Top Banner
������
153

NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

Aug 25, 2018

Download

Documents

phamngoc
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Page 1: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

��������� �

������ ��� ������������

������� ����������

������������ ����������� ���������

�� ���� � � ������ ����

������������ ����������� ���������������������� ������������� ������������� ���

������������� �������

�� �������

������ ������������ ��������� �������� ����������� �� ���������� ��� ������� ���

Page 2: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

Constantino Tsallis

Centro Brasileiro de Pesquisas Fisicas, BRAZIL

Santa Fe Institute, New Mexico, USA

C. T., M. Gell-Mann and Y. Sato, Proc Natl Acad Sci (USA) 102, 15377 (2005)

L.G. Moyano, C. T. and M. Gell-Mann, Europhys Lett 73, 813 (2006)

S. Umarov, C.T., M. Gell-Mann and S. Steinberg, cond-mat/0603593, 0606038, 0606040

W. Thistleton, J.A. Marsh, K. Nelson, L.G. Moyano and C. T. (2006), in progressTrieste, July 2006

NONEXTENSIVE STATISTICAL MECHANICS: THEORETICAL, EXPERIMENTAL, OBSERVATIONAL

AND COMPUTATIONAL ASPECTS

Page 3: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS

Nonextensive Statistical Mechanics and Thermo-dynamicsSRA Salinas and C Tsallis, edsBrazilian Journal of Physics

Nonextensive Sta

29, Number 1 (1999)

tistical

Mechanics and Its ApplicationsS Abe and Y Okamoto, edsLectures Notes in Physics

Non Extensive Thermodynamics and Physical ApplicationsG Kaniadakis, M Lissia and A Rapisarda,

(Springer, Berlin, 2001) eds

Physica A 305, Issue 1/2 (2002)

Classical and Quantum Complexity and Nonextensive ThermodynamicsP Grigolini, C Tsallis and BJ West, edsChaos, Solitons and Fractals 13, Issue 3 (2002)

Nonadditive Entropy and Nonextensive Statistical Mechanics M. Sugiyama, ed Continuum Mechanics and Thermodynamics 16 (Springer, Heidelberg, 2004)

Nonextensive Entropy -Interdisciplinary ApplicationsM Gell-Mann and C Tsallis, eds(Oxford University Press, New York, 2004)

Anomalous Distributions, Nonlinear Dynamics, and NonextensivityHL Swinney and C Tsallis, edsPhysica D 193, Issue 1-4 (2004)

News and Expectations in ThermostatisticsG Kaniadakis and M Lissia, edsPhysica A 340, Issue 1/3 (2004)

Trends and Perspectives in Extensive and Non-Extensive Statistical MechanicsH Herrmann, M Barbosaand E Curado, edsPhysica A 344, Issue 3/4 (2004)

Complexity, Metastability and NonextensivityC Beck, G Benedek, A Rapisarda and C Tsallis, eds(World Scientific, Singapore, 2005)

Nonextensive Statistical Mechanics: New Trends, New Perspectives JP Boon and C Tsallis, eds EurophysicsNews (European Physical Society, 2005)

Fundamental Problems of Modern Statistical Mechanics G Kaniadakis, A Carbone and M. Lissia, eds Physica A 365, Issue 1 (2006)

Page 4: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

Recent minireviews(Europhysics News, Nov-Dec 2005, European Physical Society)

http://www.europhysicsnews.com

Full bibliography (29 July 2006: 1935 manuscripts)

http://tsallis.cat.cbpf.br/biblio.htm

Page 5: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

UBIQUITOUS LAWS IN COMPLEX SYSTEMS

ORDINARY DIFFERENTIAL EQUATIONS

ENTROPY Sq (Nonextensive statistical mechanics)

PARTIAL DIFFERENTIAL EQUATIONS (Fokker-Planck, fractional derivatives, nonlinear, anomalous diffusion, Arrhenius)

STOCHASTIC DIFFERENTIAL EQUATIONS (Langevin, multiplicative noise)

NONLINEAR DYNAMICS (Chaos, intermittency, entropy production, Pesin, quantum chaos, self-organized criticality)

CENTRAL LIMIT THEOREMS (Gauss, Levy-Gnedenko)

q-ALGEBRA

CORRELATIONS IN PHASE SPACE

GEOMETRY (Scale-free networks)

LONG-RANGE INTERACTIONS (Hamiltonians, coupled maps)

SIGNAL PROCESSING (ARCH, GARCH)

IMAGE PROCESSING

GLOBAL OPTIMIZATION (Simulated annealing)

q-TRIPLETTHERMODYNAMICS

FURTHER APPLICATIONS (Physics, Astrophysics, Geophysics, Economics, Biology, Chemistry, Cognitive psychology, Engineering, Computer sciences, Quantum information, Medicine, Linguistics …)

AGING (metastability)

SUPERSTATISTICS (Other generalizations)

Page 6: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

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

= =

Page 7: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

ENTROPIC FORMS

Page 8: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

C.T., M. Gell-Mann and Y. Sato Europhysics News 36 (6), 186 (2005) [European Physical Society]

Page 9: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

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

Page 10: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive
Page 11: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

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.

Page 12: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

Laszlo TISZA

Generalized Thermodynamics(MIT Press, Cambridge, Massachusetts, 1961)

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.

Page 13: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

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.

Page 14: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS 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.

Page 15: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

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.]

Page 16: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

//

: -

:

, ( )( )

(

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

Page 17: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive
Page 18: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

(in the sense of x = y symmetry)

Page 19: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive
Page 20: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive
Page 21: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive
Page 22: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive
Page 23: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive
Page 24: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive
Page 25: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive
Page 26: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

CONCAVITY:

then

Page 27: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive
Page 28: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive
Page 29: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

q=0.5 (QC)

q=2 (QEP)

q=0.5 (QEP)

q=2 (QC)

Page 30: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

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

Page 31: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

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

=

=

−⎛ ⎞= = ⇒ = −⎜ ⎟− ⎝ ⎠

∑∑

Page 32: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

( , ) qS N t versus t

Page 33: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

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)

Page 34: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

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

ΔΔΔ

λ

ξ

→∞

=

−=

Page 35: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

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

Page 36: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

(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

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+

= =

⎡ ⎤= − ⇒ = − = −⎢ ⎥−⎣ ⎦

Page 37: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive
Page 38: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

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)

Page 39: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

[G. Casati, C.T. and F. Baldovin, Europhys Lett 72, 355 (2005)]

Page 40: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

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)

Page 41: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

( , ) qS N t versus N

Page 42: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

( 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)

Page 43: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

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:

Page 44: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

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)

Page 45: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

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)

Page 46: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

. .,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)

Page 47: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

Continental Airlines

Page 48: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

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]

Page 49: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

, . ,

) , ( , )

( , ):

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)

Page 50: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

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

Page 51: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

1( ) ( )

(

( ) ( ) ( )

( )

,

. ., ,

) ( ) ( )

( ) ( 1)

(

q q q q

BG B

q

A B A Bij i

G BG

q

j

B

q

independent

qS A B S A S

If A and B are

i e if p p

B S A S Bk

S A S B if

p

then

whereas

But i

q

especiallyf A and glB ar

S A B A S

e

S B

+

−+ = + +

≠ +

=

+ =

+

( ) ( )

)

( )

,

( ) ( )

( )q q q

BG BG BG

then

wher

obally correlated

S A B S A S Bea

A S Bs

S B A S+

+ ≠ +

= +

Page 52: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive
Page 53: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

q - CENTRAL LIMIT THEOREM:

2 ( , ) [ ( , )] (0 2; 3)| |

qp x t p x tD qt x

γ

γ γ−∂ ∂

= < ≤ <∂ ∂

globally correlated variables;finite q-variance; q-Gaussian attractor

C.T., Milan J. Math. 73, 145 (2005)

independent variables; divergent variance; Levy attractor

independent variables; finite variance; Gaussian attractor

q - CENTRAL LIMIT THEOREM (conjecture)

Page 54: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

D. Prato and C. T, Phys Rev E 60, 2398 (1999)

q-GAUSSIANS:

Page 55: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

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

= =

Page 56: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

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)

Page 57: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

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⎛ ⎞⎜ ⎟≠ ≠⎝ ⎠

Page 58: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

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!)±

Page 59: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

,

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:

Page 60: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

2 2(1 )

1

1

1

1

22

1 3

3 8

121

13(3 ) (1 )

2(1 )

qqq

q

q

tq qq F

qwhere qq

qandC

qif q

ourierTransform

qq qq

with C

Ce e βββ ω

ββ

πΓ

Γ

π

− −

−−⎡ ⎤=⎢ ⎥

⎢ ⎥⎣ ⎦+

=−

−=

⎛ ⎞⎜ ⎟−⎝ ⎠ <

⎛ ⎞−− − ⎜ ⎟−⎝ ⎠

=

1

32( 1)

1 311

1

if q

qq

if qq

q

πΓ

Γ

⎧ ⎫⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪=⎨ ⎬⎪ ⎪⎛ ⎞−⎪ ⎪⎜ ⎟⎪ ⎪−⎝ ⎠ < <⎪ ⎪⎛ ⎞⎪ ⎪− ⎜ ⎟⎪ ⎪−⎝ ⎠⎩ ⎭

Page 61: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive
Page 62: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive
Page 63: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive
Page 64: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

, | |

- ( 0, 0, 0 2)

( , )

-

( )

q

bq

A random variable X is said to have a

if its q Fourier transform has the

q stable dist

form a

ribution

b

L x

a e αα

ξ

α

α− > > < ≤

S. Umarov, C. T., M. Gell-Mann and S. Steinberg (2006)

cond-mat/0606038

cond-mat/0606040

1[ ( )],, , ,

1,2

1,

,2

| |

. .,

( ) = ( ) =

)

)

[ ](

)

( ) ( ) ( )

( ) ( ) (

( ) ( ) (

qix

L xqq

xqqi

q q q q

qq

bq

stable Levy distribution

q Gaus

i e if

L x dx L x dx

s n

L

a

F

i

L x G x Gaussian

L x L x

L x G x

e e a eξ

αξα αα

α

αα

ξξ

α

−∞ ∞

−∞ −∞

⊗−

∫ ∫

Page 65: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

1 [ ]q independent= 1 ( . ., 2 1 1) [ ]q i e Q q globally correlated≠ ≡ − ≠

1

, ( )

( ) (

Classic CL

)

T

with same ofx Gaussian G x

f xσ=F

<

( 2)Qσ

α

=

(0 2)

Q

α

σ< <

→∞

2

1

, | |

( )

| | (1, )L ( )

( ) / | | | | (1, )

lim (

( ) L

1,

)

)

(

c

c

c

with same xasymptotic behavior

x Levy di

G xif x x

xf x C x

i

stri

f x xwit

b on x

h

u

x

ti

α

α

α

α αα

αα→

+

→∞

⎧ ⎫⎪ ⎪<<⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪>>⎩ ⎭

=

=

∼ ∼

F

Levy-Gnedenko CLT

1/[ (2- )] -CENTRAL LI MIT 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

+ −

+ − −∼

Page 66: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

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

Page 67: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

'

'

' '

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∂

= −∂

∂ ∂ ∂≡ = = −

∂ ∂ ∂

Page 68: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive
Page 69: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive
Page 70: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

ON THE NATURE OF q-CORRELATIONS:

W. Thistleton, J.A. Marsh, K. Nelson, L.G. Moyano and C. T. (2006)

⎥⎥⎥⎥⎥⎥

⎢⎢⎢⎢⎢⎢

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

− ≤ ≤

= ⇒= ⇒

Page 71: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

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)

Page 72: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

-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= =

Page 73: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

-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= =

Page 74: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

-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)

Page 75: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

-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)

Page 76: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

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

−=

Page 77: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

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

α

1 (2) (3) ... ( -1) ( )(2) 1 (2) ... ( - 2) ( -1)(3)

N N covariance matrix of the Normal distributionsgiven by

N NN N

ρ ρ ρ ρρ ρ ρ ρρ

×

(2) 1 ... ( -3) ( - 2) ... ... ... ... ... ...

( -1) ( - 2) ... (2) 1 (2)( )

N N

N NN

ρ ρ ρ

ρ ρ ρ ρρ ( -1) ... (3) (2) 1

( 1 1; 0; 2, 3,..., )

( )

N

with

r N

rrα

ρ ρ ρ

ρ α

ρρ

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

− ≤ ≤ ≥ =

=

q

W. Thistleton, J.A. Marsh, K. Nelson, L.G. Moyano and C. T. (2006)

INFLUENCE OF THE RANGE OF CORRELATIONS DECAYING FAR FROM THE DIAGONAL OF THE COVARIANCE MATRIX:

(N=35 )

513

1q

ρ

ρ

Page 78: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

Connections with Hamiltonianand more complex systems

Page 79: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

0 1 2 3 4 50

1

2

3

4

5

α

d

EXTENSIVE

NONEXTENSIVE

dipole-dipole

Newtonian gravitation

dipole-monopole (ti

des)

d-dimensional g

ravitatio

n

( ) ( )

( 0, 0)

AV r rr

A

α

α

− → ∞

> ≥

Page 80: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

BOLTZMANN-GIBBS STATISTICAL MECHANICS(Maxwell 1860, Boltzmann 1872, Gibbs ≤ 1902)

Entropy

Internal energy

Equilibriumdistribution

Paradigmaticdifferential equation

1

W

B G i ii

U p E=

= ∑1

lnW

B G i ii

S k p p=

= − ∑

/iEi BGp e Zβ−=

1

jW

EBG

jZ e β−

=

⎛ ⎞≡⎜ ⎟

⎝ ⎠∑

( 0 ) 1

d y a yd xy

⎫= ⎪ ⇒⎬⎪= ⎭

x a y(x)Equilibrium distribution Ei -β Z p(Ei)Sensitivity toinitial conditions t λTypical relaxation of observable Ο t -1/τ

SBG → extensive, concave, Lesche-stable, finite entropy production

( 0 ) 0

( )lim(0 )

t

x

x t ex

λξΔ →

Δ≡ =

Δ

/( ) ( )(0) ( )

tO t O eO O

τ−− ∞Ω ≡ =

− ∞

axy e=

Page 81: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

NONEXTENSIVE STATISTICAL MECHANICS(C. T. 1988, E.M.F. Curado and C. T. 1991, C. T., R.S. Mendes and A.R. Plastino 1998)

Entropy

Internal energy

Stationary statedistribution

Paradigmaticdifferential equation

x a y(x)Stationary state distribution

Ei

Sensitivity toinitial conditions tTypical relaxation of observable Ο t

11 /( 1)

Wq

q ii

S k p q=

⎛ ⎞= − −⎜ ⎟

⎝ ⎠∑

( ) /q i qE Ui q qp e Zβ− −=

( )

1

E Uq j qW

q qj

Z eβ− −

=

⎛ ⎞≡⎜ ⎟

⎝ ⎠∑

(0) 1

qdy a ydxy

⎫⎪⎬⎪⎭

=⇒

=

Sq → extensive, concave, Lesche-stable, finite entropy production

qsen

sen

tqeλξ =

/ qrel

rel

tqe τ−

Ω =

[ ]1

11 (1 ) qq

a xq a xy e −+ −= ≡

1 1

/W W

q qq i i j

i j

U p E p= =

= ∑ ∑

(typically 1)senq ≤

1 /r e lqτ−

senqλ

(typically 1)relq ≥

statqβ− ( )statq iZ p E

C. T., Physica A 340,1 (2004)

(typically 1)statq ≥

Page 82: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

Prediction of the Prediction of the q q -- triplet:triplet: C. T., Physica A 340,1 (2004)

Page 83: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

L.F. Burlaga and A. F.-Vinas (2005) / NASA Goddard Space Flight Center; Physica A 356, 375 (2005)

[Data: Voyager 1 spacecraft (1989 and 2002); 40 and 85 AU; daily averages]

SOLAR WIND: Magnetic Field Strength

0.6 0.2senq = − ±

3.8 0.3relq = ± 1.75 0.06statq = ±

Page 84: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

( 2 ) ( 1/ )

1 2

(

)

relsen

Playing with additive dualityand with multiplicative dualityand using numerical results related to the q generalized central limit theorem

we conject

q qq q

qq

ure

→ −→

+

=

!

1 2

1 1 3 2

( )

statrel

statsen

stat

stat

and

hence

Burlaga and Vinas NASA most precise value of the q tripl

hence onl

et

y one independe

qq

qqq

t

is

n

q−

+ =

−− =

( 0.6 0.2 !)(

1.75 7 / 4 0.5 1/ 2

4 3.8 0.3 ! ) sen

re

sen

rel l

hencea

consistent with qconsistent withnd

qq q

= − ±= ±

= == − = −=

C.T., M. Gell-Mann and Y. Sato Proc Natl Acad Sc USA 102, 15377 (2005)

Page 85: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

Connections withasymptotically scale free networks−

Page 86: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

(1) Locate site i=1 at the origin of say a plane

(2) Then locate the next site with

(3) Then link it to only one of the previous sites using

2

( )

1/ ( 0) GG G

r distance to the baricenter of the pre existing cluster

P r α α+

≡ −

∝ ≥

4) Repeat

A

( )

( )

/ ( 0) Ai i A

i

i

k links already attached to site i

r distance to site i

k rp α α≡

∝ ≥

GEOGRAPHIC PREFERENTIAL ATTACHMENT GROWING NETWORK:

THE NATAL MODELD.J.B. Soares, C. T. , A.M. Mariz and L.R. Silva, Europhys Lett 70, 70 (2005)

Page 87: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

G

( 1; 1; 250)A Nα α= = =

D.J.B. Soares, C. T. , A.M. Mariz and L.R. Silva Europhys Lett 70, 70 (2005)

Page 88: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive
Page 89: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

/

1 / ( 1 )

P ( k ) /P (0 )=

1 / [1 ( 1) / ]

kq

q

e

q k

κ

κ

−≡ + −D.J.B. Soares, C. T. , A.M. Mariz and L.R. Silva Europhys Lett 70, 70 (2005)

Page 90: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive
Page 91: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

0.526

q=1+(1/3) ( ) A

G

e α

α

Barabasi-Albert universality class

D.J.B. Soares, C. T. , A.M. Mariz and L.R. Silva Europhys Lett 70, 70 (2005)

= 0 .0 8 3 + 0 .0 9 2 Aκ α

( )Gα∀

Page 92: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

( 0)

( )

1

ij

ijij

Merging probability

d shortest path chemical distance connecting nodes i and j on the network

pd α α ≥

(Kim, Trusina, Minnhagen and Sneppen, . . . 43 (2005) 369)

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= = =

Page 93: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

2000 4000 6000 8000 10000 12000 140000

10

20

30

40

50

60

70

80

time

k max

; k i

ki

kmax

Page 94: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

S. Thurner, Europhys News 36, 218 (2005)

Page 95: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

[ ] [ ]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)

Page 96: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

- ( -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)

Page 97: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

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)

Page 98: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

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)

Page 99: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

Hydra viridissima: A. Upadhyaya, J.-P. Rieu, J.A. Glazier and Y. Sawada Physica A 293, 549 (2001)

q=1.5

Page 100: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

1.24 0.12

3

slope

hence is satisfiedq

γ

γ

= ±

=−

Page 101: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

Defect turbulence:K.E. Daniels, C. Beck and E. Bodenschatz, Physica D 193, 208 (2004)

Page 102: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

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≈ ≈ =−

Page 103: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

XY FERROMAGNET WITH LONG-RANGE INTERACTIONS:

A. Rapisarda and A. Pluchino, Europhys News 36, 202 (2005) (European Physical Society)

Page 104: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

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

Page 105: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

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

= ≡

+

Page 106: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

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

= +

Page 107: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

EARTHQUAKES

Page 108: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive
Page 109: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive
Page 110: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive
Page 111: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive
Page 112: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive
Page 113: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive
Page 114: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive
Page 115: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

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)

Page 116: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive
Page 117: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

U. Tirnakli, in Complexity, Metastability and Nonextensivity, eds. C. Beck, G. Benedek, A. Rapisarda and C. T. (World Scientific, Singapore, 2005), page 350

Page 118: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

ASTROPHYSICS

Page 119: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive
Page 120: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive
Page 121: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

A. Bernui, C. T. and T. Villela, Phys Lett 356, 426 (2006)

(Band Q: 22.8 GHz) (Band V: 60.8 GHz) (Band W: 93.5 GHz)

1.045 0.005 (99 % confidence level)q = ±(Data after using Kp0 mask)

Page 122: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

A. Bernui, C. T. and T. Villela, Phys Lett 356, 426 (2006)

1.045 0.005 (99 % confidence level)q = ±

Page 123: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

ECONOMICS

Page 124: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive
Page 125: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

J de Souza, SD Queiros and LG Moyano, physics/0510112 (2005)

STOCK VOLUMES:

Page 126: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

q-GENERALIZED BLACK-SCHOLES EQUATION:L Borland, Phys Rev Lett 89, 098701 (2002), and Quantitative Finance 2, 415 (2002)L Borland and J-P Bouchaud, cond-mat/0403022 (2004)L Borland, Europhys News 36, 228 (2005) See also H Sakaguchi, J Phys Soc Jpn 70, 3247 (2001)

C Anteneodo and CT, J Math Phys 44, 5194 (2003)

31

[ :

] nn

REMARK Student t distributions are the particular case

of q Gauss q with n intians whe rn ege++

=

Page 127: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

LONDON STOCK EXCHANGE (Block market):

Data: I.I. Zovko; Fitting: E.P. Borges (2005)

VODAPHONE stocks (31 May 2000 to 31 December 2002)

Daily net exchange of shares (between all pairs of two institutions)

4

6'

3.28 ; 1.1 10

' 1.45 ; 1.1 10q

q

q

q

β

β

Cum

ulat

ive

dist

ribut

ion

= = ×

= = ×

Page 128: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive
Page 129: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

FINGERING

Page 130: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

P. Grosfils and J.P. Boon, 2005

Page 131: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

P. Grosfils and J.P. Boon, 2005

Page 132: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

P. Grosfils and J.P. Boon, 2005

Page 133: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

GENERALIZED SIMULATED ANNEALING AND RELATED ALGORITHMS

Page 134: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

HYBRID LEARNING OF NEURAL NETWORKS

A.D. Anastasiadis and G.D. Magoulas, Physica A 344, 372 (2004)

(Hybrid Learning Scheme = HLS; q > 1)

HLS Rprop

Page 135: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

Rprop

Sarprop

HLS

Page 136: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

Anastasiadis and Magoulas (2004)

Page 137: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

A. Plastino and O.A. RossoEurophysics News 36 (6), 224 (2005) [European Physical Society]

ELECTROENCEPHALOGRAMS (tonic-clonic transition in epilepsy):

starts

ends

TONIC CLONIC

(INC

LUD

ING

MU

SCU

LAR

AC

TIVI

TY C

ON

TRIB

UTI

ON

)

Page 138: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

IMAGE THRESHOLDING:

M.P. de Albuquerque, I.A. Esquef, A.R.G. Mello and M.P. de Albuquerque Pattern Recognition Letters 25, 1059 (2004)

Page 139: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive
Page 140: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

IMAGE EDGE DETECTION [A. Ben Hanza, J. Electronic Imaging 15, 013011 (2006)]

Original image

q = 1.5

q = 1

(Jensen-Shannon)

Canny edge detector

Page 141: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

IMAGE EDGE DETECTION [A. Ben Hanza, J. Electronic Imaging 15, 013011 (2006)]

Original image

q = 1.5

Canny edge detector

q = 1

(Jensen-Shannon)

Page 142: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

IMAGE EDGE DETECTION [A. Ben Hanza, J. Electronic Imaging 15, 013011 (2006)]

Original image

Canny edge detector

q = 1.5

q = 1

(Jensen-Shannon)

Page 143: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

IMAGE EDGE DETECTION [A. Ben Hanza, J. Electronic Imaging 15, 013011 (2006)]

Original image

Canny edge detector

q = 1.5

q = 1

(Jensen-Shannon)

Page 144: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

q-GENERALIZED SIMULATED ANNEALING (GSA):

C.T. and D.A. Stariolo, Notas de Fisica / CBPF (1994); Physica A 233, 395 (1996)

:

:

VGeneralized machine q GaussBoltzmann machine Gaussian

Boltzmann machine Boltzmann

Visiting algo

weight

rithm

Acceptance aian

Generalized mac

lgorithm

hi

→ −→

1

1

T(t) ln 2 T(1) ln(1 )

T(t) 2 1 T(1)

[ :

(1 ) 1

1 3

:

] 1

A

V A

V

V

q

q

ne q exponential weight

Gen

Bo

er

ltz

alized machi

mann machinet

Typ

Cool

ica

net

ing algori

l values and q

thm

q

→ =

→ −

−→ =

+ −

< < <

+

Page 145: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

C. T. and D.A. Stariolo, Physica A 233, 395 (1996)

Page 146: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

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!)

Page 147: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

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= ⇒ ≈

Page 148: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

qthan

Page 149: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

6 August ROUND TABLE

(Panelists: Nauenberg, Rapisarda, Robledo, Ruffo)

Page 150: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

HMF MODEL: ABOUT THE ZEROTH PRINCIPLE OF THERMODYNAMICS

L.G. Moyano, F. Baldovin and C. T., cond-mat/0305091

Page 151: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

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:

Page 152: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

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.

Page 153: NONEXTENSIVE STATISTICAL MECHANICS: …indico.ictp.it/event/a05215/session/2/contribution/1/material/0/3.pdf · NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS Nonextensive

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]