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It is the natural (or artificial or social) system itself which, through its geometrical-dynamical properties, determines the specific informational tool --- entropy ---to be meaningfully used for the study of its (thermo) statistical properties.
TRIADIC CANTOR SET:
0.6309... 0.6309
Hence the interesting measure
ln 2 0.63
is given by
09...ln 3
(10 ) 4.27 5
fd
cm cm
= =
≅
ENTROPIC FORMS
C.T., M. Gell-Mann and Y. Sato Europhysics News 36 (6), 186 (2005) [European Physical Society]
Vorlesungen uber Gastheorie (Leipzig, 1896)Lectures on Gas Theory, transl. S. Brush (Univ. California Press, Berkeley, 1964), page 13
The forces that two molecules impose one onto the other during an interaction can be completely arbitrary, only assuming that their sphere of action is very small compared to their mean free path.
Ludwig BOLTZMANN
Enrico FERMI
Thermodynamics (Dover, 1936)
The entropy of a system composed of several parts is very oftenequal to the sum of the entropies of all the parts. This is true if the energy of the system is the sum of the energies of all the parts and if the work performed by the system during a transformation is equal to the sum of the amounts of work performed by all the parts. Notice that these conditions are not quite obvious and that in some cases they may not be fulfilled. Thus, for example, in the case of a system composed of two homogeneous substances, it will be possible to express the energy as the sum of the energies of the two substances only if we can neglect the surface energy of the two substances where they are in contact. The surface energy can generally be neglected only if the two substances are not very finely subdivided; otherwise, it can play a considerable role.
The situation is different for the additivity postulate Pa2, the validity of which cannot be inferred from general principles. We have to require that the interaction energy between thermodynamic systems be negligible. This assumption is closely related to the homogeneity postulate Pd1. From the molecular point of view, additivity and homogeneity can be expected to be reasonable approximations for systems containing many particles, provided that the intramolecularforces have a short range character.
Peter LANDSBERG
Thermodynamics and Statistical Mechanics(1978)
The presence of long-range forces causes important amendments to thermodynamics, some of which are not fully investigated as yet.
Is equilibrium always an entropy maximum?J. Stat. Phys. 35, 159 (1984).
[...] in the case of systems with long-range forces and which are therefore nonextensive (in some sense) some thermodynamic results do not hold. [...] The failure of some thermodynamic results, normally taken to be standard for black hole and other nonextensive systems has recently been discussed. [...] If two identical black holes are merged, the presence of long-range forces in the form of gravity leads to a more complicated situation, and the entropy is nonextensive.
L.G. TAFF
Celestial Mechanics(John Wiley, New York, 1985)
This means that the total energy of any finite collection of self-gravitating mass points does not have a finite, extensive (e.g., proportional to the number of particles) lower bound. Without such a property there can be no rigorous basis for the statistical mechanics of such a system (Fisher and Ruelle 1966). Basically it is that simple. One can ignore the fact that one knows that there is no rigorous basis for one's computer manipulations; one can try to improve the situation, or one can look for another job.
David RUELLE
Thermodynamical Formalism -The Mathematical Structures of Classical Equilibrium Statistical Mechanics(page 1 of both 1978 and 2004 editions)
The formalism of equilibrium statistical mechanics -- which we shall call thermodynamic formalism -- has been developed since G.W. Gibbs to describe the properties of certain physical systems. [...] While the physical justification of the thermodynamic formalism remains quite insufficient, this formalism has proved remarkably successful at explaining facts.The mathematical investigation of the thermodynamic formalism is in fact not completed: the theory is a young one, with emphasis still more on imagination than on technical difficulties. This situation is reminiscent of pre-classic art forms, where inspiration has not been castrated by the necessity to conform to standard technical patterns.
(page 3) The problem of why the Gibbs ensemble describes thermal equilibrium (at least for “large systems”) when the above physical identifications have been made is deep and incompletely clarified. -----------------------------------------------------------------------------------------------------------------
[The first equation is dedicated to define the BG entropy form. It is introduced after the words “we define its entropy” without any kind of justification or physical motivation.]
//
: -
:
, ( )( )
(
ii
i
i
E kTE kT
ii
The values of p are determined by the followingif the energy of the system in the i th state is E and if thetemperature of the system is T then
ep where
dogm
Z T eZ T
this a
a
l
−−= =∑
1).
. ;
"
ii
i We shall giveno justification for thi
st constant is taken so that p
This choice of p is called the Gibbs distributioneven a physicist like Ruelle
disposes of this question ass dogma
de
=∑
".ep and incompletely clarified
(in the sense of x = y symmetry)
CONCAVITY:
then
q=0.5 (QC)
q=2 (QEP)
q=0.5 (QEP)
q=2 (QC)
SANTOS THEOREM: RJV Santos, J Math Phys 38, 4104 (1997)
(q - generalization of Shannon 1948 theorem)
({ }) { } ( 1/ , )
( ) ( ) ( ) ( ) ( ) ( )
({ }) ( , ) ({ / }) ({
(1 )
i i
i
A B A Bij i j
i L M L l L Mq q
S p continuous function of pS p W i monotonically increases with WS A B S A S B S A S B with p p p
k k k k kS p S p p p S
IFAND
AND
A p p S
q
ND p
+
= ∀+
= + +
+
− =
= +
1
1
1 ({ }) 1 ({
}) ln
/ }) ( 1
( ) :" ,
)
1
W
i Wi
i i i
m M
ii
M L
q
CE SHANNON The Mathematica
pS p k q S p k p
l Theory
THEN AND ONLY THEN
of
p p with
CommunicationThis theorem and th
p
pq
p
=
=
−⎛ ⎞
= = ⇒ = −⎜ ⎟− ⎝ ⎠
+ =
∑∑
,
. .
e assumptions required for its proof for the
present theory It is given chiefly to lend a certain plausibility to some of our later definitions
are in no way necessaryThe
real justification of the , , .se definitions however will reside in their implications
ABE THEOREM: S Abe, Phys Lett A 271, 74 (2000)
(q - generalization of Khinchin 1953 theorem)
1, 2 1, 2
({ }) { } ( 1/ , ) ( ,..., ,0) ( ,..., )
( ) ( ) ( | )
( ) ( |(
) ) 1
i i
i
W W
IF S p continuous function of pS p W i monotonically increases with W
S p p p S p p pS A B S A S B A S A S B A
k
ANDAND
AND
THEN AND ONLY
k k k
THE
k
N
q
= ∀=
+= + + −
1
1
1 ({ }) 1 ({ }) ln
1
(1996, 1999).
W
i Wi
i i i ii
q
The possibility of such theorem was conjecturedb
pS p k q S p k
y AR Plastino and A Plas n
p p
o
q
ti
=
=
−⎛ ⎞= = ⇒ = −⎜ ⎟− ⎝ ⎠
∑∑
( , ) qS N t versus t
LOGISTIC MAP:
21 1 (0 2; 1 1; 0,1,2,...) t t tx a x a x t+ = − ≤ ≤ − ≤ ≤ =
(strong chaos, i.e., positive Lyapunov exponent)
V. Latora, M. Baranger, A. Rapisarda and C. T., Phys. Lett. A 273, 97 (2000)
1
1 1
11
(0) 0
( )lim
(
( )( ) lim(0)
)
t
tx
We verify
w
Pesin like ide
here
S t
ntity
Kt
andx tt e
K
xλ
ΔΔΔ
λ
ξ
→∞
→
≡
=
−=
≡
q = 0.1
q = 0.2445
q = 0.5
S (t)q
t
N = W = 2.5 106
a = 1.4011552
x = 1 - a xt +1 t
2
# realizations = 15115
0
10
20
30
40
50
0 20 40 60 80
(weak chaos, i.e., zero Lyapunov exponent)
C. T. , A.R. Plastino and W.-M. Zheng, Chaos, Solitons & Fractals 8, 885 (1997) M.L. Lyra and C. T. , Phys. Rev. Lett. 80, 53 (1998) V. Latora, M. Baranger, A. Rapisarda and C. T. , Phys. Lett. A 273, 97 (2000) E.P. Borges, C. T. , G.F.J. Ananos and P.M.C. Oliveira, Phys. Rev. Lett. 89, 254103 (2002) F. Baldovin and A. Robledo, Phys. Rev. E 66, R045104 (2002) and 69, R045202 (2004) G.F.J. Ananos and C. T. , Phys. Rev. Lett. 93, 020601 (2004) E. Mayoral and A. Robledo, Phys. Rev. E 72, 026209 (2005), and references therein
(0) 0
min
( ) lim sup
( )
1 1 1
( ) sup lim(0)
( )
1
qq t
x
q q
q tq
We verify
whereS t
Kt
and
q generalized Pesin like id
x ttx
with
entiK
q
ty
eΔ
λΔξΔ
λ
α
→∞
→
⎧ ⎫≡ ⎨ ⎬
⎩ ⎭
⎧ ⎫≡ =⎨ ⎬
⎩ ⎭
=−
− −=
−
1min max
max
1 1 1 ln ( )1 | | ( 1)1 ( ) ( ) (
ln 1 l
) n 2
2 1
l
n
z F
F
t t
q
zx a x zq z z
dq
an
z
α λα
αα α+
= =
⎡ ⎤= − ⇒ = − = −⎢ ⎥−⎣ ⎦
−
CASATI-PROSEN TRIANGLE MAP [Casati and Prosen, Phys Rev Lett 83, 4729 (1999) and 85, 4261 (2000)](two-dimensional, conservative, mixing, ergodic, vanishing maximal Lyapunov exponent)
G. Casati, C. T. and F. Baldovin, Europhys. Lett. 72, 355 (2005)
[G. Casati, C.T. and F. Baldovin, Europhys Lett 72, 355 (2005)]
0 20 40 60 80 100n0
50
100
150
200
Sq(n) q=-0.2
q= 0
q=+0.2
(a)4000 4000 1000
100
[ 0 0.99993]
W cellsN initial conditions randomly chosen in one cellAverage done over initial cells
q linear correlation
= ×=
= → =
CASATI-PROSEN TRIANGLE MAP [Casati and Prosen, Phys Rev Lett 83, 4729 (1999) and 85, 4261 (2000)](two-dimensional, conservative, mixing, ergodic, vanishing maximal Lyapunov exponent)
0 0
00
( ) lim 1
t
n
Also eS nwith
n
λξ
λ →∞
=
= =
q - generalization of Pesin (- like) theorem
G. Casati, C. T. and F. Baldovin, Europhys. Lett. 72, 355 (2005)
( , ) qS N t versus N
( N = 0 )
( N = 1 )
( N = 2 )
1 1
1
11
1 1 2 2
1 1 2
1
3 6
×
× ×
× ×
1 3
1 1 1 1 4 1 2 1 2 4
1 1 1 1 1
1
1 3 3 1
1 4 6 4 5 2 0 3 0 2 0
( N = 3
1 5
1
)
( N = 4 )
( N
= 6
5 ) 1
×
× × × ×
× × × × ×
×
1 1 1 1 1
5 1 0 1 0 5 1
3 0
6 0 6 0 3 0 6
1 ( )NΣ
× × × × ×
= ∀
HYBRID PASCAL - LEIBNITZ TRIANGLE
Blaise Pascal (1623-1662)Gottfried Wilhelm Leibnitz (1646-1716)Daniel Bernoulli (1700-1782)
1 2
1 p
2 1- p
p 1- p 1
2p κ+ (1 )p p κ− −
(1 )p p κ− − 2(1 )p κ− +
A B
(N=2)
2 2
( 0) ( 1)
1 (
( 2)
11 1
11- )
[ ] [ (1 ) ] [(1 ) ]2 1p p
p p p
N
pNN κ κ κ
= ×= × ×
= × − −×+ − +×
EQUIVALENTLY:
100
90
80
70
60
50
40
30
20
10
01009080706050403020101
NS p q=1.0
q=0.9
q=1.1
(b) 20
10
020101
NS p
q=1.0
q=0.9
q=1.1
(c)
100
90
80
70
60
50
40
30
20
10
01009080706050403020101
NS p
q=1.0
q=0.9
q=1.1
(a)
1. ., 1 SY
STEM (
S) ( )i e such that
qS N N N∝ →∞
=
.0
1/ 2
NNp p
with p
Stretched exponentialα
α
⎛ ⎞=⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟= =⎝ ⎠
,0
1/ 2
N
Np p
N independen
with p
t coins
⎛ ⎞=⎜ ⎟⎜ ⎟⎜ ⎟=⎝ ⎠
,0
1
1N
Leibnitz triangle
pN
⎛ ⎞=⎜ ⎟+⎝ ⎠
(All three examples strictly satisfy the Leibnitz rule)C.T., M. Gell-Mann and Y. Sato Proc Natl Acad Sc USA 102, 15377 (2005)
C.T., M. Gell-Mann and Y. Sato Proc Natl Acad Sc USA 102, 15377 (2005)
Asymptotically scale-invariant (d=2)
d+1
(It asymptotically satisfies the Leibnitz rule)
. .,1 S
YSTEM( ) ( )
S qi e such that
qS N N N∝ →∞
≠
11qd
= −
0 2000 4000 6000 8000 10000
2000
4000
6000
8000
10000
N
S p q=0.0
q=-0.1
q=+0.1
(a)
(d =1) (d = 2) (d = 3)
(All three examples asymptotically satisfy the Leibnitz rule)
0 2000 4000 6000 8000 10000
2000
4000
6000
8000
10000
N
S p
q=1/2
q=1/2-0.1
q=1/2+0.1
(b)
0 2000 4000 6000 8000 10000
2000
4000
6000
8000
10000
N
S p
q=2/3
q=2/3-0.1
q=2/3+0.1
(c)
C.T., M. Gell-Mann and Y. Sato Proc Natl Acad Sc USA 102, 15377 (2005)
Continental Airlines
0 10 20 30 40TIME
0
100
200
300
400
500
600
EN
TR
OP
Y
0 10 20 30 400
2
4
6
0.0 0.5 1.00.0
0.5
1.0
q=1
q=0.05
q=0.2445
q=0.5
(a)
(b)
q
R
a=1.40115519
t
q=1
(c)
S p
(d)
S p
q=0.8
q=0.2445
t
q=0.5
q=0.05
q=1.2
q =1sen q <1sen
0 2000 4000 6000 8000 10000
2000
4000
6000
8000
10000
N
S p
(b)
q=1/2-0.1
q=1/2
q=1/2+0.1
q <1sen
100
90
80
70
60
50
40
30
20
10
01009080706050403020101
N
S p
q=1
q=0.9
q=1.1
(a)q =1sen
C.T., M. Gell-Mann and Y. Sato Europhysics News 36 (6), 186 (2005) [European Physical Society]
, . ,
) , ( , )
( , ):
sen se
sen
q q
q
For q q N and t play essentially the same roleIn particular
i Under conditions of graining in phase spaceS N t K
infinitely fine
A conjecture for S N t
= →∞ →∞
∼
( )1
( )1 1
/ 0
( )
) ,
( 1 , . ., Pesin )
lim ( , )
n
se
j
nt q
jsen
j
N t N t
ii Under con
q K i e like identity for fini
ditions of graining in phas
te N
e spaceS N t N
finite
λ
λ
→∞
>
= ⇒ =
∝
∝
−∑
(Cl
aus
ius)
C.T., M. Gell-Mann and Y. Sato Europhysics News 36, 186 (2005)
fixed N’>N
fixed N
S p
q<q sen
q=q sen
q>q sen
00
t
q=q sen
ε ’< ε
ε’ < ε
ε ’< ε
ε
ε
ε
ε 0ε 0
ε 0
RELAXATION
SENSITIVITY
STATIONARY STATE
ε 0
C.T., M. Gell-Mann and Y. Sato Europhysics News Special Issue Nonextensive Statistical Mechanics – New Trends, New Perspectives (European Physical Society, Nov-Dec 2005), in press
q - CENTRAL LIMIT THEOREM (q-product and de Moivre-Laplace theorem):
11 1 1
1
:) )
[ ln ( ) ln ln (1
1
)(ln )(ln )ln ( ) n ln
]l
q
q q q q q
q qq
q q q q
Propertiesiii
whereas x y
x y x
x
y
x y x yx y
qx y
y x y
− − −
= +
⎡ ⎤⊗ ≡ + −⎣ ⎦
⊗ =
+
⊗ =
−
+
The q- product is defined as follows:
[L. Nivanen, A. Le Mehaute and Q.A. Wang, Rep. Math. Phys. 52, 437 (2003); E.P. Borges, Physica A 340, 95 (2004)]
The de Moivre-Laplace theorem can be constructed with
,0 1/ 2
NNp p with p
Leibnitzand
rule
= =
0 0.2 0.4 0.6 0.8 1q
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
q e
qe=2-1/q
0.5 0.6 0.7 0.8 0.9 1q
0
0.2
0.4
0.6
0.8
1
q e
0 0.2 0.4 0.6 0.8 1
x2
-0.4
-0.3
-0.2
-0.1
0
ln-4
/3[p
(x)/
p(0)
]
N=50N=80N=100N=150N=200N=300N=400N=500N=1000
0 0.02 0.04 0.06 0.081/N
0.41
0.42
0.43
0.44
β(N)
L.G. Moyano, C. T. and M. Gell-Mann, Europhys. Lett. 73, 813 (2006)
q - CENTRAL LIMIT THEOREM: (numerical indications)
,0
11 1
,0
. .
1 1 1 1 ... ( )
( 1)
,
( 1/ 2)
q qN
q qN
We q generalize the de Moivre Laplace theorem with
i
N termsp p p p
p N p N i
e
w th p− −
⎛ ⎞ ⎛ ⎞ ⎛ ⎞= ⊗ ⊗⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠
⎡ ⎤= − − =⎦
−
⎣
−
[Hence q 2 – q (additive duality) and q 1/q (multiplicative duality) are involved]
(q = 3/10)
q - GENERALIZED CENTRAL LIMIT THEOREM: (mathematical proof)
S. Umarov, C.T. and S. Steinberg [cond-mat/0603593]
1[ ( )] (nonline
q-Fourier transform:
q-correlation
[ ]( ) ( ) = ( )
[ ( )]
ar
: [ ( )]
!)
X Y
qix
f xixq q q qF f f x dx f x dx
Two random variables X with density f x and Y with density f yare said
e eξ
ξξ∞ ∞
−∞ −∞
−⊗≡ ∫ ∫
- [X+Y]( ) = [X]( ) [Y]( ) ,
. .,
( ) ( ) ( ) ,
( ) ( , ) ( ) ( , )
q q q q
X Y X q Y
X Y
q q qiyiz ix
q q q
q correlated ifF F F
i e if
dz f z dx f x dy f y
with f z dx dy h x y x y z dx h x z x dy h
e e e ξξ ξ
ξ ξ ξ
δ
∞ ∞ ∞
+−∞ −∞ −∞
∞ ∞ ∞ ∞
+ −∞ −∞ −∞ −∞
⊗
⊗⊗ ⊗ ⊗⎡ ⎤ ⎡ ⎤= ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
= + − = − =
∫ ∫ ∫
∫ ∫ ∫ ∫
- 1 , . ., ( , ) ( ) ( )
( , )
( , )
.
X Yq correlation means independence if q i e h
z y y
where h x y is the joint densix y f x f y
global correlati
t
on
y= =
−
1 , ( , ) ( ) ( )X Yif q hence h x y f x f y⎛ ⎞⎜ ⎟≠ ≠⎝ ⎠
1 0
- - ( )1( ) (- ,3)3-
2 (1 )( ) ( ( )
-
)
( 0, 1, 2,...; )2 (1
Closure:
Iterat
(the
ion
same as in
:
)
n n n
qThe q Fourier transform of a q Gaussian is a z q Gaussia
z qq
q n qq z q z z q n q
n wit
qn
h
q−
+= ∈ ∞
+ −≡ ≡ = = ± ± =
+ −
11
(i)
R.S. Mendes and C.T. [Phys Lett A 285, 273 (2005)] when calculating marginal probabilities!)
(the same as in L.G. Moyano, C.T. and
(1) 1 ( ), ( ) 1 ( ),1(ii) 2 ,
n
nn
q n q q qn
q
he ce
q
±∞
−+
= ∀ = ∀
= −
( ) 2
M. Gell-Mann (2005)!) (the same as in A. Robledo [Physica D 193, 153 (2004)] for pitchfork and
(1 )(iii) 2 =0, 2, 4,...
tangent bifurcations!)
(
1 (1 )
the sa
m myi q m qn m q qm
eldsq
+ −= ± ± ≡ =
+ −me obtained in C.T., M. Gell-Mann and Y. Sato [Proc Natl Acad Sci (USA) 102, 15377 (2005)],
by combining additive and multiplicative dualities, and which was conjectured only to be a possible explanation for the NASA-detected q-triangle for m = 0, 1!)±
,
1 1
( 0, 1, 2,...)
nn
q q
n
α
α α= +
− −
= ± ±
S. Umarov, C.T., M. Gell-Mann and S. Steinberg (2006), cond-mat/0606040
ALGEBRA ASSOCIATED WITH qALGEBRA ASSOCIATED WITH q--GENERALIZED CENTRAL LIMIT THEOREMS:GENERALIZED CENTRAL LIMIT THEOREMS:
q SCALED ATTRACTOR WHEN SUMMING NCORRELATED IDENTICAL RANDOM VARIABLES WITH SYMMETRIC DISTRIBUTION
xNq f x
α →∞F
2
3 11
3 11
( 1)/( 1)
3 1
1,
[ ( )] / [ ( )] ( )
( ) | | ( , 2)
( )
( ) / | | | | (
( )
( )
-
Q Q
c
cq
q q
qq
Q
q q
q
q
with same dx x f x dx f x of f x
G x if x x q
f x C x if x
x
x q
G x Ga
G
sian
x
us
σ
−+
−+
+ −
−
+=
⎡ ⎤≡⎣ ⎦<<
>>
≡
∫ ∫
∼ ∼
F
1
S. Umarov, C. T. and S. Steinberg
, 2)
(2006
lim ( , 2)
) [cond-mat/
060 5 3]
9
3q cwith x q→
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
= ∞
( )
,
,
2 31
1
2
1
3
,
,1
(1 ) / (1 )( ) / | |
( )
( )
,
~
,
Lq
q
qq
q
q
q
q
qw i th L
o
x L s ta b le
f x C x
d i s t r i b u t io n
x L s ta b le d i s t rr
w i th
i t i o n
L
b u
αα α αα
α αα
α
α
α
α α
α α α
−
− ++
−
+
−−
+
+
++
+ + −
=
∼
=
F
F
( * )
, ,11
2 ( 1 ) / ( 1 )
S . U m a r o v , C . T . , M . G e l l -M a n n a n d S . S t e i n b e r g ( 2 0 0 6 ) [ c o n d -m a t /0 6 0 6 0 3 8 ] a n d [ c o n d -m a t /0 6 0
(
6
~
0
) / |
4 0 ]
|qq
q qf x C xα α
α
α α
+ −
+ − −∼
NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS (CANONICAL ENSEMBLE):
1
1
1
1
1[ ]
1
1
Wqi
iq i
Wqi iW
ii qW
qii
i
qi
Extremization of the functional
with the constraints an
pS p k
q
p Ed
yi
p
el
p U
dse
p
=
=
=
=
−≡
−
= =
=
∑
∑∑
∑
( )
1
1
( )
, ,
q i qW
E Uq q qW
q ii
i
q qiE U
q
energy Lagwith andrange parameter ep
β
β
ββ β − −
=
=
− −
≡ ≡ ≡∑∑
Z
Z
'
'
' '
1
'
1 (1 )
1 1( )
1( ) ln
,
ln ln
(
q i
q
WEq
q q qiq q
q
q
q q q q q q q q q q
i
q
Eq
iWe can rewrite
with and
And we can
Z eq U
Si T
T U k
ii F U T
p
S Z Z
rove
with
U
i
h e
Z
w er
ep
β
β
ββ
β
β
ββ
−
=
−
≡ ≡+ −
∂= ≡∂
≡ − = − = −
=
∑
Z
2
2
( . .,
) ln
( )
-
!)
q q q
q q qq
i e the Legendre structure of Thermod
ii U Z
S U Fiv C T T
T Tynamics is q invariant
T
β∂
= −∂
∂ ∂ ∂≡ = = −
∂ ∂ ∂
ON THE NATURE OF q-CORRELATIONS:
W. Thistleton, J.A. Marsh, K. Nelson, L.G. Moyano and C. T. (2006)
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
=Σ
11
11
1
ρρρ
ρρρ
Z
1 1/ 2 1/ 2 ( )
0
Let us consider N correlated randomvariables that generate N correlated uniform distributions
if xf x
otherwise
The N N covariance matrix of
No
the N
rm
N a
l
orm l
a
− ≤ ≤⎧ ⎫= ⎨ ⎬⎩ ⎭
×
( 1 1)
0 1
independencefull correlat
distributions is giv
n
b
io
en y
ρ
ρρ
− ≤ ≤
= ⇒= ⇒
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.2
0
0.2
0.4
0.6
0.8
1
1.2Correlation Coefficient of Uniform RVs vs. Correlation Coefficient of Normal RVs
r1,2 Normal RVs
r 1,2 U
nifo
rm R
Vs
copmuted valuesy=x
Bivariate normal data with a specified correlation coefficient is generated. The data (red dots) corresponding to the resulting marginal distributions are transformed to uniformly distributed random variables on (-1/2,1/2). The correlation coefficient of the bivariateuniform marginal data is presented as a function of the normal correlation coefficient. The bisector (continuous black line) is indicated as well as a guide to the eye.
(N=2)
-0.5
0
0.5
-0.5
0
0.5
0
0.2
0.4
0.6
0.8
1
Marginal U1
Bivariate Uniform Distribution, ρ=0
Marginal U2
frequ
ency
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50
0.5
1
Marginals U1, U2
x
f U 1(x),f
U 2(x)
U1 Histogram
U2 Histogram
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10
0.5
1
Usum=U1+U2, r=0.011257
x
f U sum(x
)
U1 + U2 Histogram
-0.5
0
0.5
-0.5
0
0.5
0
0.5
1
1.5
2
Marginal U1
Bivariate Uniform Distribution, ρ=0.1
Marginal U2
frequ
ency
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50
0.5
1
Marginals U1, U2
x
f U 1(x),f
U 2(x)
U1 Histogram
U2 Histogram
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10
0.5
1
Usum=U1+U2, ρ=0.1
x
f U sum(x
)
U1 + U2 Histogram
( 2; 0)N ρ= =
W. Thistleton, J.A. Marsh, K. Nelson, L.G. Moyano and C. T. (2006)
( 2; 0.1)N ρ= =
-0.5
0
0.5
-0.5
0
0.50
1
2
3
4
5
6
7
Marginal U1
Bivariate Uniform Distribution, ρNormal=0.5
Marginal U2
frequ
ency
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50
0.5
1
Marginals U1, U2
x
f U 1(x),f
U 2(x)
U1 Histogram
U2 Histogram
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10
0.5
1
Usum=U1+U2, ρ=0.5
x
f U sum(x
)
U1 + U2 Histogram
-0.5
0
0.5
-0.5
0
0.50
5
10
15
20
25
Marginal U1
Bivariate Uniform Distribution, ρ=0.9
Marginal U2
frequ
ency
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50
0.5
1
Marginals U1, U2
xf U 1(x
),fU 2(x
)
U1 Histogram
U2 Histogram
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10
0.5
1
Usum=U1+U2, ρ=0.9
x
f U sum(x
)
U1 + U2 Histogram
W. Thistleton, J.A. Marsh, K. Nelson, L.G. Moyano and C. T. (2006)
( 2; 0.5)N ρ= =
( 2; 0.9)N ρ= =
-1.5 -1 -0.5 0 0.5 1 1.50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Empirical PDF of Xsum with Asymptotic Fitted qGaussian PDFsystem size=2 q
∞=-0.27112 β∞=0.675
EmpiricalFitted
-2 -1.5 -1 -0.5 0 0.5 1 1.5 20
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Empirical PDF of Xsum with Asymptotic Fitted qGaussian PDFsystem size=3 q
∞=-0.042292 β∞=0.39472
EmpiricalFitted
-3 -2 -1 0 1 2 30
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Empirical PDF of Xsum with Asymptotic Fitted qGaussian PDFsystem size=5 q
∞=0.15505 β∞=0.18051
EmpiricalFitted
-4 -3 -2 -1 0 1 2 3 40
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Empirical PDF of Xsum with Asymptotic Fitted qGaussian PDFsystem size=6 q
∞=0.19471 β∞=0.13266
EmpiricalFitted
-15 -10 -5 0 5 10 150
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
Empirical PDF of Xsum with Asymptotic Fitted qGaussian PDFsystem size=20 q
∞=0.31718 β∞=0.014581
EmpiricalFitted
-15 -10 -5 0 5 10 150
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Empirical PDF of Xsum with Asymptotic Fitted qGaussian PDFsystem size=24 q
∞=0.32112 β∞=0.010238
EmpiricalFitted
(N=2 ) (N=3 )
(N=5 )
(N=6 )
(N=20 ) (N=24)
W. Thistleton, J.A. Marsh, K. Nelson, L.G. Moyano and C. T. (2006)
-40 -30 -20 -10 0 10 20 30 4010
-6
10-5
10-4
10-3
10-2
10-1
Empirical PDF of Xsum with Asymptotic Fitted qGaussian PDF: UNIFORM system size=100 ρ=0.2 q
∞=0.8347 β∞=0.0024114
EmpiricalFitted
(N=100 )(q = 0.8347)
W. Thistleton, J.A. Marsh, K. Nelson, L.G. Moyano and C. T. (2006)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-10
-8
-6
-4
-2
0
Fitted Asymptotic q∞
versus Normal Correlations, ρ System Size = 25 number deviates = 5e+006
ρ
q ∞
Fitted q∞
y=(1-5/3ρ)/(1-ρ)
W. Thistleton, J.A. Marsh, K. Nelson, L.G. Moyano and C. T. (2006)
513
1q
ρ
ρ
−=
−
00.2
0.40.6
0.81
0
0.5
1
1.5-6
-4
-2
0
2
ρ
Fitted q as a function of Normal Correlation (ρ) and Scaling Exponent (α)System Size = 35
0 and recover th random neighbore and the schemes respectivelyEur Phys J B
α α= →∞
GAS-LIKE (NODE COLLAPSING) NETWORK:
S. Thurner and C. T., Europhys Lett 72, 197 (2005)Number N of nodes fixed (chemostat); i=1, 2, …, N
Degree of the most connected node Degree of a randomly chosen node
7( 2 ; 0; 2)N rα= = =
2000 4000 6000 8000 10000 12000 140000
10
20
30
40
50
60
70
80
time
k max
; k i
ki
kmax
S. Thurner, Europhys News 36, 218 (2005)
[ ] [ ]1
( )
( 1.84
1( ) ln ( )
1)
q
q q
c
P kZ k
optimal
P kq
q
−
=
> −≡ > ≡
−
( ;α →∞
( ; 8)rα →∞ < >=
S. Thurner and C. T., Europhys Lett 72, 197 (2005)
- ( -2)/( ) ( 2, 3, 4,...)ck
qP k ke κ≥ = =
[0.999901,0.999976]linear correlation∈
9( 2 ; 2)N r= =
S. Thurner and C. T., Europhys Lett 72, 197 (2005)
9( 2 )N =
( 2)r =
[ ]( ) ( ) (0) ( ) c c c cq q q q e αα −= ∞ + − ∞
S. Thurner and C. T., Europhys Lett 72, 197 (2005)
2 2 /(3 )
2 2
2
1/(1 )2 2 /(3 ) / ( )
2 2
( , ) [ ( , )] [ ( ,0) (0)] ( 3)
( , ) 1 (1 ) / ( ) ( )
( . .,
)
q
q
qq x tq
The solution ofp x t p x tD p x q
t xis given by
p x t q x t e D
hence
x scales like t e g x t
with
Γ
γ γ
δ
Γ Γ−
−
−− −
∂ ∂= = <
∂ ∂
⎡ ⎤∝ + − ≡ ∝⎣ ⎦
∝
2 3 q
γ =−
PREDICTION:
C.T. and D.J. Bukman, Phys Rev E 54, R2197 (1996)
Hydra viridissima: A. Upadhyaya, J.-P. Rieu, J.A. Glazier and Y. Sawada Physica A 293, 549 (2001)
q=1.5
1.24 0.12
3
slope
hence is satisfiedq
γ
γ
= ±
=−
Defect turbulence:K.E. Daniels, C. Beck and E. Bodenschatz, Physica D 193, 208 (2004)
K.E. Daniels, C. Beck and E. Bodenschatz, Physica D 193, 208 (2004)
21.5 4 / 3 3
q and are consistent withq
γ γ≈ ≈ =−
XY FERROMAGNET WITH LONG-RANGE INTERACTIONS:
A. Rapisarda and A. Pluchino, Europhys News 36, 202 (2005) (European Physical Society)
8Wt =
2.35 ? 1.5 ?
0 :rel
stat
qq
modelα =
(q=2.35)
F.A. Tamarit and C. AnteneodoEurophysics News 36 (6), 194 (2005) [European Physical Society]
t
COLD ATOMS IN DISSIPATIVE OPTICAL LATTICES:
Theoretical predictions by E. Lutz, Phys Rev A 67, 051402(R) (2003):
(i) The distribution of atomic velocities is a q-Gaussian;
(ii) 0
0
where recoil energy
potential depth
441 RREq
UE
U
= ≡
≡
+
Experimental and computational verificationsby P. Douglas, S. Bergamini and F. Renzoni, Phys Rev Lett 96, 110601 (2006)
(Computational verification:quantum Monte Carlo simulations) (Experimental verification)
0
441 REqU
= +
EARTHQUAKES
n / nw1.05
10-4 10-3 10-2 10-1 100 101 102
D(n
+n w
, n w
)
0.1
1.0 nw=250
nw=1000
nw=500
nw=2000
nw=5000
n / nw
1.050 2 4 6 8 10 12 14 16
lnq[D
(n+
n w , n
w)]
-12
-10
-8
-6
-4
-2
0
S. Abe, U. Tirnakli and P.A. VarotsosEurophysics News 36 (6), 206 (2005) [European Physical Society]
MODEL FOR EARTHQUAKES (OMORI REGIME):
(q=2.98)
U. Tirnakli, in Complexity, Metastability and Nonextensivity, eds. C. Beck, G. Benedek, A. Rapisarda and C. T. (World Scientific, Singapore, 2005), page 350
ASTROPHYSICS
A. Bernui, C. T. and T. Villela, Phys Lett 356, 426 (2006)
C. T. and D.A. Stariolo, Physica A 233, 395 (1996)
q-GENERALIZED PIVOT METHOD:
P. Serra, A.F. Stanton and S. Kais, Phys Rev E 55, 1162 (1997)
(Branin function) (Lennard-Jones clusters)
Genetic algorithm
Present with q=2.7slo
pe 4
.7
slope 2.
9
Num
ber o
f fun
ctio
n ca
lls
Recently: M.A. Moret, P.G. Pascutti, P.M. Bisch, M.S.P. Mundim and K.C. MundimClassical and quantum conformational analysis using Generalized Genetic AlgorithmPhysica A (2006), in press (presumably better than both!)
4 42 2
1 2 3 41 1
( , , , ) ( 8) 5
(15 )
: i ii i
E x x x x x x
local minima and one global minimum
Illustration= =
= − +∑ ∑
q-GENERALIZED SIMULATED ANNEALING (GSA):
( 1 50000)Vq mean convergence time= ⇒ ≈
qthan
6 August ROUND TABLE
(Panelists: Nauenberg, Rapisarda, Robledo, Ruffo)
HMF MODEL: ABOUT THE ZEROTH PRINCIPLE OF THERMODYNAMICS
L.G. Moyano, F. Baldovin and C. T., cond-mat/0305091
1) For u = 5, and d=1, 2, 3 : maximal Lyapunov exponent vanishes like 1 / N^deltawith delta(alpha/d) decreasing from 1/3 to zero for alpha/d increasing from zero to 1, and remaining zero for alpha/d >1;
2) For u = 0.69, at QSS, for d = 1: maximal Lyapunov exponent vanishes like 1 / N^delta' with delta'(alpha/d) decreasing from 1/9 to zero for alpha/d increasing from zero to 1, and remaining zero for alpha/d > 1;
3) For u = 0.69, for alpha = 0 and d=1: T(t) - T(infinity) ~ exp_q ( - t / tau), with qdifferent from unity;
4) For u = 0.69 and d = 1: t_QSS ~ ln_(alpha / d) N for 0 < alpha < 1 ;
5) For u = 0.69, at QSS, for alpha = 0, and d = 1: the marginal probability of the velocityof one particle is not Maxwellian, and its central part decays like exp_q ( - B p^2) forM_0 =1, with q different from unity;
6) For u = 0.69, at QSS, for alpha = 0 and d = 1: The autocorrelation function of velocities presents scalable aging and decays like exp_q [ - A t / (t_W^rho)] with different from unity;
Facts that make think of q-statistics for the HMF and similar models:
7) For u=5, alpha = 0 and d = 1: The autocorrelation function of velocities presents noaging and decays like exp_q [ - A t ] with q different from unity which coincides withthat of point (6) ;
8) For various N, various alpha, various M_0, and d=1: gamma = 2 / (3-q)
9) For u = 0.69, N >>1, and t >>> 1, alpha = 0 and d = 1: marginal probability for theangles of one particle ~ exp_q ( - C theta^2) with q different from unity;
10) For u = 0.69, alpha = 0, d=1 and finite N: the system has long memory as exhibitedby the relevant influence of the initial conditions (dependence on M_0, and dependence on initial condition Catania-type or Rio de Janeiro-type);
11) For u = 0.69, alpha = 0, d=1 and finite N: the system has long memory as exhibitedby the nonvanishing glassy polarization versus N along some time regime;
12) There is a manner of presenting the recent results by Baldovin and Orlandini in thecanonical ensemble (instead of microcanonical) which enables them to beconsistent with any value of q between unity and say 2. This is a consequence ofthe fact that their variation of computational total energy is only of 8 %, and of thefact that, in first order, the q-exponential function does not depend on q;
13) Werner Braun is not sure whether one can take all those derivatives in the Braun-Hepp theorem, in the case 0 < alpha / d < 1. This suggests that his intuition tells him that something quite unusual might occur in such a case.
Forse mi e scapata qualche altra raggione.
Posso qualificare con piu detagli (valori di N, valori di t, condizioniiniziali precise, algoritmo di calcolo in dinamica moleculare, etc) le condizioni sui qualli ogni una di queste raggioni e valida. Ho tutte le referenze alla tua disposizione.
Si come non abbiamo ancora una prova irrefutabile, il mio argomentoe temporariamente che quello che ha il sapore di pizza, odore di pizza, e rotondo come pizza, ha pomodoro come pizza, ha muzarella come pizza, e venduto nelle pizzerie ... probabilmente e pizza!
[Fragmenti della lettera di Constantino a Stefano sul tema]