FUNDAMENTOS DA MECÂNICA ESTATÍSTICA: PARADIGMAS E CONSEQUÊNCIAS

Constantino Tsallis

Centro Brasileiro de Pesquisas Físicas

e Instituto Nacional de Ciência e Tecnologia de Sistemas Complexos Rio de Janeiro, Brasil

e

Santa Fe Institute, New Mexico, USA

Campinas, Junho 2016

FUNDAMENTOS DA MECÂNICA ESTATÍSTICA:

PARADIGMAS E CONSEQUÊNCIAS

World champion of complexity!

(Photo taken in London by Luciano Pietronero)

THERMODYNAMICS

MECHANICS (classical, quantum, relativistic, QCD) LANGEVIN EQUATION FOKKER-PLANCK EQUATION

STATISTICAL MECHANICS LIOUVILLE EQUATION

VON NEUMANN EQUATION VLASOV EQUATION

BOLTZMANN KINETIC EQUATION BBGKY HIERARCHY

Braun and Hepp theorem

MASTER EQUATION

N 

THEORY OF PROBABILITIES ENTROPY FUNCTIONAL ENERGY

Enrico FERMI Thermodynamics (Dover, 1936)

The entropy of a system composed of several parts is very often equal 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.

ENTROPIC FORMS

Concave Extensive Lesche-stable

Finite entropy production per unit time

Pesin-like identity (with largest entropy production) Composable

Topsoe-factorizable (unique) Amari-Ohara-Matsuzoe

conformally invariant geometry (unique) Biro-Barnafoldi-Van thermostat universal independence (unique)

ENTROPIC FUNCTIONALS

nonadditive (if q  1) additive

Entropy Sq (q real) BG entropy (q =1)

Possible generalization of Boltzmann-Gibbs statistical mechanics C.T., J. Stat. Phys. 52, 479 (1988)

 

1

1 1

1 1

: :

ln 1 ( 0; ln ln )

1

1 (1 ) ( )

:

q

q q

x q x x

DEFINITIONS q logarithm q exponential

x x x x x

q

q x e e

e









   



   

W i=1

i=

, :

ln 1 ln

ln 1 ( 1)

ln

( )

q

i

i

q i q

i

Hencethe entropies can be rewritten

equal probabilities generic probabilities

BG en k W k p

p

k W

tropy q

entropy S q R

k p

p







W



1

p

_i^q

 p  if  q  1     (aristocratic :  minority controls) p

_i^q

 p  if  q  1     (demagogic :   majority controls)

p

_i^q

 p  if  q  1     (democratic :   equal opportunities)     Boltzmann­Gibbs 1985   Ciudad de Mexico + Rio de Janeiro

1987   Maceio + Rio de Janeiro 1987   Notas de Física ­ CBPF

1988   Journal of Statistical Physics              3869 citations/ISI Web of Knowledge

1991   Journal of Physics A (with E.M.F. Curado)             483 citations/ISI Web of Knowledge

1998   Physica A (with R.S. Mendes and A.R. Plastino)        830 citations/ISI Web of Knowledge

      [2 June 2016]

TYPICAL SIMPLE SYSTEMS:

Short-range space-time correlations

Markovian processes (short memory), Additive noise

Strong chaos (positive maximal Lyapunov exponent), Ergodic, Riemannian geometry Short-range many-body interactions, weakly quantum-entangled subsystems

Linear and homogeneous Fokker-Planck equations, Gausssians à Boltzmann-Gibbs entropy (additive)

à Exponential dependences (Boltzmann-Gibbs weight, ...)

TYPICAL COMPLEX SYSTEMS:

Long-range space-time correlations

Non-Markovian processes (long memory), Additive and multiplicative noises

Weak chaos (zero maximal Lyapunov exponent), Nonergodic, Multifractal geometry Long-range many-body interactions, strongly quantum-entangled sybsystems

Nonlinear and/or inhomogeneous Fokker-Planck equations, q-Gaussians à Entropy S_q (nonadditive)

à q-exponential dependences (asymptotic power-laws)

W( N )  

^N

⁽  ^{ 1)}

e.g., ( ) W N ^ N

^

(  ^ 0)

ADDITIVITY:

additive probabilistically independent

( ) ( ) Therefo

An entropy is if, for any two systems an

( )

re

( ) ( ) ( ) (1

, since d ,

q q q

S A B S A S B S A B S A

A B

S B

  

   

and ( ) are additive, an

) ( ) d ( 1) is nonadditiv

( , e .

)

Renyi

BG q q

q q

S S q S

q S A S B q 



 

EXTENSIVITY:

1 2

1 2

Consider a system made of (not necessarily independent) identical elements or subsystems and , ..., .

An en

...

extensive

trop

y is

0 l m i i

f

N

N

N

N

A A A

A A A



^{   }

 ( )

, . ., ( ) ( ) S N i e S N N N

N

   

Foundations of Statistical Mechanics: A Deductive Treatment (Pergamon, Oxford, 1970), page 167

O. Penrose,

EXTENSIVITY OF THE ENTROPY (N  )

W  total number of possibilities with nonzero probability, assumed to be equally probable

If W( N ) ∼ 

^N

⁽   1)

 S

_BG

( N )  k

_B

lnW( N)  N OK!

If W( N ) ∼ N

^

(   0)

 S

_q

( N )  k

_B

ln

_q

W( N )  [ W( N )]

^1q

 N

^(1q)

 S

_{q11 }

( N )  N OK!

If W( N ) ∼ 

^N

⁽   1; 0    1)

 S

_

( N )  k

_B

⎡⎣ lnW( N ) ⎤⎦

^

 N

^{ }

 S

_{1 }

( N )  N OK!

IMPORTANT: 

^N

 

^N

 N

^

if N  1

EXTENSIVE SYSTEMS

   W (N)

EXTENSIVE (q  1  1 /  ⁾

EXTENSIVE   (   1 /  ⁾

NONEXTENSIVE

NONEXTENSIVE NONEXTENSIVE

NONEXTENSIVE

NONEXTENSIVE

NONEXTENSIVE

ENTROPY  S

_BG

  (ADDITIVE)

   ENTROPY  S

_q

         (q  1)

(NONADDITIVE)

   ENTROPY  S

_

         (   1)

(NONADDITIVE)

e.g.,  

^N

 (  ^{ 1)}

e.g., N

^

 (  ^{ 0)}

e.g.,  

^{N }

(  ^{ 1;}

0   ^{ 1)}

(ADDITIVE) (NONADDITIVE)(NONADDITIVE)

A theory is the more impressive the greater the

simplicity of its premises is, the more different kinds of things it relates, and the more extended is its area of applicability. Therefore the deep impression that classical thermodynamics made upon me. It is the only physical theory of universal content concerning which I am convinced that, within the framework of applicability of its basic concepts, it will never be overthrown.

Albert Einstein (1949)

King Thutmosis I 18^th Dynasty

circa 1500 BC

The relation between S and W given in Eq. (1) is the only reasonable given the proposition that the entropy of a system consisting of subsystems is equal to the sum of entropies of the subsystems.

(Free translation by Tobias Micklitz) Annalen der Physik 33 (1910)

(21 March 2016: 624 downloads)

q  1:       S

_BG

 k

_B

lnW      hence        

^p_i

 ^{ e}

^{SBG  pi /kB}

    ^hence 

(A  B)  e

^{SBG( AB)/kB}

 e

^{SBG( A)/kBSBG( B)/kB}

 e

^SBG( A)/kB

e

^SBG( B)/kB

 (A)  (B)          OK!

q :       S

_q

 k

_B

ln

_q

W      hence         

^p_i

 ^{ e}

q

S_{q p}  _i ^/kB

    hence 

(A  B)  e

_q^{Sq( AB)/kB}

 e

_q^{Sq( A)/kBSq( B)/kB(1q)[Sq( A)/kB][Sq( B)/kB]}

        e

_q^Sq( A)/kB

e

_q^Sq( B)/kB

 (A)  (B)       OK   q !

Einstein 1910 (reversal of Boltzmann formula): 

      For any two independent systems  A and B,        the likelihood function should satisfy

   

       (A  B) (A)  (B)  (Einstein principle)

Full bibliography (regularly updated):

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

5654 articles by 12116 scientists (93 countries)

[2 June 2016]

CONTRIBUTORS (5654 MANUSCRIPTS)

[Updated 2 June 2016]

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AUSTRIA ⁹² MALAYSIA ¹⁷ NIGERIA ⁴ 12116 SCIENTISTS

HUNGARY ⁸⁷ CUBA ¹⁶ SENEGAL ⁴ 93 COUNTRIES

(1956)

Because scientists are reasonable men, one or another argument will ultimately persuade many of them. But there is no single argument that can or should persuade them all. Rather than a single group conversion, what occurs is an increasing shift in the distribution of

professional allegiances.

The historian of science may be tempted to claim that when

paradigms change, the world itself changes with them. Led by a new paradigm, scientists adopt new instruments and look in new

places. Even more important, during revolutions, scientists see new and different things when looking with familiar instruments in places they have looked before. It is rather as if the professional community had been suddenly transported to another planet where familiar objects are seen in a different light and are joined by unfamiliar ones as well.

Well-established theories collapse under the weight of new facts and observations which cannot be explained, and then accumulate to the point where the once useful theory is clearly obsolete.

E debbasi considerare come non è cosa più difficile a trattare, né più dubia a riuscire, né più pericolosa a maneggiare, che farsi capo ad introdurre nuovi ordini. Perché lo introduttore ha per nimici tutti quelli che delli ordini vecchi fanno bene, et ha tepidi defensori tutti quelli che delli ordini nuovi farebbono bene. La quale tepidezza nasce, parte per paura delli avversarii, che hanno le leggi dal canto loro, parte dalla incredulità delli uomini; li

quali non credano in verità le cose nuove, se non ne veggono nata una ferma esperienza.

Niccolò Machiavelli (1469-1527)

Il Principe (Capitolo VI)

We must consider that nothing is harder to implement, of more uncertain success, nor more dangerous to deal with, than to initiate a new order of things.

Because the one who introduces the novelties finds enemies in all those who profit from the old order and tepid defenders in all those who would profit from the new order. This tepidity comes in part from their fear of their adversaries, who have the laws on their side, and in part from the incredulity of people, who

do not really believe in new things until they have solid experience of them.

(Translation: C. Tsallis and M. Gell-Mann)

SPIN ½ XY FERROMAGNET WITH TRANSVERSE MAGNETIC FIELD:

|  ^{| } 1  Ising ferromagnet

0  |  ^{|  1 } anisotropic XY ferromagnet

 ^ 0  isotropic XY ferromagnet

 ^{ transverse magnetic field}

L  length of a block within a N → ∞ chain 

F. Caruso and C. T., Phys Rev E 78, 021101 (2008)

F. Caruso and C. T., Phys Rev E 78, 021101 (2008)

ISING MODEL (c=1/2)

0 0.2 0.4 0.6 0.8 1.0

0 0.5 1.0 1.5 2.0

q

1/ c

BG

XY Ising

Z(  ) m





SU(4)

9 c

²

3

q c

  

Block entropy for the d=1+1 model, with central charge c, at its quantum phase transition at T=0 and critical transverse “magnetic” field

analytically obtained from first principles

Self­dual Z(n) magnet (n 1, 2,...)       [FC Alcaraz, JPA 20 (1987) 2511]

       c  2(n 1)

n 2 [0, 2]    

SU(n) magnets (n 1, 2,...; m 2,3,...)   [FC Alcaraz and MJ Martins, JPA 23 (1990) L1079]

       c  (n 1) 1  n(n 1)

(m  n2)(m  n1)

⎡

⎣⎢ ⎤

⎦⎥[0, n1] 

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

q -GAUSSIANS:

2

( / )2

1 -1

( ) 1 ( 3)

1 ( -1) ( / )

q x

q

q

p x q

q x

e

^







 

⎡  ⎤

⎣ ⎦

See also: H.J. Hilhorst, JSTAT P10023 (2010) M. Jauregui and C. T., Phys Lett A375, 2085 (2011) M. Jauregui, C. T. and E.M.F. Curado, JSTAT P10016 (2011) A. Plastino and M.C. Rocca, Physica A and Milan J Math (2012) A. Plastino and M.C. Rocca, Physica A392, 3952 (2013)

1 [ ]

q independent q1 ( . ., i e Q2q 1 1) [globally correlated]

1

( )

( ) (

Classic CL

)

,

T with same of

x Gaussian G x

 f x



< F

( 2)

Q







(0 2)

Q



 



1/[ (2- )]

CENTRAL LIMIT THEOREM -

-

( )

( )

q scaled attractor when summing N independent identical random variables with symmetric distribut

x

ion w

N q

f x

 F 

2

1

[ ( )] / [ ( )] 2 1, 1

3

Q Q

Q

ith dx x f x dx f x Q q q q

 ^{ ⎛}_⎜ ^{   } q^⎞_⎟

⎝  ⎠

 

2

1

| |

( )

L

evy

| | (1, ) L ( )

( ) / | |

| | (

(

-Gned

1, )

lim (

enk

) ( )

o CLT 1 )

,

,

c

c c

with samex behavior

G x

if x x

x f x C x

if x x

with x

x Levy distribution Lx



 









 





⎧⎪ 

⎪⎨

⎪ 

⎪ 

⎩



 F

 ¹ 1  1

1

3 / 1

2/( 1)

( ) ( ) | | ( , 2)

( ) / | | | | ( , 2)

( ) ,

lim ( )

( ) ( )

c c q

q q

c q

Q

q q

q

with same of f x

G x if x x q

f x C x if x x q

with x

x G x

G x G _{ } x 





⎧  ⎫

⎪ ⎪

⎨ ⎬

 

⎪



⎩ ⎪⎭

 F

S. Umarov, C. T. and S. Steinberg, Milan J Math 76, 307 (2008)

( , 2)

q 

   

   

   

2 1 3

* 2(1 )

2 1 ,

1 ,

2 1 3

,

2 1 3, 2 , ,

(1 )/(1 )

| |

( ) / | |

( ) ,

(

~

( ) / | |

)

L

q q

q

q q

q q

q

q q q

q

q

x L

intermediate regime

with samex asymptotic behavior

G x C x

L

G x C x



 

 

  





   

  

   

  

 

  







 

F

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

J Math Phys 51, 0335

02

(2010)

(

)

distant regime

⎧⎪

⎪⎪

⎪⎪

⎨⎪

⎪⎪

⎪⎪

⎩

[1877]

(area-preserving)

STANDARD MAP (Chirikov 1969)

Particle confinement in magnetic traps, particle dynamics in accelerators,

comet dynamics,

ionization of Rydberg atoms, electron magneto-transport

p

_i1

 p

_i

 K sin x

_i

 (mod  2  ⁾

x

_i1

 x

_i

 p

_i1

        (mod  2  ⁾ 

      (i  0,1, 2,...)

Tirnakli and Borges Nature / Scientific Reports 6, 23644 (2016)

Tirnakli and Borges Nature / Scientific Reports 6, 23644 (2016)

y

Tirnakli and Borges

Nature / Scientific Reports 6, 23644 (2016)

J.W. GIBBS

Elementary Principles in Statistical Mechanics - Developed with Especial Reference to the Rational Foundation of Thermodynamics

C. Scribners Sons, New York, 1902; Yale University Press, New Haven, (1981), page 35

In treating of the canonical distribution, we shall always suppose the multiple integral in equation (92) [the partition function, as we call it nowadays] to have a finite valued, as otherwise the coefficient of

probability vanishes, and the law of distribution becomes illusory. This will exclude certain cases, but not such apparently, as will affect the value of our results with respect to their bearing on thermodynamics.

It will exclude, for instance, cases in which the system or parts of it

can be distributed in unlimited space […]. It also excludes many cases in which the energy can decrease without limit, as when the system

contains material points which attract one another inversely as the squares of their distances. […]. For the purposes of a general

discussion, it is sufficient to call attention to the assumption implicitly

involved in the formula (92).

0 1 2 3 4 5

0 1 2 3 4 5



d

EXTENSIVE SYSTEMS

NONEXTENSIVE SYSTEMS

dipole-dipole

Newtonian gravitation

V(r)  A

r ^ (r ) (A 0,



^{ 0)}

integrable if



^{/ d}1 (short-ranged) non-integrable if 0 



^{/ d}1 (long-ranged)

HMF (inertial XY model CLASSICAL LONG-RANGE-INTERACTING MANY-BODY HAMILTONIAN SYSTEMS

 ^{-XY and}

 -FPU models

 ^{ 2} q   1

 ^{ 0.9} q   1.58

L.J.L. Cirto, V.R.V. Assis and C. T., Physica A 393, 286 (2014)

( ^ _i )/ ₀

 _i   ₀

U = 0.69

N = 1 000 000

  ^= 0.90 n = 1

t = 500 000

Condição Inicial:

     {p

_i

} = Uniforme      { 

_i

} = 0  i

(c)

(q

_n

,  

_qn

) = (1.53, 11.3)

10 ⁻⁵ 10 ⁻⁴ 10 ⁻³ 10 ⁻² 10 ⁻¹ 10 ⁰

−3 −2 − 0  2 3

 ^{ 0.9}

q   1.53

FERMI-PASTA-ULAM MODEL WITH LONG-RANGE INTERACTIONS

 ^{ 0.7} yields

q   1.25

 ^{ 1.4}

yields

q   1

q

_{ }^1.48

H. Christodoulidi, C. T. and T. Bountis, EPL 108, 40006 (2014)

(d=2)



_max

 N

^{( ,d)}

 L

^{d( ,d)}

10^­2 10^­1 10⁰ 10¹ 10² 10³

10¹ 10² 10³ 10⁴ 10⁵ 10⁶

x 2⁰ x 2¹ x 2² x 2³ x 2⁴ x 2⁵ x 2⁶ x 2⁷ x 2⁸ x 2⁹ x 2¹⁰



 max

N

0.00 0.25 0.50 0.75 1.00 1.20 1.50 1.80 2.00 2.50 3.00

 0  0.1  0.2  0.3

 0  1  2  3  4

(,d)

 0 

 0.2  0.4  0.6  0.8  1

 0  0.2  0.4  0.6  0.8  1  1.2

1 ­ x² 1 + x²/6

(d+2) (, d)

/d

d = 1 d = 2 d = 3

!!

FPU!model!!2dN !degrees!of!freedom whereas

XY !!!model!!2N !!!degrees!of!freedom

D. Bagchi and C. T., PRE (2016), in press 1509.04697 [cond-mat.stat-mech]

P( k )  P(0) e

_q^k/

 P(0) 1  ( q  1)k / 

⎡⎣ ⎤⎦

^q11

S.G.A. Brito, L.R. da Silva and C. T., Nature/Scientific Reports 6, 27992 (2016)

S.G.A. Brito, L.R. da Silva and C. T. Nature/Scientific Reports6, 27992(2016)

S.G.A. Brito, L.R. da Silva and C. T., Nature/Scientific Reports 6, 27992 (2016)

LHC (Large Hadron Collider)

CMS, ALICE, ATLAS and LHCb detectors

~ 4000 scientists/engineers from ~ 200 institutions of ~ 50 countries

(q = 1+1/n)

C.Y. Wong, G. Wilk, L.J.L. Cirto and C. T.,

EPJ Web of Conferences 90, 04002 (2015), and PRD 91, 114027 (2015) SIMPLE APROACH: TWO-DIMENSIONAL SINGLE RELATIVISTIC FREE PARTICLE

dydpdN _T   = A e_q^{− ET / T} [A] = GeV⁻²c³

(A, q, T) = (38, 1, 0.13)      Boltzmann−Gibbs

2  p_T

A / 10⁰ A / 10¹ A / 10² A / 10⁴ A / 10⁶

[T] = GeV 1

0

(A, q, T) = (38, 1.150, 0.13) (A, q, T) = (43, 1.151, 0.13) (A, q, T) = (30, 1.127, 0.13) (A, q, T) = (32, 1.125, 0.13) (A, q, T) = (27, 1.124, 0.13) CMS  s = 7  TeV ATLAS  s = 7  TeV CMS s = 0.9 TeV ATLAS s = 0.9 TeV ALICE s = 0.9 TeV

10⁻¹⁴ 10⁻¹³ 10⁻¹² 10⁻¹¹ 10⁻¹⁰ 10⁻⁹ 10⁻⁸ 10⁻⁷ 10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰ 10¹

data/fit

p_T [GeV/c]

0.5 1.0 1.5

10⁻¹ 10⁰ 10¹ 10²

p [GeV/c]

E _{ }

mc

²

^− 1

Einstein (1905)

E = (m²c⁴ + p²c⁴)^1/2

Newton

E = mc² + p²/2m

E

_p

 = 7 TeV E

_e

 = 7 TeV

E

_p

 = 10

⁸

 TeV (EECR)

Extreme Energy Cosmic Rays

electron

proton

m

_p

  938.3 MeV/c

²

m

_e

  0.511 MeV/c

²

10

⁻⁴

10

⁻³

10

⁻²

10 10

⁻¹⁰

10

¹

10

²

10

³

10

⁴

10

⁵

10

⁶

10

⁷

10

⁸

10

⁹

10

¹⁰

10

¹¹

10

¹²

10

⁻⁵

10

⁻⁴

10

⁻³

10

⁻²

10

⁻¹

10

⁰

10

¹

10

²

10

³

10

⁴

10

⁵

10

⁶

10

⁷

10

⁸

10

⁹

10

¹⁰

10

¹¹

Combe, Richefeu, Stasiak and Atman PRL 115, 238301 (2015)

CT and DJ Bukman, PRE 54 (1996) R2197 Combe, Richefeu, Stasiak and Atman

PRL 115, 238301 (2015)

1/

Constantino Tsallis

Centro Brasileiro de Pesquisas Físicas

e Instituto Nacional de Ciência e Tecnologia de Sistemas Complexos Rio de Janeiro, Brasil

e

Santa Fe Institute, New Mexico, USA

Campinas, Junho 2016

FUNDAMENTOS DA MECÂNICA ESTATÍSTICA:

PARADIGMAS E CONSEQUÊNCIAS

World champion of complexity!

(Photo taken in London by Luciano Pietronero)

THERMODYNAMICS

MECHANICS (classical, quantum, relativistic, QCD) LANGEVIN EQUATION FOKKER-PLANCK EQUATION

STATISTICAL MECHANICS LIOUVILLE EQUATION

VON NEUMANN EQUATION VLASOV EQUATION

BOLTZMANN KINETIC EQUATION BBGKY HIERARCHY

Braun and Hepp theorem

MASTER EQUATION

N 

THEORY OF PROBABILITIES ENTROPY FUNCTIONAL ENERGY

Enrico FERMI Thermodynamics (Dover, 1936)

The entropy of a system composed of several parts is very often equal 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.

ENTROPIC FORMS

Concave Extensive Lesche-stable

Finite entropy production per unit time

Pesin-like identity (with largest entropy production) Composable

Topsoe-factorizable (unique) Amari-Ohara-Matsuzoe

conformally invariant geometry (unique) Biro-Barnafoldi-Van thermostat universal independence (unique)

ENTROPIC FUNCTIONALS

nonadditive (if q  1) additive

Entropy Sq (q real) BG entropy (q =1)

Possible generalization of Boltzmann-Gibbs statistical mechanics C.T., J. Stat. Phys. 52, 479 (1988)

 

1

1 1

1 1

: :

ln 1 ( 0; ln ln )

1

1 (1 ) ( )

:

q

q q

x q x x

DEFINITIONS q logarithm q exponential

x x x x x

q

q x e e

e









   



   

W i=1

i=

, :

ln 1 ln

ln 1 ( 1)

ln

( )

q

i

i

q i q

i

Hencethe entropies can be rewritten

equal probabilities generic probabilities

BG en k W k p

p

k W

tropy q

entropy S q R

k p

p







W



1

p

_i^q

 p  if  q  1     (aristocratic :  minority controls) p

_i^q

 p  if  q  1     (demagogic :   majority controls)

p

_i^q

 p  if  q  1     (democratic :   equal opportunities)     Boltzmann­Gibbs 1985   Ciudad de Mexico + Rio de Janeiro

1987   Maceio + Rio de Janeiro 1987   Notas de Física ­ CBPF

1988   Journal of Statistical Physics              3869 citations/ISI Web of Knowledge

1991   Journal of Physics A (with E.M.F. Curado)             483 citations/ISI Web of Knowledge

1998   Physica A (with R.S. Mendes and A.R. Plastino)        830 citations/ISI Web of Knowledge

      [2 June 2016]

TYPICAL SIMPLE SYSTEMS:

Short-range space-time correlations

Markovian processes (short memory), Additive noise

Strong chaos (positive maximal Lyapunov exponent), Ergodic, Riemannian geometry Short-range many-body interactions, weakly quantum-entangled subsystems

Linear and homogeneous Fokker-Planck equations, Gausssians à Boltzmann-Gibbs entropy (additive)

à Exponential dependences (Boltzmann-Gibbs weight, ...)

TYPICAL COMPLEX SYSTEMS:

Long-range space-time correlations

Non-Markovian processes (long memory), Additive and multiplicative noises

Weak chaos (zero maximal Lyapunov exponent), Nonergodic, Multifractal geometry Long-range many-body interactions, strongly quantum-entangled sybsystems

Nonlinear and/or inhomogeneous Fokker-Planck equations, q-Gaussians à Entropy S_q (nonadditive)

à q-exponential dependences (asymptotic power-laws)

W( N )  

^N

⁽  ^{ 1)}

e.g., ( ) W N ^ N

^

(  ^ 0)

ADDITIVITY:

additive probabilistically independent

( ) ( ) Therefo

An entropy is if, for any two systems an

( )

re

( ) ( ) ( ) (1

, since d ,

q q q

S A B S A S B S A B S A

A B

S B

  

   

and ( ) are additive, an

) ( ) d ( 1) is nonadditiv

( , e .

)

Renyi

BG q q

q q

S S q S

q S A S B q 



 

EXTENSIVITY:

1 2

1 2

Consider a system made of (not necessarily independent) identical elements or subsystems and , ..., .

An en

...

extensive

trop

y is

0 l m i i

f

N

N

N

N

A A A

A A A



^{   }

 ( )

, . ., ( ) ( ) S N i e S N N N

N

   

Foundations of Statistical Mechanics: A Deductive Treatment (Pergamon, Oxford, 1970), page 167

O. Penrose,

EXTENSIVITY OF THE ENTROPY (N  )

W  total number of possibilities with nonzero probability, assumed to be equally probable

If W( N ) ∼ 

^N

⁽   1)

 S

_BG

( N )  k

_B

lnW( N)  N OK!

If W( N ) ∼ N

^

(   0)

 S

_q

( N )  k

_B

ln

_q

W( N )  [ W( N )]

^1q

 N

^(1q)

 S

_{q11 }

( N )  N OK!

If W( N ) ∼ 

^N

⁽   1; 0    1)

 S

_

( N )  k

_B

⎡⎣ lnW( N ) ⎤⎦

^

 N

^{ }

 S

_{1 }

( N )  N OK!

IMPORTANT: 

^N

 

^N

 N

^

if N  1