Towards a large deviation theory for statistical mechanical complex systems

Towards a large deviation theory

for statistical mechanical

complex systemsG. Ruiz LópezG. Ruiz López1,21,2, C. Tsallis, C. Tsallis1,31,3

11Centro Brasileiro de Pesquisas Fisicas. BrazilCentro Brasileiro de Pesquisas Fisicas. Brazil22Universidad Politécnica de Madrid. Spain.Universidad Politécnica de Madrid. Spain.

33Santa Fe Institute, USASanta Fe Institute, USA

Towards a large deviation theory

for statistical mechanical

complex systemsG. Ruiz LópezG. Ruiz López1,21,2, C. Tsallis, C. Tsallis1,31,3

11Centro Brasileiro de Pesquisas Fisicas. BrazilCentro Brasileiro de Pesquisas Fisicas. Brazil22Universidad Politécnica de Madrid. Spain.Universidad Politécnica de Madrid. Spain.

33Santa Fe Institute, USASanta Fe Institute, USA

Large deviation theory and Statistical Large deviation theory and Statistical MechanicsMechanics

Rare events:

Tails of probability distributions

Rates of convergence to equilibrium

BG: lies on LDT NEXT: ¿ q-LDT ?

Large deviation theory and Statistical Large deviation theory and Statistical MechanicsMechanics

G. Ruiz & C. Tsallis, Phys. Lett .A 376 (2012) 2451-2454.

G. Ruiz & C. Tsallis, Phys. Lett. A 377 (2013) 491-495.

Physical scenario of a possible LDT Physical scenario of a possible LDT generalizationgeneralization

a) Standard many-body Hamiltonian system in thermal equilibrium a) Standard many-body Hamiltonian system in thermal equilibrium ((TT))

BG weight:BG weight: (short-range + ergodic = (short-range + ergodic =

extensiveextensive energy) energy)

LDT probability:LDT probability:

(( BG relative entropy BG relative entropy per per particleparticle))

1, NN

B

H N NH k Te e

1 1 rate function( ) , r NP N e r

1 r

1 ( ) r NP N e

-rate function( ) , qr N

q q qP N e r LDT probability: LDT probability:

( ) 1V r rb) b) dd-dimensional classical system: 2-body interactions-dimensional classical system: 2-body interactions Large ranged ( )Large ranged ( )

( ( intensiveintensive variable) variable)

, ln, N N dN N N N N NH H NH H H N N

1

1 1

1

11

ln ln ln1

1 (1 ) ; q

z z

qq

z zq

ze e z z z

qq ze e

0 1:d

NH N

Outcomes: : 22 (each toss) (each toss) 22NN ((NN tosses) tosses)

Number of heads, n:

Containing Containing nn heads: heads:

Probability of Probability of nn heads: heads:

LDT standard results: LDT standard results: NN uncorrelated coinsuncorrelated coins

1

0, 1, ...

N

n n n N

,

!

!( )!N n

N NC

n n N n

,, 2

N nN n N

Cp

Weak Law of large numbers:Weak Law of large numbers:

Rate at which limit is attained:Rate at which limit is attained:

Large Deviation Principle (Large Deviation Principle (rr1 1 : rate function): rate function)

1 1lim ; 0 0 lim ; 0

2 2N N

n nP N P N x x

N N

1

1lim ln ; ln (1 ) ln(1 ) ln 2 (x) N

nP N x x x x x r

N N

1 ( ); Nr xnP N x e

N

(0 1/ 2)x

Average number of heads per toss in a range:

,

: :

1; 0 1/ 2

2N n Nn n

n x n xN N

NnP N x p x

nN

Outcomes: : 2 (each toss)2 (each toss)22N N (N tosses)(N tosses)

Number of heads

Containing Containing nn heads: heads:

Probability of Probability of nn heads: heads:

1

0,1,...N

N

,

!

!( )!N n

N NC

n n N n

,, 2

N nN n N

CP

Weak Law of large numbers:Weak Law of large numbers:

Rate at which limit is attained:Rate at which limit is attained:

Large Deviation Principle (Large Deviation Principle (rr1 1 : rate function): rate function)

1 1lim ; 0 0 lim ; 0

2 2N N

n nP N P N x x

N N

1

1lim ln ; ln (1 ) ln(1 ) ln 2 (x) N

nP N x x x x x r

N N

1 ( ); Nr xnP N x e

N

(0 1/ 2)x

Average number of heads per toss in a range:

,

: :

1; 0 1/ 2

2N n Nn n

n x n xN N

NnP N x p x

nN

LDT standard results: LDT standard results: NN uncorrelated coinsuncorrelated coins

a) Independent random variables2

2

( )

21

2

( )x

p x e

N ��Standard CLT

:

Rate function and relative entropyRate function and relative entropy

( )p n

(0)

11

lnW

ii

i i

pI p

p

11

1ln ln

W

ii i

I W pp

11 ln

B

SI W

k 1

1

lnW

B i ii

S k p p

Relative entropy:(0) 1

ipW

N uncorrelated coins (W=2, p1=x, p2=1-x): 1 1(x)= ln 2 ln (1 ) ln(1 ) (x)I x x x x r

q-Generalized relative entropy:(0) 1

ipW(0)

1

lnW

iq i q

i i

pI p

p

1 1

1ln

1

Wqi

q iq

pW W

q

1 ln qqq q

B

SI W W

k

1

1

1

Wqi

iq B

pS k

q

C. Tsallis, Phys. Rev. E 58 (1998) 1442-1445.

:q b) Strongly correlated random variables

N ��q-CLT 2( )( ) x

q qp x Ae ( )p n

S.Umarov, C. Tsallis, S. Steinberg, Milan J. Math. 76 (2008) 307.S. Umarov, C. Tsallis, M. Gell-Mann, S. Steinberg, J. Math. Phys. 51 (2010) 033502.

Non-BG: Non-BG: NN strongly correlated coins strongly correlated coins

2

2( )y

q

yq

ep y

e dy

,( ) N nN

NP y p

(0 1)

N

��

Histograms:

A. Rodriguez, V. Schwammle, C. Tsallis, J. Stat. Mech (2008)P09006.

Discretization:

Suport:

,,

,0

2 1 1

0 2( )

( )

( ) [1 ( 1) ]

Q N nN n N

Q N nn

QQ

Qp y

pp y

p z Q z

( 1) ( 0;0 1)N N

,

1,

2 2 2N N

N n N

ny

N

( 0,..., )n N

Average number of heads per toss :

:

Large deviations in Large deviations in (Q, (Q, )-)-modelmodel

,,

: : ,0

( );

( )

Q N nN n N

n nn x n x Q N nN N

n

p ynP N x p

N p y

(0 1/ 2)x

( , ), Q

Large deviations in Large deviations in (Q, (Q, )-)-modelmodelAverage number of heads per toss :

:

,,

: : ,0

( );

( )

Q N nN n N

n nn x n x Q N nN N

n

p ynP N x p

N p y

(0 1/ 2)x

( , ), Q

11 (0 1/ 2; )

(3 )

Qq x

Q

00 1/ 2

12x

x

qq

1 1:Q q

q 1

1( ) ( )

; ;uncorr

QNr x Nr x

q

n n

N NN x N xPP e e

1 1:Q q

1/( 1); 1/ qnP N x N

N

Large Deviation Principle in Large Deviation Principle in (Q, (Q, )-)-modelmodel


:

,,

: : ,0

( );

( )

Q N nN n N

n nn x n x Q N nN N

n

p ynP N x p

N p y

(0 1/ 2)x

( , ), Q

( )(0 1 / 2)

( , ) :

; qN r x

q x

q q Q

nP N x e

N

Generalized q-rate function:

•

•

• What about q-generalized relative entropy?

1/

1/N

( ; ; ; ) ( ; ; ;1) ( , ), 0

q q

N

r x Q r x Q Q

N

0(0) lim ( ) q q

xr r x

( 2)1 11

ln [1 2 [ (1 ) ]]1

Wqq q q q

q qB

SI W W x x

k q

( ) ( ) q qr x I x


11 (0 1/ 2; )

(3 )

Qq x

Q

00 1/ 2

12x

x

qq


:

,,

: : ,0

( );

( )

Q N nN n N

n nn x n x Q N nN N

n

p ynP N x p

N p y

(0 1/ 2)x

( , ), Q

( )(0 1 / 2)

( , ) :

; qN r x

q x

q q Q

nP N x e

N

;( ) ( ) 1 (3 )

1 ( ) 0; ( ) 0 , were 01 1

n

NN x

B x C x QP B x C x

N N q Q

; / ( ) 0 / lim 1 ( )

( )N

N n N xP Nx B x N C x

B x


max1,2,..., ( )

! ( ) ( , , )

j j j x

B x xQ

C

maxj ( 0.1) =10max

j ( 0.15) = 20

4.57660794B

Asymptotic numerical behavior

max1,..., ( )

;( )( ) 1 (3 )

1 ! ( ) 0; 0 where 01 1

jj j j x

n

NN x

C xB x QP B x C

N N q Q

Numericaly known calculation ( )( ) , ( )jj qa x r x

( )

max

( )

1,...,( )

jqr x N

j qj j

a x e


Asymptotic expansion of q-exponential :

( , , ) (3 / 2,1/ 2,1)Q

Bounding numerical results:


( ) ( )( ) ; / ( )lower upperq qr x N r x Nlower upper

q q q qa x e P N n N x a x e

( )2 ( )

1( ) ( ) 1

1( )

( 1) ( )

( ) ( ) ( 1) ( )

lower upperq lower upper

lower upper lower upper qq q

r xq C x

a x B x q r x

Bounding numerical results:


( ) ( )( ) ; / ( )lower upperq qr x N r x Nlower upper

q q q qa x e P N n N x a x e

( )2 ( )

1( ) ( ) 1

1( )

( 1) ( )

( ) ( ) ( 1) ( )

lower upperq lower upper

lower upper lower upper qq q

r xq C x

a x B x q r x

4.69lowerC

18.75upperC


For all strongly correlated systems which have Q-Gaussians (Q>1) as attractors in the sense of the central limit theorem, a model-dependent set [q>1, B(x)>0,rq

(low)(x)>0, rq(up)(x)>0] might exists such that

P(N;n/N<x) satisfies these inequalities:

1 12 21 1

( ) 1 1 1 ( ) 1 1 11 ; / 1

( 1) ( ) ( 1) ( )lower upperq qq q

B x B xo P N n N x o

q r x N N q r x N NN N

4.69lowerC

18.75upperC

( , )x N


For all strongly correlated systems which have Q-Gaussians (Q>1) as attractors in the sense of the central limit theorem, a model-dependent set [q>1, B(x)>0,rq

(low)(x)>0, rq(up)(x)>0] might exists such that

P(N;n/N<x) satisfies these inequalities:

1 12 21 1

( ) 1 1 1 ( ) 1 1 11 ; / 1

( 1) ( ) ( 1) ( )lower upperq qq q

B x B xo P N n N x o

q r x N N q r x N NN N

( , )x N

ConclusionsConclusions We address a family of models of strongly correlated variables of a certain class We address a family of models of strongly correlated variables of a certain class

whose attractors, in the probability space, are whose attractors, in the probability space, are QQ-Gaussians (-Gaussians (Q>1Q>1). They ). They illustrate how the classical Large Deviation Theory can be generalized. illustrate how the classical Large Deviation Theory can be generalized.

We conjecture that for all strongly correlated systems that have We conjecture that for all strongly correlated systems that have QQ-Gaussians -Gaussians ((Q>1Q>1) as attractors in the sense of the central limit theorem, a model-dependent ) as attractors in the sense of the central limit theorem, a model-dependent set [set [q>1q>1, , B(x)>0B(x)>0,,rrqq

(low)(low)(x)>0(x)>0, , rrqq(up)(up)(x)>0(x)>0] might exists such that] might exists such that P(N;n/N<x) P(N;n/N<x)

satisfies:satisfies:

The argument of the The argument of the qq-logarithmic decay of large deviations remains -logarithmic decay of large deviations remains extensiveextensive in in our model. This reinforces the fact that, according to NEXT for a wide class of our model. This reinforces the fact that, according to NEXT for a wide class of systems whose elements are strongly correlated, a value of index systems whose elements are strongly correlated, a value of index qq exists such exists such thar thar SSqq preserves extensivity. preserves extensivity.

Our models open the door to a Our models open the door to a qq-generalization of virtually many of the classical -generalization of virtually many of the classical results of the theory of large deviations.results of the theory of large deviations.

The present results do suggest the mathematical basis for the ubiquity of The present results do suggest the mathematical basis for the ubiquity of qq--exponential energy distributions in nature. exponential energy distributions in nature.

( ) ( ), ( ), ( ) ; / , ( ), ( )lower upperq qr x N r x Nlower upper

j q j qa q B x r x e P N n N x a q B x r x e

Kaniadakis’ Kaniadakis’ logarithm and logarithm and --exponentialexponential

(back)

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Towards a large deviation theory for statistical mechanical complex systems