon patterns. In our experiment, the disk was covered with a thin smooth layer of white plastic sheet. It is worth noting that the roughness of the disk affects the contact angle between the disk and the fluid; this can delay the formation of the pattern. However, from our earlier observation in many experiments, roughness of the disk does not seem to influence prominently the transition mechanism.
Fig. 1. Experimental setup.
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0.1
1
10
100
1000
10000
0 20 40 60 80 100
Gylcerol concentration (%wt)
Vis
cosi
ty (m
Pa ⋅
s)
Fig. 2. The variation of dynamic viscosity as a function of glycerol
concentration (by weight %wt). The disk speed, liquid initial height and viscosity were the control parameters in this study. The motor speed, therefore the disc’s speed, was controlled using a PID controller loop implemented on LABVIEW environment. Experiments with tape water and aqueous glycerol mixtures, as the working fluids, were conducted at three different initial liquid heights of 20, 30 and 40 mm above the rotating disc. The viscosity values of the used mixtures were obtained through technical data provided by a registered chemical company (Dow Chemical Company 1995-2010). Eight different aqueous glycerol mixtures were used in the experiments with viscosity varying from 1 to 22 (0 ~ 75% glycerol) times the water’s at room temperature (21°C). The detailed points of study were: 1, 2, 4, 6, 8, 11, 15 and 22 times the water’s viscosity (μwater) at room temperature. Although the viscosity of the mixture varied exponentially with the glycerol concentration (see Fig. 2), closer points of study were conducted at low concentration ratios since significant effects have been recognized by just doubling the viscosity of water as it will be discussed later. The temperature variation of the working fluid was measured using a mercury glass thermometer and recorded before and just after typical experimental runs and was found to be stable and constant (i.e. room temperature). Therefore, the viscosity of the mixture was ensured to be constant and stable during the experiment. Phase diagrams had been conducted and showed great approximation in defining the different regions for existing patterns in terms of disc’s speed and initial height within the studied viscosity range. A digital CMOS high-speed camera (pco.1200hs) with a resolution of 1280 x 1024 pixels was placed vertically above the cylinder using a tripod. Two types of images were captured: colored and 8-bit gray scale images, at 30 frames per second, for the top view of the formed polygonal patterns (see Fig. 3 for example). The colored images were used as illustration of the observed stratification of the hollow vortex core where each colored layer indicates a water depth within the vortex core. It is worth noticing that the water depth increases continuously as we move away from the center of the disk (due to the
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applied centrifugal force). The continuous increase in the water depth, depicted in the Fig. 3 by the colored layers, indicates momentum stratification in the radial direction (i.e., starting with the central white region which corresponds to a fully dry spot of the core and going gradually through different water depth phases until reaching the black color region right outside the polygonal pattern boundary layer). For subsequent quantitative analysis, the data was conducted with grey images as those are simpler for post-processing. The transition mechanism is investigated using image processing techniques. First the images were segmented; the original 8-bit gray-scale image is converted into a binary image, using a suitable threshold, to extract the polygonal contours (Gonzalez et al. 2004). This threshold value is applied to all subsequent images in a given run. In the image segmentation process, all the pixels with gray-scale values higher than the threshold were assigned 1’s (i.e. bright portions) and the pixels with gray-scale values lower than or equal to the threshold were assigned 0’s (i.e. dark portions). The binary image obtained after segmentation is filtered using a low-pass Gaussian filter to get rid of associated noises. In the next step, the boundaries of the pattern were extracted using the standard edge detection procedure. The pattern contours obtained from the edge detection procedure were then filtered using a zero-phase filter to ensure that the contours have no phase distortion. The transformations of the vortex core are analyzed using Fast Fourier Transform (FFT) of the time series of the radial displacement for a given point on the extracted contour, defined by its radius and its angle in polar coordinates with origin at the centroid of the pattern; see Ait Abderrahmane et al. (2008, 2009) for further details.
Fig. 3. Polygonal vortex core patterns. The inner white region is the dry part of the disk and the dark spot in the middle of the image is the bolt that fixes the disk to the shaft. The layers with different colors indicate the variation of water depth from the inner to the outer flow region.
N=2
N=6N=5
N=3
N=4
N=2
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(a) (b)
(c) (d)
Fig. 4. (a), (b), (c) Oval pattern progression and corresponding Power spectra; and (d) oval to triangular transition N = 2 to N = 3 and corresponding power spectrum.
3. Results and Discussion We first discuss results obtained at an initial height hi = 40 mm where transitions from N = 2 → N = 3 and N = 3 → N = 4 were recorded and analyzed using power spectral analysis. Starting with stationary undisturbed flow, the disc speed was set to its starting point of 50 RPM and was then increased with increments of 1 RPM. Sufficient buffer time was allowed after each increment for the flow to equilibrate. At a disc speed of 2.43 Hz the first mode shape (oval) appeared on top of the disc surface. At the beginning of the N = 2 equilibrium state, the vortex core is fully flooded. While increasing the disc speed gradually, several sets of 1500 8-bit gray-scale images were captured and recorded. Recorded sets ranged 3 RPM in between. Systematic tracking of the patterns speed and shape evolution were recorded and the recorded images were processed. The evolution of the oval equilibrium state shape and rotating frequency is shown in Figs. 4a to 4d. Starting with a flooded core at fp = 0.762 Hz in figure 4a where the vertex of the inverted bell-like shape free surface barely touched the disc surface, Fig. 4b then shows the oval pattern after gaining more centrifugal force by increasing the disc speed by 9 RPM. The core became almost dry and the whole pattern gained more size both longitudinally and
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transversely with a rotating frequency of fp = 0.791 Hz. It is clearly shown that at this instance, one of the two lobes of the pattern became slightly fatter than the other. Fig. 4c shows shape development and rotational speed downstream the N = 2 range of existence. It is important to mention that once the oval pattern is formed, further increase in the disc speed, therefore the centrifugal force applied on the fluid, curved up the oval pattern and one of the lobes became even much fatter giving it a quasi-triangular shape. Fig. 4d features the end of the oval equilibrium pattern in the form of a quasi-triangular pattern and therefore the beginning of the first transition process (N = 2 to N = 3). The transition process is recorded, processed and the corresponding power spectrum was generated (see Fig. 4d). The power spectral analysis revealed two dominant frequencies from the extracted time series function of the captured images; frequency fm corresponds to the original oval pattern and frequency fs corresponds to the growing subsequent wave N = 3, which is a travelling soliton-like wave superimposed on the original oval pattern therefore forming the quasi-triangular pattern (Ait Abderrahmane et al. 2009). Further increase of the disc speed resulted in the forming and stabilizing of the triangular mode shape (N = 3) with a flooded core; both the troughs and apexes of the polygonal pattern receded and the core area shrank significantly. Following the same procedure, the development of the triangular pattern and its transition to square (N = 4) shape were recorded, image processed and analyzed. Figs. 5a to 5e show the power spectra plots and their corresponding sample image from the set recorded and used in generating each of the power spectra. The behaviour of the oval pattern’s shape development and transition was also respected for the triangular pattern evolution. Ait Abderrahmane et al. (2009) described the transition process in the form of a rotating solid body N shape associated with a traveling “soliton”-like wave along the vortex core boundary layer. The evidence of such soliton-like wave is revealed here. Fig. 6 shows a sample set of colored RGB images during the transition process described above; these images feature the quasi-periodic state during N = 3 to N = 4 transition described earlier. Giving a closer look at the sequence of images, one could easily figure out the following: the three lobes or apexes of the polygonal pattern are divided into one flatten apex and two almost identical sharper apexes. Keeping in mind that the disc, therefore the polygonal pattern, is rotating in the counter clockwise direction and that the sequence of images is from left to right, by tracking the flatten lobe, one could easily recognize that an interchange between the flatten lobe and the subsequent sharp lobe (ahead) takes place (see third row of images). In other words, now the flattened apex receded to become a sharp stratified apex and the sharp lobe gained a more flattened shape. Such phenomenon visually confirms the fact that transition takes place through a soliton-like wave travelling along the vortex core boundary but with a faster speed than the parent pattern. This first stage of the transition process was referred to as the quasi-periodic stage by Ait Abderrahmane et al. (2009). The quasi-periodic stage takes place in all transitions until the faster travelling soliton-like wave synchronizes with the patterns rotational frequency forming and developing the new higher state of
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equilibrium pattern. Vatistas et al. (2008) found that the synchronization process takes place when the frequencies ratio of both pattern (N) and the subsequent pattern developed by the superimposed soliton wave (N+1) lock at a ratio of (N-1)/N. Therefore, for transition from N = 2 to N = 3, the synchronization takes place when the frequencies ratio is rationalized at 1/2. And the transition N = 3 to N = 4, takes place when the ratio between both frequencies are equal to 2/3. In the above illustrated two transition processes, the frequency ratio for first transition was equal to fN / fN+1 = fm / fs = 1.69/3.04=0.556 ≈ 1/2. On the other hand, the second transition took place when fN / fN+1 = fm / fs = 3.28/4.92=0.666 ≈ 2/3.
(a) (b)
(c) (d)
(e)
Fig. 5. (a), (b), (c) Triangular pattern progression and corresponding power spectra; (d) Transitional process from triangular to square pattern; and (e) square pattern and corresponding power spectra.
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Fig. 6. Quasi-periodic state during triangular to square transition. Following the same trend, the second experiment was conducted using water at an initial height of 20 mm. At this low aspect ratio, transition between higher mode shapes was tracked and recorded. Using similar setup and experimental procedure, the transition from square mode (N = 4) to pentagonal pattern (N = 5) and from pentagonal to hexagonal pattern (N = 6) were recorded and image-processed for the first time in such analysis. Following the same behavior, the transition occurred at the expected frequency mode-locking ratio. Fig. 7a shows the third polygonal transition, from N = 4 to N = 5. The frequency ratio of the parent pattern to the soliton-like wave is fm/fs = 4.102/5.449 = 0.753 ≈ 3/4. Similarly, Fig. 7b shows the transition power spectrum for the last transition process observed between polygonal patterns, which is from N = 5 to N = 6 polygonal patterns. The frequency ratio fm/fs = 5.625/6.973 = 0.807 which is almost equal to the expected rational value 4/5. With these two experimental runs, the explanation of the transition process between polygonal patterns observed within hollow vortex core of swirling flows within cylinder containers under shallow water conditions is confirmed for all transitional processes.
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(a) (b)
Fig. 7. (a) Square to pentagonal transition; and (b) pentagonal to hexagonal transition.
Initial height (hi ) hi = 20 mm hi = 30 mm hi = 40 mm Transition (N) - ( N+1) 3 - 4 4 - 5 5 - 6 2 - 3 3 - 4 4 - 5 2 - 3 3 - 4
%error Table 1. Transition mode-locking frequencies for different liquid viscosities.
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Fig. 8. Power spectrum for N = 2 pattern replica
The influence of the liquid viscosity on the transitional process from any N mode shape to a higher N+1 mode shape is also investigated. As described earlier, eight different liquid viscosities were used in this study ranging from 1 up to 22 times the viscosity of water. All transitional processes between subsequent mode shapes were recorded, and acquired images were processed. Using the same procedure as in the last section, the frequency ratio of the parent pattern N and the subsequent growing wave N+1 has been computed and tabulated in Table 1. As shown in Table 1, the maximum deviation from the expected mode-locking frequency ratio (fm/fs) always appeared in the first transition (N = 2 to N = 3). A reasonable explanation for such induced error is the fact that, the higher the number of apexes per full pattern rotation, the more accurate is the computed speed of the pattern using the image processing technique explained before. Therefore, throughout the conducted analysis, the most accurate pattern’s speed is the hexagon and the least accurate is the oval pattern. Apart from that significant deviation, one can confidently confirm that even at relatively higher viscous swirling flows, the transition between polygonal patterns instabilities takes place when the parent pattern (N) frequency and the developing pattern (N+1) frequency lock at a ratio of (N-1)/N (Vatistas et al. 2008). As explained earlier, transition has been found to occur in two main stages being the quasi-periodic and the frequency-locking stages (Ait Abderrahmane et al. 2009). It is also confirmed that frequency mode-locking does exist in polygonal patterns transition irrelative of the mode shapes, liquid heights and the liquid viscosity (within the studied region). In this section, the quasi-periodic phase will be further elucidated and confirmed. Earlier in this paper the quasi-periodic state in the transition of N = 3 to N = 4, using water as the working fluid, was observably described in Fig. 6. To further analyze the quasi-periodic stage, a technique has been developed which animates the actual polygonal patterns instabilities but without the existence of the speculated travelling soliton-like wave along the patterns boundary layer. Using MAPLE plotting program, all mode shapes replica have been plotted and printed. Table 2 shows the plots and their corresponding plotting functions. Printed images were glued to the rotating disc under dry conditions one at a time. The disc was rotated with corresponding pattern’s expected speeds under normal working conditions. Such
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technique gave full control of the rotating pattern. Therefore, both speed and geometry of the patterns were known at all times. Sets of 1500 8-bit images were captured and processed using similar computing procedure.
N Pattern plot Plot function
2
r =1+ 0.2 sin(2 θ)
2 - 3
r =1+ 0.2 sin(2 θ) + 0.1 sin(3 θ +1)
3
r =1+ 0.1 sin(3 θ)
3 - 4
r =1+ 0.1 sin(3 θ) + 0.15 sin(4 θ +1)
4
r =1+ 0.15 sin(4 θ )
Table 2. Patterns replica with corresponding functions.
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(a) (b)
Fig. 9. Power spectrum of transition processes using patterns replica (a) N = 2 to N = 3; and (b) N = 3 to N = 4 Power spectra of the processed sets of images revealed similar frequency plots. Starting with the oval-like shape, the disc was rotated at a constant speed of 1 Hz and the power spectrum was generated from the extracted images and plotted as shown in Fig. 8. Since the oval pattern speed is controlled in this case (by disc speed), the frequency extracted could have been presumed to be double the disc frequency (2 Hz). The actual frequency extracted is shown in Fig. 8, fm = 1.934 Hz (3.3% error). Following the same procedure, other polygonal patterns replica were printed to the disc, rotated, captured and processed subsequently. Figs 9a and 9b show the power spectra generated from rotating the quasi-triangular and the quasi-square patterns, respectively. Fig. 9a shows a power spectrum generated from the set of pictures featuring a quasi-triangular pattern captured at 30 fps. The power spectrum revealed two dominant frequencies being fm = 3.809 Hz and fs = 5.742 Hz corresponding to the oval and triangular patterns, respectively. Since the quasi-triangular pattern is stationary and under full control, it could have been presumed that the frequency ratio would have a value of 2/3 since the replica pattern is generated by superimposing the oval and triangular functions. The actual extracted frequency was fm/fs = 3.81/5.74 = 0.663 ≈ 2/3. Comparing this frequency ratio with the real polygonal patterns mode-locking ratio of 1/2 described earlier, it is clear that the ratio is totally different which proves that both patterns are not behaving equivalently although having generally similar instantaneous geometry. Therefore, the actual rotating pattern does not rotate rigidly as the pattern replica does, but rather deforms in such a way that the ratio of the two frequencies is smaller which confirms the idea of the existence of the fast rotating soliton-like wave (fs). Moving to the second transition process, triangular to square, as shown in Fig. 9b, the frequency ratio was found to be 3/4 as expected since the function used to plot the quasi-square pattern is the superposition of both functions used in plotting the pure triangular and square patterns given in Table 2. Comparing this ratio with the actual mode-locking ratio of 2/3 observed with real polygonal patterns, it is obvious that the ratio is still smaller which respects the existence of a faster rotating wave along the
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triangular pattern boundary that eventually develops the subsequent square pattern as visualized earlier using the colored images. From these two experiments, along with the visual inspection discussed earlier, the existence of the fast rotating soliton-like wave (N+1) along the parent pattern boundary layer (N) is verified, therefore, the quasi-periodic stage.
4. Conclusion Through the analysis of the present experimental results from different initial conditions, we confirmed with further evidences and generalized the mechanism leading to transition between two subsequent polygonal instabilities waves, observed within the hollow vortex core of shallow rotating flows. The transition follows the universal route of quasi-periodic regime followed by synchronization of the two waves’ frequencies. We shows, for the first time, all observed transitions from N-gon to a subsequent (N+1)-gon occur when the frequencies corresponding to N and N+1 waves lock at a ratio of (N-1)/N. The effect of varying the working fluid viscosity on the transitional processes between subsequent polygonal patterns was also addressed in this paper. Both stages of the transitional process were further explored in this work. The quasi-periodic stage was first tackled using two different techniques, a visual method and an animated method. The deformation of the colored stratified boundary layers of polygonal patterns were inspected during transition process of polygonal patterns and the existence of a fast rotating wave-like deformation was recognized which confirms the idea of the co-existence of a soliton-like wave that initiates the quasi-periodic stage at the beginning of the transition. In order to further materialize this observation, experiments were re-conducted using fixed patterns replica featuring the quasi-periodic geometry of polygonal patterns under dry conditions. Such technique allowed full control of the patterns geometry and speed at all time, therefore working as a reference to the real experiment performed under wet conditions. The experiments revealed an interesting basic idea that was useful when addressing the significant difference in behavior associated with the real patterns transitions. The second part of the transition process included the frequency mode-locking ratio of subsequent patterns. Dealing with the first part of the transition process as being a bi-periodic state or phase, in order for such state to lose its stability, a synchronization event has to occur (Bergé et al. 1984). This synchronization has been confirmed to occur when the frequency ratio of the parent pattern N to the subsequent pattern N+1 rationalized at (N-1)/N value (Vatistas et al. 2008). The frequency mode-locking phenomenon was found to be respected even at relatively higher viscosity fluids when mixing glycerol with water. Acknowledgment This work is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).
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References 1. H. Ait Abderrahmane. Two Cases of Symmetry Breaking of Free Surface Flows. Ph.D.
Thesis, Concordia University, Montreal, Canada, 2008. 2. H. Ait Abderrahmane, M.H.K. Siddiqui and G.H. Vatistas. Transition between
Kelvin's equilibria. Phys. Rev. E 80, 066305, 2009. 3. H. Ait Abderrahmane, K. Siddiqui, G.H. Vatistas, M. Fayed and H.D. Ng.
Symmetrization of polygonal hollow-core vortex through beat-wave resonance. Submitted to Phys. Rev. E. Oct. 8, 2010.
4. P. Bergé, Y. Pomeau and C. Vidal. Order Within Chaos Hermann, Paris, 1984. 5. J.M. Chomaz, M. Rabaud, C. Basdevant and Y. Couder. Experimental and numerical
investigation of a forced circular shear layer J. Fluid Mech. 187:115-140, 1988. 6. Dow Chemical Company (1995-2010). 7. M.P. Escudier. Observations of the flow produced in a cylindrical container by a
rotating endwall. Experiments in Fluids 2:189-196, 1984. 8. R.C. Gonzalez, R.E. Woods and S.L. Eddins. Digital Image Processing Using
MATLAB 7th edition. Prentice Hall, 2004. 9. R. Hide and C.W. Titman. Detached shear layers in a rotating fluid J. Fluid Mech.
29:39-60, 1967. 10. T.R.N. Jansson, M.P. Haspang, K.H. Jensen, P. Hersen and T. Bohr. Polygons on a
rotating fluid surface Phys. Rev. Lett. 96: 174502, 2006. 11. H. Niño and N. Misawa. An experimental and theoretical study of barotropic
instability J. Atmospheric Sciences 41:1992-2011, 1984. 12. M. Rabaud and Y. Couder. Instability of an annular shear layer. J. Fluid Mech.
136:291–319, 1983. 13. S. Poncet and M.P. Chauve. Shear-layer instability in a rotating system J. Flow
Visualization and Image Processing 14:85-105, 2007. 14. H.U. Vogel. Experimentelle Ergebnisse über die laminare Strömung in einem
zylindrischen Gehäuse mit darin rotierender Scheibe MPI für Strömungsforschung Bericht 6, 1968.
15. G.H. Vatistas. A note on liquid vortex sloshing and Kelvin's equilibria J. Fluid Mech. 217:241-248, 1990.
16. G.H. Vatistas, H. Ait Abderrahmane and M.H.K. Siddiqui. Experimental confirmation of Kelvin’s equilibria Phys. Rev. Lett. 100, 174503, 2008.
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The Influence of Charge Traps in Semiconductor
Diode on Complex Dynamics in Non-autonomous
RL-Diode Circuit
Manturov A.O., Akivkin N.G., Glukhovskaya E.E.
Saratov State Technical University, Saratov, Russia
Abstract. Landslides are a recurrent phenomenon in many regions of Italy: in
particular, the rain-induced shallow landslides represent a large percentage of this
type of phenomenon, responsible of human life loss, destruction of assets and in-
frastructure and other major economical losses. In this paper a theoretical com-
putational mesoscopic model based on interacting particles has been developed to
describe the features of a granular material along a slope. We use a Lagrangian
method similar to molecular dynamic (MD) for the computation of the movement
of particles after and during a rainfall. In order to model frictional forces, the MD
method is complemented by additional conditions: the forces acting on a particle
can cause its displacement if they exceed the static friction between them and the
slope surface, based on the failure criterion of Mohr-Coulomb, and if the resulting
speed is larger that a given threshold. Preliminary results are very satisfactory;
in our simulations emerging phenomena such as fractures and detachments can be
observed. In particular, the model reproduces well the energy and time distribution
of avalanches, analogous to the observed Gutenberg-Richter and Omori distribu-
tions for earthquakes. These power laws are in general considered the signature of
self-organizing phenomena. As in other models, this self organization is related to a
large separation of time scales between rain events and landslide movements. The
main advantage of these particle methods is given by the capability of following the
trajectory of a single particle, possibly identifying its dynamical properties.
Keywords: Landslide, molecular dynamics, lagrangian modelling, particle based
method, power law.
1 Introduction
Predicting natural hazards such as landslides, floods or earthquake is one ofthe challenging problems in earth science. With the rapid development ofcomputers and advanced numerical methods, detailed mathematical modelsare increasingly being applied to the study of complex dynamical processessuch as flow-like landslides and debris flows.
The term landslide has been defined in the literature as a movement of amass of rock, debris or earth down a slope under the force of gravity (Varnes[1958], Cruden [1991]). Landslides occur in nature in very different ways.It is possible to classify them on the bases material involved and type ofmovement (Varnes [1978]).
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Landslides can be triggered by different factors but in most cases the trig-ger is an intense or long rain. Rainfall-induced landslides deserved a largeinterest in the international literature in the last decades with contributionsfrom different fields, such as engineering geology, soil mechanics, hydrologyand geomorphology (Crosta and Frattini [2007]). In the literature, two ap-proaches have been proposed to evaluate the dependence of landslides onrainfall measurements. The first approach relies on dynamical models whilethe second is based on the definition of empirical rainfall thresholds overwhich the triggering of one or more landslides can be possible(Segoni et al.[2009]). At present, several methods has been developed to simulate thepropagation of a landslide; most of the numerical methods are based on acontinuum approach using an Eulerian point of view (Crosta et al. [2003],Patraa et al. [2005]).
An alternative to these continuous approaches is given by discrete meth-ods for which the material is represented as an ensemble of interacting butindependent elements (also called units, particles or grains). The modelexplicitly reproduces the discrete nature of the discontinuities, which cor-respond to the boundaries of each element. The commonly adopted termfor the numerical methods for discrete systems made of non deformable ele-ments, is the discrete element method (DEM) and it is particularly suitableto model granular materials, debris flows and flow-like landslide (Iordanoffet al. [2010]). The DEM is very closely related to molecular dynamics (MD),the former method is generally distinguished by its inclusion of rotationaldegrees-of-freedom as well as stateful contact and often complicated geome-tries. As usual, the more complex the individual element, the heavier is thecomputational load and the “smaller” is the resulting simulation, for a givencomputational power. On the other hand, the inclusion of a more detaileddescription of the units allows for more realistic simulations. However, theaccuracy of the simulation has to be compared with the experimental dataavailable. While for laboratory experiments it is possible to collect very ac-curate data, this is not possible for real-field landslides. And, finally, theproposed model is just an approximation of a much more complex dynamics.These arguments motivated us in exploring the consequences of reducing thecomplexity of the model as much as possible.
In this paper we present a simplified model, based on the MD approach,applied to the study of the starting and progression of shallow landslides,whose displacement is induced by rainfall. The main hypothesis of the modelis that the static friction decreases as a result of the rain, which acts as alubricant and increases the mass of the units. Although the model is stillschematic, missing known constitutive relations, its emerging behavior isquite promising.
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2 The model and simulation methodology
We limit the study to two-dimensional simulations (seen from above) alonga slope, modeling shallow landslides. We consider N particles, initially ar-ranged in a regular grid (Fig. 5), all of radius r and mass m.
The idea is to simulate the dynamics of these particles during and after arainfall. In the model the rain has two effects: the first causes an increase inthe mass of particles, while the second involves a reduction in static frictionbetween the particle and the surface below.
The equation of Mohr-Coulomb,
τf = c + σ tan(φ), (1)
says that the shear stress τf on the sliding surface is given by an adhesive partc plus a frictional part tan(φ). In the our model we want to find a triggercondition of the particle that is based on the law of Mohr-Coulomb (Eq. (1)).The coefficient of cohesion, c in the Eq. (1), has been modeled by a randomcoefficient that depends on the position of the surface. On the other hand,the term σ tan(φ) in the Eq.(1), has been modeled by a theoretical force of
static friction F (s)i which is described later.
The static-dynamic transition is based on the following trigger conditions:
|F (a)i | < F (s)
i + c,
|vi| < v(threshold)i → 0,
(2)
then the motion of the single block will not be triggered until the active
forces F (a)i (gravity forces + contact forces) do not exceed the static friction
F (s)i plus the cohesion term c and until the velocity |vi| not overcomes the
threshold velocity v(threshold)i (Eq. (2)). The irregularities of the surface are
modeled by means of the friction coefficients, which depends stochasticallyon the position (quenched disorder).
In Eq. (2), the force F (a)i is given by the sum of two components: the
gravity F (g)i and the interaction between the particles F (i)
i .
F (a)i = F (g)
i + F (i)i . (3)
The gravity F (g)i is given by
F (g)i = g sin(α)(mi + wi(t)), (4)
where g is the acceleration of gravity, α the slope (supposed constant) of thesurface, mi the dry mass of block i and wi the absorbed water cumulated intime. The quantity wi(t) is a stochastic variable (corresponding to rainfallevents σ(w)(t)),
wi(t) =
σ(w)i (t) dt. (5)
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(a) (b)
Fig. 1. (a) Particles in the computational domain: the maximum radius of iteration
defined in the algorithm is equal to the side L of the cell. Considering the black
particle in the center of the circumference, it can interact only with the neighboring
blue particles.
Fig. 2. (b) Cells considered when calculating the forces: if a particle is in cell (x, y),the interaction forces will be calculated considering only the particles located in cells
(x+ 1, y), (x+ 1, y + 1), (x+ 1, y) and (x− 1, y). This method halves the number
of interactions because it calculates 4 cells instead of 8.
The interaction force between two particles is defined trough a potentialthat, in the absence of experimental data, we modeled after the Lennard-
Jones one. The corresponding interaction force F (i)ij that acts on block i due
to block j is given by
F (i)ij = −F (i)
ji = −∇V (Rij) = −∇4ε ·
r
Rij
−12
−
r
Rij
−6
, (6)
where Rij is the distance between the particles,
Rij =(xj − xi)2 + (yj − yi)2, (7)
r is the radius of the particles and is a constant.The computational strategy for calculating the interaction forces between
the particles is similar to the Verlet neighbor list algorithm (Verlet [1967]).In the code the computational domain is divided in square cells of side L(see Fig. 1), corresponding to the length at which the interaction force istruncated. The truncation has a very little effect on the dynamics, so we didnot correct the potential by setting V (L) = 0, as usual in MD.
Thanks to the Newton’s third law it is possible reduce the number in-teraction and consider the only particle that has not been considered in theprevious step (see Fig. 2).
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0 100 200 300 400 500 600 700 800 900 10000
0.2
0.4
0.6
0.8
1
1.2
Static friction coefficient
Time
µs
30 35 40 45 500
0.5
1
1.5
2
2.5
3
3.5x 10
5
Degree of slope (°)
T
Trigger Time VS Slope
(a) (b)
Fig. 3. (a) Static friction coefficient µs vs. time, with µ(0)s = 1.2 and µ(∞)
s = 0.4.
Fig. 4. (b) Triggering time vs. slope, Eq. 18 with m = 0.01, c = 0.1, µ(0)s = 1.15
and µ(∞)= 0.45.
The condition of motion for a given particle is governed by Eq. 2. The
static friction F (s)i is given by
F (s)i = µs(mi + wi(t)) cos(α). (8)
The Equation 8 is expressed by the friction’s coefficient µs. We assumedthat the rain has a lubricating effect between the particles and underlyingsurface; the friction coefficient has therefore been defined as,
µs = µ(∞)s + (µ(0)
s − µ(∞)s ) exp(−w0t), (9)
where µ(0)s0 and µ(∞)
s are, respectively, the initial (dry) friction coefficient att = 0 (starting of rainfall) and the final (wet) for t → ∞. The effect of rainfallis to lubricate the sliding surface of the landslide, at a constant speed w0 inthis example.
When the active forces exceed the static friction plus the quenched stochas-tic coefficient of cohesion c, the particle start to move. In this case the forceacting on the particle i is given by
F i = F (a)i − F (d)
i , (10)
where F (a)i are the active forces, and F (d)
i is the force of dynamic friction,
F (d)i = µd(mi + wi(t)) cos(α). (11)
Eq .(11) is of the same type as Eq. (8); the coefficient of dynamic frictionµd is defined similarly to the static one (Eq. (9)). The friction coefficients(static and dynamic) varies from point to point of the computational domainthis choice serves to model the sliding surface like a rough surface.
When a particle exceed the threshold condition (Eq. 2), it moves on theslope with an acceleration a equal to
a =F i
(mi + wi(t)). (12)
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10 20 30 40 50 60 70 8020
25
30
35
40
45
50
55
60
65
70
Fig. 5. Initial configuration of simulations. The 2500 particles are arranged on a
regular grid of 50x50 cells of size 1× 1.
In MD the most widely used algorithm for time integration is the Verletalgorithm. This algorithm allows a good numerical approximation and isvery stable. It also does not require a large computational power because theforces are calculated once for each time step. The model was implementedusing the second-order Verlet algorithm. We first compute the displacementof particles, and half of the velocity updates,
ri = ri + vi∆t+F i
2mi∆t2,
vi = vi +
F i
2mi∆t,
(13)
then compute the forces F i as function of the new positions ri, and finally
compute the second half of velocities,
vi = v
i +F
i
2mi∆t. (14)
We have to define a landslide-triggering time, for instance the time of thefirst moving block. In this case it is very simple to obtain the trigger timetheoretically for an uniform rain of intensity w0. We can write, in equilibriumconditions, for a given mass
|F i| = F (s)i + c
F i = F (g)i + F (i)
i
(15)
We assume that the first movement of the particle is only due to theeffect of gravity, so that we can set the interaction forces equal to zero, andtherefore the equilibrium condition is given by
|F i| = F (g)i + c, (16)
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where m = m+ w(t) = m+ w0t.Using Eq. 17, we can define the trigger time T as
T = − 1
w0· log
tan(α)− c
mg cos(α) − µ(∞)s
µ(0)s − µ(∞)
s
. (18)
3 Results
In order to simulate a landslide along an inclined plane, we use the theoreticalmodel as described above with different parameters.
In the Table 1 we illustrate the parameters used in different simulations,where Sim is the number of simulation, m and r are respectively the mass
and the radius of the particles, µ(0)s , µ(∞)
s , µ(0)d , µ(∞)
d are the coefficients ofstatic and dynamic friction and c’ is the coefficient of cohesion. In the oursimulations the time dt of simulation is set to 0.01: then the effective time tis different from the simulation time T.
3.1 Simulation 1
The position of the particles at t = 3000 is reported in Fig. 6. The rain startswith the particles at rest. We suppose that the speed of the landslide is muchbigger than the rain flux, so that the computation of sliding is performedwithout the contribution of rain (i.e., instantaneously). The rain increasesthe mass of the particle with a factor between 0 and 0.0001. The graph ofthe kinetic energy (Fig. 7) shows a ”stick-slip” dynamic. The distributionf(x) the kinetic energy (Fig. 8) is well approximated by an exponential
f(x) = a · ebx, (19)
with a 3.2 · 104 and b −0.1042.In Fig. 9 the statistical distribution of the intervals between trigger times
is reported. This distribution is well fitted by a power law
f(x) = a · xb, (20)
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10 20 30 40 50 60 70 8020
25
30
35
40
45
50
55
60
65
70
0 0.5 1 1.5 2 2.5 3
x 105
0
1
2
3
4
5
6x 10
!4 Kinetic Energy
Time
E
(a) (b)
Fig. 6. (a) Position of particles in Simulation 1 at t = 3000.
Fig. 7. (b) Kinetic energy vs. time.
10 20 30 40 50 60 70 80 90 10010
0
101
102
103
104
Frequency
Ke
100
101
102
103
102
103
Frequency
T
(a) (b)
Fig. 8. (a) Frequency distribution of the kinetic energy in Simulation 1. The plot
in semi-log axes shows an exponential distribution.
Fig. 9. (b) Frequency distribution of trigger intervals in Simulation 1. The plot in
log-log axes shows a power-law distribution.
with a 691.1 and b −0.4295.Several authors (Turcotte and Malamud [2004], Turcotte [1997], Malamud
et al. [2004]) have observed that some natural hazards such as landslides,earthquakes and forest fires exhibit a power law distribution.
3.2 Simulation 1b
In this simulation we use the same parameters as in simulation 1, but westop the rain event at time t = 20. This is a special case: we want to studythe effect of a steady rain until a fixed time. Fig. 10 shows the arrangementof the particles and Fig. 11 the kinetic energy at t = 300.
One can note that the maximum kinetic energy is much greater in thissimulation. In the case 1 the maximum value of kinetic energy is 5.74 · 10−4
while here it is 2.6 · 10−3. Many small events are observed in the first casewhile in the present one we observe a single large event.
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Fig. 10. Position of particles in Simulation 1b at t = 300.
Fig. 11. Kinetic energy versus time. We observe that the ”stick-slip” events disap-
pear and the fixed duration of precipitation changes the dynamics of the system:
in particular, there is peak at t = 20 at the end of the rain event.
3.3 Simulation 2
In order to explore the dependence of the system behavior on the coefficientof cohesion c, we wary it from 0.01 to 1. The other parameters are thesame of Simulation 1. We observe that the final disposition of the particles(Fig. 12) is not too different from Simulation 1 (Fig. 6), however, it occursat time t = 7500 versus t = 3000 of Simulation 1.
As reported in Fig. 13, the increase of the cohesion coefficient c causes atime dilatation, i.e., a translation of the time at which similar events occur.
10 20 30 40 50 60 70 8020
25
30
35
40
45
50
55
60
65
70
0 1 2 3 4 5 6 7 8
x 105
0
1
2
3
4
5
6
7
8x 10
!4
Time
E
Kinetic Energy
Sim1Sim2
(a) (b)
Fig. 12. Position of particles in Simulation 2 at t = 7000. We observe that to
have a spatial arrangement of particles similar to those of the previous simulation
(Fig. 6) a larger time is needed.
Fig. 13. Kinetic energy of the systems versus time. The black line is the kinetic
energy of Simulation 2. Comparing it with Fig. 7 of Simulation 1, we observe that
an increase in the cohesion coefficient induces a translation of the events.
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3.4 Simulation 3
We explore here the behavior of the system as a function of coefficients ofstatic and dynamic friction µs and µd. Their values are shown in Table 1.The other parameters are the same of Simulation 1. The consequence of thereduction of friction causes an immediate movement of particles. Moreoverthe number of particles involved during the event are larger then in theprevious simulations (Fig. 15).
0 50 100 150 200 250 30020
25
30
35
40
45
50
55
60
65
70
0 0.5 1 1.5 2 2.5 3
x 105
0
50
100
150
200
250
300
350
TimeN
Number of Particles
Sim1
Sim3
(a) (b)
Fig. 14. (a) Position of particles in Simulation 3 at t = 3000. The gray area
represents the particle position of Simulation 1 (Fig. 6).
Fig. 15. (b) Number of particles involved. The decrease of the friction coefficients
leads to an increase in the number of particles in motion.
0 0.5 1 1.5 2 2.5 3
x 105
0
0.002
0.004
0.006
0.008
0.01
0.012
Time
V
Kinetic Energy
Sim1
Sim3
0 1 2 3 4 5 6 7 8 9 10
x 104
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Time
V
Velocity
Sim1Sim3
(a) (b)
Fig. 16. (a) Kinetic energy of the systems vs. time. The black line is the kinetic
energy of Simulation 3. In the last simulation the value of the kinetic energy is
greater than that in Simulation 1. This is due by the number of particles involved
in the event (Fig. 15).
Fig. 17. (b) Mean velocity of the system versus time after t = 1000 for Simulations
1 and 3. We can observe that the two values are not too different between the two
simulations. The difference of the kinetic energy is due to the number of particle
in movement.
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1 2 3 4 5 6 7 8 9 10 11
x 10!3
102
103
104
Time
Ke
0.5 1 1.5 2 2.5 3
102
103
104
105
Frequency
E
(a) (b)
Fig. 18. (a) Statistical distribution of kinetic energy in Simulation 3. It follows an
exponential distribution like in Simulation 1.
Fig. 19. (b)The blue line refers to Simulation 3 with parameters a3 2.88 · 105and b3 −2.365. The black line refers to Simulation 1 with parameters a1 2.83·105 and b1 −3.078. The dots represent the normalized value of the respective
simulations.
Fig. 18 shows that also in this case the statistical distribution of thekinetic energy follows an exponential distribution. The data fit of Eq. (19)gives a 2.592 · 104 and b −0.091.
4 Conclusions
In this article we presented a theoretical model that may be useful for study-ing the effect of precipitation on granular materials. The main hypothesis isthat the rain acts as a lubricant between the terrain and the granular: thiseffect has been modeled by a preliminary report that includes the reductionof static (or dynamic) friction when we simulate the rainfall (Eq. (8) andEq. (11)). The reduction in friction allows to follow the evolution and changein the position of the particles during and after a rainfall. The results ob-tained are very encouraging as regards both the displacement and evolutionof the particles and in the statistical properties of the system. The next stepwill be to develop an experimental setup where granular material (sand orgravel) will be placed on a sloping surface: through liquid lubricant (soapand water) we will study the dynamics of these particles. The comparison ofexperimental and computational model will be very useful for the analysis ofthe effect of lubrication of the soil caused by rainfall.
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University Press, Cambridge, (2nd Edition), 1997.D.L. Turcotte and B.D. Malamud. Landslides, forest fires, and earth-quakes:examples of self-organized critical behavior. Physica A, (340):580–589, 2004.
D.J. Varnes. Landslide type and processes. In: Eckel E.B., ed., Landslides
and engineering practice. National Research Council Highway Research
Board Spec. Rept., Washington, D.C., (29):20–47, 1958.D.J. Varnes. Slope movement types and processes. In: Schuster R.L., Krizel
R.J., eds., Landslides analysis and control. Transp. Res. Board., Special
Such an equilibrium configuration has zero bulk velocity and magnetization,
while pressure and gravity equilibrates the thermodynamic free energy of the
gas.
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4. Conclusions In metriplectic formalism friction forces, acting within isolated systems, are
algebrized. The dissipative terms in the equations of motion are given by a
suitable symmetric, semi-definite bracket of the variables with the entropy of
the degrees of freedom to which friction drains energy.
Two simple “textbook” examples are reported: the point particle moving
through a viscous medium; a piston, moved by a spring against a viscous gas in
a rigid cylinder. In both the examples the evolution is generated via the
metriplectic bracket with the free energy F = H + αS, where H is the conserved
Hamiltonian and S is the monotonically growing entropy. α appears to coincide
with the equilibrium temperature.
The same formalism is then applied to an isolated magnetized plasma,
represented by the dissipative (i.e. viscous and resistive) MHD with suitable
boundary conditions. A Hamiltonian scheme already exists for the non-
dissipative limit; furthermore, the full MF had been introduced for the neutral
fluid version. In this paper, we report the extension of the latter formalisms to
include the magnetic forces and the dissipation due to Joule Effect [9]. The
“macroscopic” level of plasma physics is described by the fluid variable v, but a
“microscopic” level exists too, encoded effectively in the thermodynamical field
s. The energy attributed to the macroscopic degrees of freedom v is passed to the
microscopic ones by friction, while the electric dissipation of Joule Effect
consumes the energy pertaining to the magnetic degrees of freedom B. Notice
that the metriplectic formulation for dissipative MHD that we found, does not
require div.B = 0.
Dissipative MHD is mathematically much more complicated than the two
“textbook” examples, nevertheless its essence is rigorously the same: the MF
algebraically generates asymptotically stable motions for closed systems. At the
equilibrium, mechanical and electromagnetic energies are turned into internal
energy of the microscopic degrees of freedom: the asymptotic equilibria found
here for the three examples are essentially entropic deaths.
Let’s conclude with few more observations.
MF is a deterministic description, but it must be possible to obtain it as an
effective representation of a scenario where the superposition of the
Hamiltonian and the entropic motion mirrors the Physics of a deterministic
Hamiltonian system under the action of noise [8].
The appearance of MF offers potentially great chances because it drives the
algebraic Physics out of the realm of Hamiltonian systems: many interesting
processes in nature (as the apparent self-organization of space physics systems
[12], not to mention biological or learning processes) are not expected to be
even conceptually Hamiltonian. It is very stimulating to imagine dealing with
algebraic formalisms describing them. MF, however, is not able to compound
such processes, because it pertains to complete, i.e. closed, systems, while the
processes just mentioned take place in open ones. Adapting MF to open systems
will then be a necessary step to face this challenge.
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Before concluding, let’s underline again the dynamical role of entropy in MF:
entropy may be interpreted as an information theory quantity [13, 14], and here
we find information directly included in the algebraic dynamics. Furthermore:
irreversible biophysical processes appear to have something in common with
learning processes [15], i.e. processes in which the information is constructed or
degraded, and having a formalism where “information” is an essential function
appears to offer hopes in this branch.
References
1. L. D. Landau, E. M. Lifshitz, Mechanics. Course of Theoretical Physics.Vol.1,
Butterworth-Heinemann, 1982.
2. B. Misra, I. Prigogine, M. Courbage, From deterministic dynamics to probabilistic
description, Physica A, vol. 98, 1-26, 1979.
3. P. J. Morrison, Some Observations Regarding Brackets and Dissipation, Center for
Pure and Applied Mathematics Report PAM--228, University of California,
Berkeley (1984).
4. P. J. Morrison, Thoughts on brackets and dissipation: old and new, Journal of Physics
Conference Series, 169, 012006 (2009).
5. P. J. Morrison, A paradigm for joined Hamiltonian and dissipative systems, Physics D,
vol. 18, 410-419, 1986.
6. A. Raichoudhuri, The Physics of Fluids and Plasmas – an introduction for
astrophysicists, Cambridge University Press, 1998.
7. D. Biskamp, Nonlinear Magnetohydrodynamics, Cambridge University Press, 1993.
8. P. J. Morrison, J. M. Greene, Noncanonical Hamiltonian Density Formulation of
Hydrodynamics and Ideal Magnetohydrodynamics, Physical Review Letters, vol.
45, 10 (1980).
9. M. Materassi, E. Tassi, Metriplectic Framwork for the Dissipative Magneto-
Hydrodynamics, submitted to Physica D.
10. P. J. Morrison, Hamiltonian description of the ideal fluid, Reviews of Modern
Physics, Vol. 70, No. 2, 467-521, April 1998.
11. T. D. Frank, T. D., Nonlinear Fokker-Planck Equations, Springer-Verlag Berlin-
Heidelberg, 2005.
12. T. Chang, Self-organized criticality, multi-fractal spectra, sporadic localized
reconnections and intermittent turbulence in the magnetotail, Phys. Plasmas, 6,
4137-4145, 1999.
13. E. T. Janes, Information Theory and Statistical Mechanics, Phys. Rev., 106, 4, 620-
630, 1957.
14. E. T. Janes, Information Theory and Statistical Mechanics II, Phys. Rev., 108, 2,
171-190, 1957.
15. G. Careri, La fisica della vita (Physics of Life), Sapere, Agosto 2002.
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Fractal Market Time
James McCulloch
Macquarie University, North Ryde, Sydney, Australia.(E-mail: [email protected])
Abstract. The no arbitrage condition requires that market returns are martingaleand the existence of long range dependence in the squared and absolute value ofmarket returns (Granger et al. [9]) is consistent with Fractal Activity Time (Heyde[12]). We model the market clock as the integrated intensity of a Cox point processof the transaction count of stocks traded on the New York Stock Exchange (NYSE).A comparative empirical analysis of a self-normalized version of the integrated in-tensity is consistent with a fractal market clock with a Hurst exponent of 0.75.Keywords: Time Deformation, Long Range Dependent, Stochastic Clock, Frac-tal Activity Time, New York Stock Exchange, Doubly Stochastic Binomial PointProcess.
1 Introduction
Clark [7] observed that returns appear to follow a conditional Gaussian Dis-tribution where the conditioning is taken on a latent stochastic informationflow process. As a consequence, the unconditional returns r(t) will be gener-ated by a mixture where the returns are a Wiener process W (.) subject to atime deformation or subordination process Λ1(t).
r(t) = W[Λ1(t)
](1)
Ane and Geman [1] show that the market unconditional return distri-bution is generated from conditioning an ordinary Brownian diffusion by astochastic clock based on cumulative trade count N(t). We model cumulativetrade count as a Cox [8] (doubly stochastic) point process and assume thatthe associated integrated intensity Λ(t) can be modelled as a time acceleratedbaseline integrated intensity Λ(t) = Λ1(Kt) which is an empirical proxy forthe stochastic market clock.
The empirical analysis uses intra-day cumulative trade counts from theNew York Stock Exchange (NYSE) to explore the characteristics of the inte-grated intensity as the time deformation process by self-normalizing cumu-lative trade count R(t) and modelling the self-normalized trade count as adoubly stochastic binomial point process [22].
R(t) =N(t)
N(1), t ∈ [0, 1] (2)
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We then show that the scaling between final trade count K and the vari-ance of the self-normalized integrated intensity Λ1(Kt)/Λ1(K) is different fordifferent mathematical models of stochastic market time Λ1(Kt).
1. If Λ1(t) is modelled as a finite variance Levy subordinator then the vari-ance of the self-normalized integrated intensity will vary approximately asthe inverse of trade count 1/K.
Var
[Λ1(Kt)
Λ1(K)
]∝ 1
K(3)
2. If Λ1(t) is modelled as Fractal Activity Time (FAT) proposed by Heyde [12]and Heyde and Liu [14] then the variance of the self-normalized integratedintensity will vary approximately with trade count K as a power of theHurst exponent H of the FAT.
Var
[Λ1(Kt)
Λ1(K)
]∝ K2H−2 (4)
3. If Λ1(t) is modelled as an α-stable Levy subordinator then the variance ofthe self-normalized integrated intensity will not vary with trade count K.
Var
[Λ1(Kt)
Λ1(K)
]∝ 1 (5)
The variance of the normalized integrated intensity is found to scale pro-portionally to the inverse square root of final trade count 1/
√K. This implies
the Hurst exponent of the integrated intensity Λ1(t) is H = 0.75 and thusmarket time is fractal. This is consistent with the FAT model and excludesthe Levy subordinator models examined above.
1.1 Self-Normalized Integrated Intensity
The problem with using the stochastic integrated intensity Λ(t) of differentstocks to determine the aggregate statistical properties of the market stochas-tic clock is that stocks trade at different rates. The solution is to re-scale theintra-day trade count to between 0 and 1 by the simple expedient of dividingthe intra-day count (N(t) = k) by the final trade count (N(1) = K). Thisdefines the self-normalized trade count process R(t) which is formally namedthe random relative counting measure.
R(t) =N(t)
N(1)=
k
K= a, a ∈ 0, 1
K. . . ,
K − 1
K, 1 (6)
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It is unsurprising that the random relative counting measure R(t) is de-scribed by a binomial point process directed by the self-normalized integratedintensity. This point process is related to a binomial point process in a waydirectly analogous to the relationship between a Cox point process and thePoisson point process. Formally, the probability distribution of the randomrelative counting measure, R(t) conditioned on the final value of the inte-grated intensity Λ(1) is a binomial point process directed by the stochasticself-normalized integrated intensity of the related Cox process (McCulloch[22]).
PrR(t) = a |Λ(1) = PrN(t) = aK |N(1) = K,Λ(1)
=
(K
aK
)[Λ(t)
Λ(1)
]aK [1− Λ(t)
Λ(1)
](1−a)K
a ∈ 0, 1
K, . . . ,
K − 1
K, 1 , t ∈ [0, 1]
(7)
We can now calculate the moments of the self-normalized intensity byexamining stock trade count trajectories in a 2-d histogram [22].
2 Fractal Activity Time
A stochastic process T is called wide-sense self-similar (Sato [25]) if, foreach c > 0, there are a positive number a and a function b(t) such that
T (ct)d= aT (t) + b(t) have common finite-dimensional distributions. A wide
sense self-similar stationary increment model of market activity time wasintroduced by Heyde [12] and Heyde and Liu [14] as consistent with empiri-cally observed market behaviour, which they termed ‘Fractal Activity Time’(FAT). Heyde and Leonenko [13] developed a FAT with an inverse gammamarginal distribution implying Student-t distributed returns and Finlay andSeneta [11] have defined a FAT with gamma marginal distribution implyingvariance-gamma distributed returns.
T (t) − td= tH
(T (1) − 1
),
1
2≤ H < 1 (8)
E[T (t)] = t + tH(E[T (1)] − 1
)= t , t ∈ [0, 1] (9)
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2.1 Self-Normalized Fractal Activity Time Moments
The Taylor series approximation of the expectation of the FAT model ofthe time accelerated self-normalized integrated intensity has terms that scalewith trade count as K2H−2.
E[T (Kt)
T (K)
]≈ t +
(t − t2H + 1 − (1− t)2H
2
)K2H−2 Var[T (1)] (10)
We arbitrarily model the exogenous ‘S’ shaped non-linear variation indaily market time seasonality (‘U’ shaped daily trading activity) as a deter-ministic function with the same functional form as the expectation of the FATmodel of the self-normalized integrated intensity (eqn 10). Thus market timeas integrated intensity is formulated as Λ1(t) = T (∆(t)) where ∆(t) is thedeterministic function defined below with constant a D that determines themagnitude of the ‘S’ shaped non-linear variation with ∆(0) = 0, ∆(0.5) = 0.5and ∆(1) = 1.
∆(t) = t +(t − t2H + 1 − (1− t)2H
2
)D , t ∈ [0, 1] (11)
If the baseline intensity/stochastic clock is defined as Λ1(t) = T (∆(t))then it is obvious that a stationary increment version of the baseline in-tensity/stochastic is Λ1(∆
−1(t)) = T (∆−1(∆(t))) = T (t) where ∆−1(t) isthe inverse function of ∆(t). For a stock with K observed final trades theintegrated intensity is modelled using the FAT as:
Λ(t) = Λ1(Kt) = T (K∆(t)) (12)
Self-Normalized Fractal Activity Time Variance The Taylor seriesapproximation of the variance of the self-normalized integrated intensity hasterms that scale with trade count as both K2H−2 and K4H−4. However, witha nominal variance of Var[T (1)] = 0.875 and Hurst exponent of H = 0.75 theK4H−4 term is small relative to the K2H−2 term.
Var
[T (Kt)
T (K)
]≈
(t2 − t ( t2H + 1 − (1− t)2H) + t2H
)K2H−2Var
[T (1)
]
−(t − t2H + 1 − (1− t)2H
2
)2
K4H−4 (Var[T (1)])2
(13)
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0.2 0.4 0.6 0.8 1.0t
-0.06
-0.04
-0.02
0.02
0.04
E@THK DHtLLTHKLD-tFAT Expectation with Determinisitic Intraday Variation
401-5022 trades
201-400 trades
101-200 trades
51-100 trades
K = 960.76
K = 284.94
K = 145.02
K = 72.96
Fig. 1. The expectation of the self-normalized FAT with intra-day seasonalityE[T (K∆(t))/T (K)]− t (linear trend removed).We model the exogenous ‘S’ shapednon-linear intra-day seasonality in market time (‘U’ shaped daily trading rate) asa deterministic function ∆(t) (eqn 11) where D = 3. For comparison the empiri-cal expected intra-day variation is also displayed as thin plot lines. The empiricalexpected intra-day variation exhibits an asymmetry between the morning and af-ternoon variations that are not captured by the formal FAT model. The slightdifference in intra-day variation amplitude between trade counts in the formal FATmodel is due to the deterministic function ∆(t) plus the functional form of eqn 10.
3 Levy Subordinators
Levy subordinators are non-decreasing Levy processes (Sato [26]). Therehas been considerable research proposing the use of subordinated Wienerprocesses, and more generally subordinated Levy processes such as stableParetian processes as models of stochastic market time. A number of differentmixtures have been put forward to account for the observed characteristicsof the unconditional return process and prominent examples of subordinatedWiener processes include the Variance Gamma model of [16], [17] and theNormal Inverse Gaussian model, [2], [6], [23], [5], [4], [3]. An example of asubordinated Levy process is the α-stable Gamma model of [21], [20].
3.1 Finite Variance Subordinators
Lemma 1. The following properties of finite variance Levy subordinators areproved by examining the time dependent structure of the first two momentsof a Levy process.
1. Levy subordinators with finite moments are not self-similar.
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0.2 0.4 0.6 0.8 1.0t
0.002
0.004
0.006
0.008
0.010
Var@THKDHtLLTHKLDVariance of Self-Normalized Fractal Activity Time by Trade Count
401-5022 trades
201-400 trades
101-200 trades
51-100 trades
K = 960.76
K = 284.94
K = 145.02
K = 72.96
Fig. 2. The variance of the FAT model of self-normalized integrated intensityVar[T (K∆(t))/T (K)] (eqn 13) for different trade count bands K. The Hurst ex-ponent is H = 0.75 and nominal variance is Var[T (1)] = 0.875. For convenientcomparison, the empirical variance Var[Λ(t)/Λ(1)] is also plotted as thin lines andthe difference between the two is shaded. The difference between the empiricalvariance of the self-normalized integrated intensity and the FAT model is largelydue to the symmetry of the functional form of the deterministic intra-day variation∆(t) (eqn 11) compared to the asymmetry of the empirical intra-day variation, seefigure 1 and related commentary.
2. Any self-normalized Levy subordinator Γ (Kt) with a finite variance scalesapproximately as a function of 1/K for values of K ≫ 1 .
Var
[Γ (Kt)
Γ (K)
]∝ 1
K, K ≫ 1, t ∈ [0, 1] (14)
We examine the closely related case where the random activity time isassumed to be an independent increment additive process (a time changedLevy subordinator, Sato [26]). Using the results in James et al. [15] thevariance of self-normalized increasing additive processes can be calculateddirectly. As an example, the variance of the Self-Normalized Gamma processand Self-Normalized Inverse Gaussian process are formulated explicitly.
Assuming subordinator Γ (t) is a Gamma process, c is constant for alltrade counts and ∆(t) is the deterministic intra-day seasonality (eqn 11),then the variance of the self-normalized Gamma process for a stock with Ktrades is:
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Var
[Γ (K∆(t))
Γ (K)
]= ∆(t)
(1 − ∆(t)
) 1
Kc+ 1(15)
Clearly for the self-normalized Gamma process the term 1/(Kc + 1) ap-proximates 1/K scaling for Kc ≫ 1. Next we assume the additive subordi-nator Γ (K∆(t)) is an inverse Gaussian process and c is constant for all tradecounts, then the variance1 of the self-normalized inverse Gaussian process is:
Var
[Γ (K∆(t))
Γ (K)
]= ∆(t)
(1 − ∆(t)
)(Kc)2 eKc
∫ ∞
Kc
e−u
u−3du (16)
The trade count term for Inverse Gaussian is less transparent than theGamma case above but can be readily shown (figure 3) to approximate 1/Kscaling for K ≫ 1.
The variances of Gamma and Inverse Gaussian self-normalized Levy sub-ordinators are scaled as a function of trade count K and compared to thescaling of the empirical self-normalized stochastic clock and self-normalizedFractal Activity Time (FAT) process. The results are graphed in figure 3and it is immediately clear from this graph that the Levy subordinators scaleclose to 1/K, whereas the FAT process with Hurst exponent H = 0.75 scalesas 1/
√K as required.
3.2 α-Stable Subordinators
Another class of Levy subordinators are α-stable processes Γα with 0 < α <1. These processes have no defined moments (all moments are infinite) and
are self-similar with Γα(t)d= t1/αΓα(1) corresponding to a Hurst exponent
H = 1/α.
Mandelbrot [18], Fama [10] and Mandelbrot and Taylor [19] introducedstable Paretian processes as models of financial market returns. These areinfinite variance symmetric distributions with 1 ≤ α < 2 (α = 2 is the Gaus-sian distribution). It is well known (Samorodnitsky and Taqqu [24]) that astandard Wiener process W (t) subordinated to an α-stable Levy subordina-tor with 0.5 ≤ α < 1 is distributed as a symmetric stable Paretian processwith index 2α.
Γ 2α(t)d= W (Γα(t)) , 0.5 ≤ α < 1 (17)
1 The integral term is the upper incomplete gamma function UΓ (−2,Kc).
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æ
æ
æ
æ
à
à
à
à
ì
ì
ì
ì
ò
ò
ò
ò
ô
ô
ô
ô
ç
ç
ç
ç
100 1000500200 300150 700Trade Count
0.0010
0.0100
0.0050
0.0020
0.0200
0.0030
0.0015
0.0150
0.0070
VarianceTrade Count Scaling of the Self-Normalized Stochastic Clock Variance
ç 1K
ô Inverse Gaussian
ò Gamma
ì 1 K
à Fractal Activity Time
æ Empirical
Fig. 3. The variance scaling of the empirical self-normalized stochastic clockΛ(0.5)/Λ(1) at different trade count bands K compared to the variance scalingof self-normalized versions the Fractal Activity Time (FAT) process and Levy sub-ordinators. It is clear from this graph that the empirical stochastic clock and FAT(H = 0.75) scale close to 1/
√K. Conversely the Gamma and Inverse Gaussian
subordinators scale close to 1/K and are misspecified.
Although α-stable processes with 0 < α < 1 have no defined moments thevariance of the corresponding self-normalized process exists and James et al.[15] show that the variance of the self-normalized time transformed α-stablesubordinator is:
Var
[Γα(K∆(t))
Γα(K)
]= ∆(t)
(1 −∆(t)
)(1− α) , 0 < α < 1 (18)
Therefore a self-normalized α-stable Levy subordinator does not scalewith trade count. However, the empirical variance of the self-normalizedmarket clock displays 1/
√K scaling (figure 3) and the α-stable Levy subor-
dinator model is not consistent with this evidence.
References
1.T. Ane and H. Geman, Order flow, transaction clock, and normality of assetreturns., The Journal of Finance. 55 (2000), no. 5, 2259–2284.
2.O.E. Barndorff-Nielsen, Normal inverse gaussian distributions and stochasticvolatility modelling, Scandinavian Journal of Statistics 24 (1996), 1–13.
3.O.E. Barndorff-Nielsen, Normal inverse gaussian distributions and stochasticvolatility modelling, Scand. J. Statist. 24 (1996), 1–13.
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Proceedings, 4th Chaotic Modeling and Simulation International Conference
31 May – 3 June 2011, Agios Nikolaos, Crete Greece
4.O.E. Barndorff-Nielsen, Proceses of the normal inverse gaussian type, Financeand Stochastics 2 (1998), 41–68.
5.O.E. Barndorff-Nielsen, Processes of normal inverse gaussian type, Finance andStochastics 2 (1998), 41–68.
6.O.E. Barndorff-Nielsen and S.Z. Levendorskii, Feller processes of the normal in-verse gaussian type, Quantitative Finance 1 (2001), no. 3, 318–331.
7.P.K. Clark, A subordinated stochastic process model with finite variance for spec-ulative prices, Econometrica 41 (1973), no. 1, 135–155.
8.D. Cox, Some Statistical Methods Connected with Series of Events (With Discus-sion), Journal of the Royal Statistical Society, B 17 (1955), 129–164.
9.Z. Ding, R. Engle, and C. Granger, A long memory property of stock marketreturns and a new model, Journal of Empirical Finance 1 (1993), 83–106.
10.E. F. Fama, Mandelbrot and the Stable Paretian Hypothesis, Journal of Businessof the University of Chicago 36 (1963), 420–429.
11.R. Finlay and E. Seneta, Stationay-Increment Student and Variance-GammaProcesses, Journal Of Applied Probability 43 (2006), 441–453.
12.C. C. Heyde, A Risky Asset Model with Strong Dependence Through FractalActivity Time, Journal of Applied Probability 36 (1999), 1234–1239.
13.C. C. Heyde and N. Leonenko, Student processes, Advances in Applied Proba-bility 37 (2005), 342–365.
14.C. C. Heyde and S. Liu, Empirical realities for a minimal description risky assetmodel. The need for fractal features., Journal of the Korean MathematicalSociety 38 (2001), no. 5, 1047–1059.
15.L. F. James, A. Lijoi, and I. Prunster, Conjugacy as a Distinctive Feature of theDirichlet Process, Scandinavian Journal of Statistics 33 (2006), no. 1, 105–120.
16.D. Madan, P. Carr, and E. Chang, The variance gamma process and optionpricing, European Finance Review. 2 (1998), 79–105.
17.D. Madan and E. Seneta., The variance gamma (VG) model for share marketreturns, Journal of Business 63 (1990), 511–524.
18.B. Mandelbrot, The Variation of Certain Speculative Prices, Journal of Businessof the University of Chicago 36 (1963), 394–411.
19.B. Mandelbrot and H. M. Taylor, On the Distribution of Stock Price Differences,Operations Research 15 (1967), 1057–1062.
20.C. Marinelli, T. Rachev, and R. Roll, Subordinated exchange rate models: Evi-dence for heavy tailed distributions and long-range dependance, Mathematicaland Computer Modelling. 34 (2001), 955–1001.
21.C. Marinelli, T. Rachev, R. Roll, and H. Goppl., Subordinated stock price models:Heavy tails and long-range dependance in the high-frequency deutche bank pricerecord., Data Mining and Computational Finance (G. Bol, ed.), Springer, 1999.
22.J. McCulloch, Relative Volume as a Doubly Stochastic Binomial Point Process,Quantitative Finance 7 (2007), 55–62.
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24.G. Samorodnitsky and M. Taqqu, Stable Non-Gaussian Random Processes,Chapman and Hall, 1994.
25.K. Sato, Self-similar processes with independent increments, Probability Theoryand Related Fields 89 (1991), no. 3, 285–300.
26.K. Sato, Levy Processes and Infinitely Indivisible Distributions, Cambridge Uni-versity Press, 2002.
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