Constantino TsallisCentro Brasileiro de Pesquisas Fisicas, BRAZIL

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

P. Douglas, S. Bergamini and F. Renzoni, Phys Rev Lett 96, 110601 (2006)

L.F. Burlaga and A.F.-Vinas, Physica A 356, 375 (2005).

A. Rapisarda and A. Pluchino, Europhysics News 36, 202 (EPS, Nov/Dec 2005)

Erice, August/September 2006

COMPLEXITY AND NONEXTENSIVE STATISTICAL MECHANICSTHEORY, EXPERIMENTS,

OBSERVATIONS AND COMPUTER SIMULATIONS

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, glass, spin-glass)

SUPERSTATISTICS (Other generalizations)

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.

Ettore MAJORANAThe value of statistical laws in physics and social sciences.Original manuscript in Italian published by G. Gentile Jr. in Scientia 36, 58 (1942); translated into English by R. Mantegna (2005).

This is mainly because entropy is an addditive quantity as the other ones. In other words, the entropy of a system composed of several independent parts is equal to the sum of entropy of each single part. [...]Therefore one considers all possible internal determinations as equally probable. This is indeed a new hypothesis because the universe, which is far from being in the same state indefinitively, is subjected to continuous transformations. We will therefore admit as an extremely plausible working hypothesis, whose far consequences could sometime not be verified, that all the internal states of a system are a priori equally probable in specific physical conditions. Under this hypothesis, the statistical ensemble associated to each macroscopic state Aturns out to be completely defined.

//

: -

:

, ( )( )

(

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

ENTROPIC FORMS

Concave

Extensive

Lesche-stable

Finite entropy production per unit time

Pesin-like identity (with largest entropy production)

Composable

Topsoe-factorizable

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

( , ) 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

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)

11- pp

1- p2

p1

212p 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

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

pthen

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+

+ ≠ +

= +

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∂

= −∂

∂ ∂ ∂≡ = = −

∂ ∂ ∂

NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS

Nonextensive Statistical Mechanics and Thermodynamics, SRA Salinas and C Tsallis, eds, Brazilian Journal of Physics 29, Number 1 (1999)

Nonextensive Statistical Mechanics and Its Applications, S Abe and Y Okamoto, eds, Lectures Notes in Physics (Springer, Berlin, 2001)

Non Extensive Thermodynamics and Physical Applications, G Kania-dakis, M Lissia and A Rapisarda, eds, Physica A 305, Issue 1/2 (2002)

Classical and Quantum Complexity and Nonextensive Thermodynamics, P Grigo-lini, C Tsallis and BJ West, eds, Chaos, Solitons and Fractals 13, Issue 3 (2002)

Nonadditive Entropy and Nonextensive Statistical Mechanics, M Sugiyama, ed, Continuum Mechanics and Thermo-dynamics 16 (Springer, Heidelberg, 2004)

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

Anomalous Distributions, Nonlinear Dynamics, and NonextensivityHL Swinney and C Tsallis, eds, Physica 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 Barbosa and E Curado, eds, Physica A 344, Issue 3/4 (2004)

Complexity, Metastability and Nonextensivity, C 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)

Complexity and Nonextensivity: New Trends in Statistical Mechanics, S Abe, M Sakagami and N Suzuki, eds, Progr. Theoretical Physics Suppl 162 (2006)

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

http://www.europhysicsnews.com

Full bibliography:(28 August 2006: 1953 manuscripts)

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

(0) 0

( ) lim sup

( )

( ) sup lim

(0)

( )

qq t

x

q q

q tq

It can be pq generalized Pesin lik

roved that

whereS t

Kt

and

x ttx

e identit

w

yK

ith

eΔλΔ

Δ

λ

ξ

→∞

→

⎧ ⎫≡ ⎨ ⎬

⎩ ⎭

⎧ ⎫≡ =⎨ ⎬

⎩ ⎭

−= −

min

1mi

ma

n max

x

1 1 1 ln ( )1 | | ( 1)1 ( ) ( ) ( )

1 1 1 ln 1 1 ln

2

2

n

1

lz F

t t

Fq

zx a x zq z z

n

z

qa

qd

αα α

α λα α

+

⎡ ⎤= − ⇒ = − =

= − = =− −

−⎢ ⎥−⎣ ⎦

1

1

( 1; 0Generic pitchfork bifurcations:

Generic tangent bifurcations:

-

)

(

1; 0)

( ) | |

| |

zt t tt

zt tt

The fixed point map is a q exponential witq z

h

and the

z

s

z

e

bx x b sign x x

x x b x b

+

+

> >

>

=

>

= +

= +

1

:

-

( 1; 0 2; 1)

3

12

1 | |

se

sen

n

tt

Exam

nsitivity to the initial conditions is a q expo

ple The logistic family of mapsa

has

z for pitchf

nential wit

or

q

h

k

q

x a x ςς

ς ς+

−

>

=

≤ ≤

=

−

>= −

5( ), 3 ;3

32 ( ), 2 .2

sen

sen

bifurcations hence q and q

z for tangent bifurcations hence q and q

ς

ς

∀ = =

= ∀ = =

A. Robledo, Physica D 193, 153 (2004)

1

1

11

1

1 ln (ln ln )1

[1 (1 ) ] ( )

:

( 1 (1 ) 0;

:

- :

q

q

x x xqq

DEFINITIONS

q logxx x x

q

e q

arithm

q exponential

x e e

if q x

−

−

−≡ =

−

≡ + − =

+ − >

−

)vanishes otherwise

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

q-GAUSSIANS:2

2( / )1-1

1( ) ( 3)1 ( -1) ( / )

qx

qq

p x qq x

e σ

σ

−∝ ≡ <⎡ ⎤+⎣ ⎦

q - CENTRAL LIMIT THEOREM:

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

qp x t p x tD qt x

γ

γ γ−∂ ∂

= < ≤ <∂ ∂

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

independent variables; divergent variance; Levy attractor

independent variables; finite variance; Gaussian attractor

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

q - CENTRAL LIMIT THEOREM (conjecture)

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

q

x 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)

de Moivre - Laplace

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⎛ ⎞⎜ ⎟≠ ≠⎝ ⎠

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

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

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 (2001)] 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 myiq m qn m q q

melds

q+ −

= ± ± ≡ =+ −

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:

, | |

- ( 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

−∞ ∞

−∞ −∞

⊗−

−

−

≡

≡

≡

≡

∫ ∫

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)

Qα

σ< <

→∞

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

) ( )

q SCALED ATTRACTOR WHEN SUMMING NCORRELATED IDENTICAL RANDOM VARIABLES WITH SYMMETRIC DISTRIBUTION

xNq f x

α →∞F

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

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

3

q 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 ib u t io n

x L s t a 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 in 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 αα

α α

+ −

+ − −∼

1 2 1 2 1

( )

( , ,..., )

( , ,..

.

? :

,

)

N N N

N body joi

It appea

nt

rs to be no proof available yet

dx h x x

WHAT

x h x x

IS IT q CORRELATIO

probability

N

x −

−

=

−

∫

. ., !

i e scale invariance

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

⎫= ⎪ ⇒⎬⎪= ⎭

-1/τtTypical relaxation of observable Ο

λtSensitivity toinitial conditions

Z p(Ei)-βEiEquilibrium distributiony(x)ax

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=

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

tTypical relaxation of observable Ο

tSensitivity toinitial conditions

EiStationary state distribution

y(x)ax

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 =

/ qrelrel

tqe

τ−Ω =

[ ]1

11 (1 ) qqa x

q a xy e −+ −= ≡

1 1/

W Wq q

q i i ji j

U p E p= =

= ∑ ∑

(typically 1)senq ≤

1 /r e lq

τ−

senqλ

(typically 1)relq ≥

statqβ− ( )statq iZ p E

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

(typically 1)statq ≥

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

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 = ±

( 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)

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

( . ., 1 1 , . ., )e g q i e normal diffu

q

sionγ

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 (European Physical Society, Nov/Dec 2005)

XY FERROMAGNET WITH LONG-RANGE INTERACTIONS:

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

Silo drainage: R. Arevalo, A. Garcimartin and D. Maza, cond-mat/0607365 (2006)

(intermediate regime)

(fully developed regime)

q=3/2q=1

4 / 32

3

slope

hence is satisfiedq

γ

γ

=

=−

R. Arevalo, A. Garcimartin and D. Maza, cond-mat/0607365 (2006)

(outlet size 3.8 d)

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

= +

HADRONIC JETS FROM ELECTRON-POSITRON ANNIHILATION:

I. Bediaga, E.M.F. Curado and J.M. de Miranda, Physica A 286 (2000) 156

Hagedorn

Beck (2000): q=11/9

(Phenomenological model for collisions in a diluted gas with probability rof forming clusters of q correlated particles)

Monte Carlo

Single-parameter fitting

Connections withasymptotically scale free networks−

(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 Gr distance to the baricenter of the pre existing cluster

P r α α+

≡ −

∝ ≥

4) Repeat

A

( )

( )

/ ( 0) Ai i Ai

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)

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)

/

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)

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∀

ASTROPHYSICS

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)

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

1.045 0.005 (99 % confidence level)q = ±

Connections with conservativeand Hamiltonian systems

0 1 2 3 4 50

1

2

3

4

5

α

d

EXTENSIVE

NONEXTENSIVE

dipole-dipole

Newtonian gravitation

dipole

-mon

opole

(tide

s)

d-dim

ensio

nal g

ravita

tion

( )

/ 1 ( - ) - 0 / 1

( )

( 0, 0)

( - ) integrable if d short ranged

non integrable if d

AV r

long

r

r

A

ange

r

d

α

αα

α

− ∞

>>≤

→

≤

≥

∼

[ ]

{ }1

( 1) ( ) sin 2 ( )2sin 2 ( ) ( )

(mod1)2

i i i

i j

ijj

N

ap t p t t

t tbN rα

πθπ

π θ θ

π=

+ = + +

⎡ ⎤−⎣ ⎦∑( 1) ( ) ( 1) (mod1)i i it t p tθ θ+ = + +

1 / 1 ; 0 ; 0 ; 0 ; 11 /

dNN a b dd

α

αα

− −≡ ≥ ≥ ≥ =

−

with

MANY COUPLED STANDARD MAPS:L.G. Moyano, A.P. Majtey and C.T., Eur Phys J B 52, 493 (2006)

L.G. Moyano, A.P. Majtey and C.T., Eur Phys J B 52, 493 (2006)

L.G. Moyano, A.P. Majtey and C.T., Eur Phys J B 52, 493 (2006)

2

1 ,

1 /

1

1 cos( )1 ( 0, 0)2

0 / 1 ln / 1

/ 1

Ni j

ii i j ij

dN

ijj

JH K V L I JI r

N if dwith r N if d

constant if d

and periodic boundary

α

α

αϑ ϑ

ααα

=

−

−

=

− −= + = + > >

⎧ ⎫≤ <⎪ ⎪≡ ∝ =⎨ ⎬⎪ ⎪>⎩ ⎭

∑ ∑

∑

A

A

[

.

0, ]The HMF model correspond

conditi

s to d

ons

α = ∀

d-DIMENSIONAL CLASSICAL INERTIAL XY FERROMAGNET:

(We illustrate with the XY (i.e., n=2) model; the argument holds however true for any n>1 and any d-dimensional Bravais lattice)

C. Anteneodo and C. T., Phys Rev Lett 80, 5313 (1998)

M. Antoni and S. Ruffo, Phys Rev E 52, 2361 (1995)

C. Anteneodo and C. T. Phys Rev Lett 80, 5313 (1998)

A. Campa. A. Giansanti, D. Moroni and C. T. Phys Lett A 286, 251 (2001)

V. Latora, A. Rapisarda and C. T., Phys Rev E 64, 056134 (2001)

C. T., A. Rapisarda, V. Latora and F. Baldovin, Lecture Notes in Physics 602 (Springer, Berlin, 2002)

V. Latora, A. Rapisarda and C. T., Phys Rev E 64, 056134 (2001)

F. Baldovin and E. OrlandiniPhys Rev Lett 96, 240602 (2006)

F. Baldovin and E. Orlandini, cond-mat/0603659

In contact with a thermostat (canonical ensemble):In contact with a thermostat (canonical ensemble):

8Wt =

2.35 ? 1.5 ?

0 :rel

stat

qq

modelα =

F.A. Tamarit and C. AnteneodoEurophysics News 36 (6), 194 (2005) [European Physical Society]

(q=2.35)

t

HMF MODEL:

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

ECONOMICS

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

STOCK VOLUMES:

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++

=

−

−

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

GENERALIZED SIMULATED ANNEALING AND RELATED ALGORITHMS

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

−

−

→ =

→ −

−→ =

+ −

< < <

+

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

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= ⇒ ≈

q-GENERALIZED PIVOT METHOD:

P. Serra, A.F. Stanton and S. Kais, Phys Rev E 55, 1162 (1997)

(Branin function) (Lennard-Jones clusters)

Num

ber o

f fun

ctio

n ca

lls Genetic algorithm

Present with q=2.7slo

pe 4

.7

slope

2.9

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 363, 260 (2006) (presumably better than both!)

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

Original image

q = 1.5

Canny edge detector

q = 1

(Jensen-Shannon)

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

Original image

q = 1.5

Canny edge detector

q = 1

(Jensen-Shannon)

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

IMAGE THRESHOLDING:

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

qthan

0

1 1 1 1 1 7 3 3 4 36

36 12 3 2 2 180 2 1 37

3 !!!

7

?

number of independent physical universal constants in conte

HOW MANY PHYSICAL UNIVERSAL CONST

mporary physics

ANT

he

S

nce

ν

ν

ν

ν π

≡

⎡ ⎤= + + =⎢ ⎥∞⎣ ⎦

⎛ ⎞= = + + =

=

+ = + = + =⎜ ⎟⎝ ⎠

Nino Constantino Gerardus

Euclid Gerardus

NINO WAS RIGHT!!!

( 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= = =

2000 4000 6000 8000 10000 12000 140000

10

20

30

40

50

60

70

80

time

k max

; k i

k

ik

max

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)

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 WS 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

=

=

−⎛ ⎞= = ⇒ = −⎜ ⎟− ⎝ ⎠

∑∑