introduction to nonextensive statistical mechanics a. rodríguez august 26th, 2009 dpto. matemática...

Introduction to Nonextensive

Statistical MechanicsA. Rodríguez

August 26th, 2009

Dpto. Matemática Aplicada y Estadística. UPM

Grupo Interdisciplinar de Sistemas Complejos

EPSRC Symposium Workshop on Quantum Simulations.

Outline General concepts

q-entropy Conexion with Thermodynamics

Some applications XY Model Scale invariant probabilistic model

Summary

q-Entropy

BG1

ln 1/W

i ii

S k p p

1

ln 1/W

q i q ii

S k p p

1 1ln ( )

1

q

q

xx

q

11

1

W qii

q

pS k

q

q-logarithm:

1q

q-exponential: 1

11 (1 )x qqe q x

1q

1q

ln x

xe

; qR

Boltzmann-Gibbs entropy Tsallis entropy

Conexion with Thermodynamics

BG1

ln 1/W

i ii

S k p p

1 1ln ( )

1

q

q

xx

q

11

1

W qii

q

pS k

q

1q

1

11 (1 )x qqe q x

1q

1q

ln x

xe

1

1W

ii

p

Boltzmann’s principle

lnS k W1

ip iW

1

1W

ii

p

lnqS k W1ip i

W


BG1

ln 1/W

i ii

S k p p

11

1

W qii

q

pS k

q

1

1W

ii

p

1

W

i ii

p U

Boltzmann distribution

1

;i

i

W

ii

ep Z e

Z

1

1W

ii

p

1

W

i i qi

P U

1

qi

i W qii

pP

p

( )

( )

1

;q i q

q i q

U WUq

i q qiq

ep Z e

Z


1 ( )

1

q

q

dx p xS k

q

1 ( ) ln ( )S k dx p x p x

Gaussian:2

1( ) xp x e

q-Gaussian:2

12 1( ) 1 (1 )x q

q qp x e q x

1( ) 1dx p x

2 2 211( )x dx x p x

( ) 1qdx p x

2

2 2[ ( )]

[ ( )]

qq

q qq

dx x p xx k

dx p x

q-Gaussians

q = 1

q-Gaussians

q = 0

q-Gaussians

q = -1

q-Gaussiansq

q-Gaussians

q = 2

q-Gaussians

q = 2.9

q-Gaussians 2 5 / 3

5 / 3 3

qx

q

Central Limit Theorem

N random variables: 1 2 NX X X X

a) Independence:2

221

2

( )x

N

N

p x e

X

N �� CLT

b) Global correlations:

XN ��q-CLT 2

( ) xq qp x e

:

: | |1

2( ) cos kL x kx e dk

XN ��

(L-G)-CLT(0 2)




Summary

XY Model

1

1 11

ˆ ˆ ˆ ˆ ˆ(1 ) (1 ) 2N

x x y y zj j j j j

j

H

g=0: XX model. | |g =1: Ising model. 0<| |<1: g XY model.

intensity of Hasymmetry

0, 1cT

0, 1T

F. Carusso and C. Tsallis, Phys. Rev. E 78, 021102 (2008)

XY Model

1

1 11

ˆ ˆ ˆ ˆ ˆ(1 ) (1 ) 2N

x x y y zj j j j j

j

H

1ˆ ˆ ˆ( ) Tr lnS k

ˆ1 Trˆ( )

1

q

qS kq

1ˆ( )LS

entˆ( )q LS L

cte;

ln ;c

cL

; L N

von Neumann Entropy:


Tsallis Entropy:

XY Model

1

1 11

ˆ ˆ ˆ ˆ ˆ(1 ) (1 ) 2N

x x y y zj j j j j

j

H

Ising (g=l=1)


XY Model

Conformal field theory: (P. Calabrese et al, J. Stat. Mech.: Theory Exp. (2004), P06002

Ising and XY : XX :

1ˆ( ) ln

3L

cS L

1

6ˆTr( )c

qqq

L L

central charge

2

ent

9 3cq

c

=

1

1/ 2c

1c ent 0.08q

ent 0.16q





Summary

Scale invariance

x1

p 1-p

1 0

N=1

1 2 1 1 2 1, , , , , ,N N N N Np x x x dx p x x x N distinguisable binary independent variables

1

A. Rodríguez, C. Tsallis and V. Schammle. J. Stat. Mech: theor. exp. P09006 (2008)

Scale invariance

x1

p2 p (1-p)

p (1-p) (1-p)2

1 0

1

0

x2

N=2

N distinguisable binary independent variables


Scale invariance

x1

p2 p (1-p)

p (1-p) (1-p)2

1 0

1

0

x2

p

1-p

p 1-p

N=2

N distinguisable binary independent variables


p3 p2(1-p)

p2(1-p) p(1-p)2

N=3

Scale invariance

p2 p(1-p)

p(1-p) (1-p)2

p2(1-p) p(1-p)2

p(1-p)2 (1-p)3

x3=1

x3=0


p3

p2(1-p) p(1-p)2

p2(1-p)

N=3

Scale invariance

p2 p(1-p)

(1-p)2p(1-p)

(1-p)3

p2(1-p) p(1-p)2

p(1-p)2 1 p 1-p

N=0

N=1 N=2


,N nr

p3 p2(1-p) p(1-p)2N=3

Scale invariance

p2 p(1-p) (1-p)2

(1-p)3

1 p 1-p

N=0

N=1 N=2

,N nr

+

+

+

, , 1 1,N n N n N nr r r Leibniz rule


p3 p2(1-p) p(1-p)2N=3

Scale invariance

p2 p(1-p) (1-p)2

(1-p)3

1 p 1-p

N=0

N=1 N=2

,N nr,N n

N

np

111

2 11

3 31 1

Pascal triangle

CLT

N

2

1

2 (1- )

2 (1 )Np p

x Np

Np pe


p3 p2(1-p) p(1-p)2N=3

Invariant triangles

p2 p(1-p) (1-p)2

(1-p)3

1 p 1-p

N=0

N=1 N=2

,N nr

0,0r

1,0r 1,1r

2,0r 2,1r 2,2r

3,0r3,1r 3,2r

3,3r


N=3

N=0

N=1 N=2

0,0r

1,0r 1,1r

2,0r 2,1r 2,2r

3,0r3,1r 3,2r

3,3r

1

1

121

12

1

4

1

4

1

3

1

31

6

1

2

1

2

(1),0

1

1Nr N

Invariant triangles

Leibniz triangle

(1),N nr


N=3

N=0

N=1 N=2

1

1

121

12

1

4

1

4

1

3

1

31

6

1

2

1

2

(1),

1

1N nrN

Invariant triangles

Leibniz triangle

(1),N n

Np

n

N

2xe

1

3 1

41

4


N=3

N=0

N=1 N=2

1

1

121

12

1

4

1

4

1

3

1

31

6

1

2

(1),0

1

1Nr N

Invariant triangles

Leibniz triangle

1

2


N=3

N=0

N=1 N=2

1

1

10

1

51

5

3

10

3

10

1

5

1

2

(2),0

2 3

( 2)( 3)Nr N N

1

10

2

73

14

2

7 5

285

28

3

283

28

(3),0

3 4 5

( 3)( 4)( 5)Nr N N N

Invariant triangles

1

2


N=3

N=0

N=1 N=2

11

2

1

22

73

14

2

7 5

285

28

3

283

28

(3),0

3 4 5

( 3)( 4)( 5)Nr N N N

( )

,0

(2 1)

( ) ( 2 1)Nr N N

Invariant triangles


( ),0

(2 1)

( ) ( 2 1)Nr N N

( ),N nr ( )

,N n

Np

n

Invariant triangles

2 NN

n

1

2 1i j

0


( ),N nr ( )

,N n

Np

n

N

2

att

xqe

att

2

1q

att 0q att 1/ 2q att 3/ 4q

att 8/ 9q

att 1q 500N

Invariant triangles


Invariant triangles

,01

( )1

N qN nn

q

pS N k

q

ent 1q




Summary

Summary Nonextensive Statistical Mechanics allows to address

non-equilibrium stationary states (in Physics, Biology, Economics…) with concepts and methods similar to those of the BG Statistical Mechanics.

The entropy (bridge between mechanical microscopic laws and classical thermodynamics) may adopt different expressions depending on the system: or others.

The value of q is determined by the microscopic dynamics of the system, which is frequently unknown, so it generally cannot be predicted by first principles.

It is a currently developing theory, with many open questions as the reslationship between scale invariance, extensivity and q-Gaussianity.

BG , ,qS S

introduction to nonextensive statistical mechanics a. rodríguez august 26th, 2009 dpto. matemática...

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