Origin of generalized entropies and generalized statistical mechanics for superstatistical multifractal systems Bahruz Gadjiev and Tatiana Progulova International University for Nature, Society and Man, 19 Universitetskaya str., Dubna, 141980, Russia Abstract. We consider a multifractal structure as a mixture of fractal substructures and introduce a distribution function f (α), where α is a fractal dimension. Then we can introduce g(p) ∼ Rµ

e−y f (y)dy and show that the distribution functions f (α) in the form of f (α) = δ (α − 1),

− ln p 1 f (α) = δ (α − θ ), f (α) = α−1 , f (y) = yα−1 lead to the Boltzmann – Gibbs, Shafee, Tsallis and Anteneodo – Plastino entropies conformably. Here δ (x) is the Dirac delta function. Therefore the Shafee entropy corresponds to a fractal structure, the Tsallis entropy describes a multifractal structure with a homogeneous distribution of fractal substructures and the Anteneodo – Plastino entropy appears in case of a power law distribution f (y). We consider the Fokker – Planck equation for a fractal substructure and determine its stationary solution. To determine the distribution function of a multifractal structure we solve the twodimensional Fokker – Planck equation and obtain its stationary solution. Then applying the Bayes theorem we obtain a distribution function for the entire system in the form of q-exponential function. We compare the results of the distribution functions obtained due to the superstatistical approach with the ones obtained according to the maximum entropy principle.

Keywords: Complex systems, Generalized maximum entropy principle, Fokker – Planck equation, Superstatistics, Fractal, Multifractal PACS: 05.20.-Dd, 05.70.Ln, 89.70.Cf, 89.75.Da, 05.90.+m

INTRODUCTION Fractality is inherent in the structure of complex systems and the processes going on in them. Typically the complexity of systems is linked to a long-term memory, longrange interactions, the non-Markovian kinetics. So formalism of fractional differential equations that is equations with fractional derivatives is one of the main formal mathematical tool allowing to describe the behavior of complex systems. Fractional integrals and derivatives are widely used in statistical mechanics. In particular the fractional generalization of the Fokker – Planck equation has been used for the description of the temporal evolution of complex systems [1]. The approach of the maximum principle of a certain generalized entropy is another formalism of the statistical mechanics allowing to describe the behavior of complex systems. In this approach maximization of the generalized entropy is performed in accord with appropriate constraints and a more general measure of information than that of Shannon is used. Further we will describe the most important measures of information and their connection with the fractality of complex system structure. The examples

treated are the Tsallis entropy, Shafee entropy, Anteneodo – Plastino entropy [2, 3, 4]. In recent years superstatistic is widely used for the description of the behavior of complex systems, caused by their inhomogeneity [5,6]. Many applications have been recently reported [7]. Besides theoretical attempts to detect the relationship of the maximum entropy principle with the superstatistics approach have been done [8]. The superstatistical approach presumes that a system consists of mesoscopic substructures and is enclosed into a fluctuating environment. Due to that some intensive parameter β (~r,t) where ~r is a space coordinate and t is time makes a difference among mesoscopic substructures. In the stationary state the distribution of the values of the parameter β in substructures is described by a certain distribution function f (β ). The conditional probability p(v|β ) in the substructure is defined from the stationary solution of the Fokker – Planck equation and p(v, β ) = p(v|β ) f (β ) is a joint probability. Then p(v) =

R∞

f (β )p(v|β )dβ is an appropriate marginal distribution.

0

As a simple example we consider the distribution of probabilities p(v|β ), which is determined from the Fokker – Planck equation [7]   ∂ p(v|β ) ∂ 1 ∂ 2 =− −γvp(v|β ) − σ p(v|β ,t) . (1) ∂t ∂v 2 ∂v A stationary solution of this equation is defined from the condition v2

∂ p(v|β ,t) ∂t

= 0 and has

the form p(ε|β ) = p0 e−β ε , where ε = 2 , β = σ2γ2 , and the parameter p0 is defined from the normalization condition p(v|β ). If the intensive variable β is distributed in accord with the distribution χ 2 in the form   1 1 1 − β  f (β ) =  β q−1 −1 e (q−1)β0 , (2) 1 (q − 1)β 0 Γ q−1 we obtain for a marginal distribution Z∞

p(ε) =

1

p(ε|β ) f (β )dβ = [1 + (q − 1)β0 ε] 1−q .

(3)

0

Thereby the distribution p(ε) is the Tsallis distribution [7]. It should be stressed that earlier on the assumption of the Tsallis entropy and the standard constraints distribution (3) was obtained within the framework of the maximum entropy method [5]. If we suppose that fast and slow variables exist in the system, the emerging the distribution in the form (2) can be interpreted within the framework of the Fokker – Planck theory [8]. In the present paper we investigate the origin of entropies for fractal and multifractal structures. We also present a superstatistical theory for multifractal structures within the framework of the Fokker – Planck theory and derive an appropriate distribution function.

ORIGIN OF GENERALIZED ENTROPIES Depending on the nature of the linkages among the elements systems may be homogeneous, fractal and multifractal. A multifractal structure can be considered as a mixture of fractal subsystems with various fractal dimensions. Homogeneous structures are characterized by the famous Boltzmann – Gibbs entropy, defined as ∞ S=−

Z

p(x) ln p(x)dx

(4)

0

with a corresponding normalization condition

R∞

p(x)dx = 1. Consider the quantity GI =

0

R1

− p ln pd p. Applying a fractional generalization of the integral we can write 0

Γ(α)GF = −

Z1

pα ln pd p.

(5)

0

Here Γ(α) is a gamma function. In the discrete version Γ(α)GF = − ∑ pαi ln pi ∆pi . We i

note that GSha f ee = − ∑ pαi ln pi is the Shafee entropy, which was introduced in [3]. When i

α = 1 the Shafee entropy coincides with that of Boltzmann – Gibbs. After integration of expression (5) over α we obtain 1 α −1

Zα 1

1 Γ(α)GF dα = − α −1

Z1

Zα

dα 1

0

1 p ln pd p = − α −1 α

Z1

(pα − p)d p,

(6)

0

W p −pα i i α−1 . We note that GT sallis i=1

or in the discrete version at ∆pi = 1 we obtain GT sallis = ∑

is

the Tsallis entropy [5]. When α = 1 the Tsallis entropy coincides with that of Boltzmann – Gibbs. Therefore, the Shafee entropy is connected with a fractal structure, whereas the Tsallis entropy corresponds to multifractals structures. This result can be generalized by considering a multifractal structure as a mixture of fractal substructures. We introduce the distribution function f (α), where α is a dimension of a fractal substructure. And we can introduce g(p) ∼

Zµ

e−y f (y)dy

(7)

− ln p

and show that the distribution functions f (α) in the form of f (α) = δ (α − 1), f (α) = 1 δ (α − θ ), f (α) = α−1 , f (y) = yα−1 lead to the Boltzmann – Gibbs, Shafee, Tsallis and Anteneodo – Plastino entropies respectively. Note that generalized entropies don’t satisfy the fourth Khinchin axiom [9].

The Anteneodo – Plastino entropy appears in case of f (y) = yα−1 and has the form     W γ+1 Sγ = ∑ sγ (pi ), where sγ (pi ) ≡ Γ γ+1 , − ln p − p Γ . Here γ is a positive numi i γ γ i=1

R∞

exp(−t) R

t

0

ber, Γ(µ,t) = yµ−1 e−y dy =

[− ln x]µ−1 dx, µ > 0 is an incomplete gamma func-

tion, and Γ(γ, 0) = Γ(γ) is a gamma function. Applying the principle of maximum entropy and the corresponding standard constraints we can obtain a distribution function for systems. In case of a fractal structure of a system applying the entropy GSha f ee = − ∑ pαi ln pi with the constraints ∑ pi = 1 and i

i

∑ pi Ei = U we obtain a distribution function i

 1 q−1 −qW (z) pi = , (8) (q − 1)(α + β Ei )   q−1 where z = − q−1 (α + β E ) exp and W (z) is the Lambert function, which is a i q q 

solution of the equation z = W (z)eW (z) [3]. W p −pα i i α−1 i=1

Applying the entropy GT sallis = ∑

with the constraints ∑ pi = 1 and ∑ pi Ei = i

i

U, in case of a multifractal structure we obtain a distribution function in the form   1 1 k 1−q Pst (k) = 1 − (1 − q) , (9) Z k0  1 R∞  1−q where Z = 1 − (1 − q) kk0 dk. Thus, we get the Tsallis distribution that has the 0

form of an exponential distribution at q → 1 while at large k it has the form of a powerlaw distribution [5]. In case of the power-law distribution f (y) for a multifractal structure the optimization W

W

(r)

(r)

Sq under the constraints ∑ pi = 1, hO (r) i ≡ ∑ pi Oi = Oγ , (r = 1, ..., R) where i=1n i=1 n o o (r) (r) O are observables and Oγ are finite known quantities, yields "  #γ !  R γ +1 (r) pi = exp − Γ + α + ∑ βr Oi , i = 1, ...,W, γ r=1

(10)

where α and {βr } are the Lagrange multipliers associated with the constraints [4]. So called generalized entropies are usually introduced in the form of ! W

Sc,d [p] = (1 − c + cd)−1 e ∑ Γ(1 + d, 1 − c ln pi ) − c ,

(11)

i

which is a slight generalization of the Anteneodo – Plastino entropy [4, 9]. This allows to classify entropies by using the parameters (c, d), and to obtain the Boltzmann –

Gibbs entropy (c = 1, d = 1), the Shafee entropy (c = β , d = 1), the Tsallis entropy (c = 1, d = 0), and the Anteneodo – Plastino entropy (c = 1, d = η1 ) [9,10]. The distribution function p(ε) = Ec,d,r (−ε) associated with S(c,d) involve the Lambert-W exponentials    d   1/d −Wk (B) , Wk B(1 − x/r) (12) Ec,d,r (x) = exp − 1−c   (1−c)r (1−c)r with the constant B ≡ 1−(1−c)r . The function Wk is the k’th branch of the exp 1−(1−c)r Lambert-W function which has only two real solutions Wk , the branch k = 0 and branch k = −1. The branch k = 0 covers the classes for d ≥ 0, the branch k = −1 those for d < 0. Important special cases of these distributions are exponentials, power-laws and stretched exponential distributions [10].

GENERALIZED STATISTICAL MECHANICS FOR SUPERSTATISTICAL MULTIFACTAL SYSTEMS Now we consider a multifractal structure consisting of fractal subsystems with various values of a certain intensive parameter β . In this case the Fokker – Planck equation for definition of the distribution function for the fractal subsystem is written as [1]   ∂ p(v|β ,t) ∂ 1 ∂ 2 α = − α −γv p(v|β ,t) − σ p(v|β ,t) , (13) ∂t ∂v 2 ∂ vα where α is a dimension of the fractal substructure and in case the fractal dimension tends to unity, the equation (13) coincides with equation (1). Here we have introduced 1−α the notation ∂∂xα = x α ∂∂x . The stationary solution of this equation is defined from the 2α

) condition ∂ p(v|β = 0 and has the form p(ε(α)|β ) = p0 e−β ε(α) , where ε(α) = v2 , ∂t β = σ2γ2 and the parameter p0 is defined from the normalization condition p(ε|β ). Fokker – Planck equation (13) implies that we have a random variable q(t), the behavior of which is defined by the Langevin equation as the following stochastic equation dq = K(qα )dt + Gdw, where w is white noise satisfying the conditions hdwi = 0 and hdw(t1 )dw(t2 )i = δ (t1 − t2 )dt. We note that the function K(qα ) satisfies the condition K(0) = 0. If the distribution of the intensive variable β in substructures of the system is characterized by the distribution χ 2 (2), we obtain for a marginal distribution in the form

Z∞

p (ε(α)) =

  1 v2α 1−q . p(ε|β ) f (β )dβ = 1 + (q − 1)β0 2

(14)

0

Suchwise the basic formula of superstatistics is defined by the expression p(ε) = R∞

p(ε|β ) f (β )dβ where p(ε|β ) is a conditional distribution and f (β ) is some normal-

0

ized distribution for the variable β . Hence the main problem of superstatistics is to define

f (β ). We solve the Fokker – Planck equation to determine the distribution function for multifractal structures 2 ∂ φ (q1 , q2 ,t) ∂ jk (q1 , q2 ,t) =∑ (15) α ∂t ∂ q k k=1 where jk (q1 , q2 ,t) = Kk (q1 , q2 )φ (q1 , q2 ,t) −

1 2 ∂ φ (q1 , q2 ,t) Qkl . ∑ 2 l=1 ∂ qαk

This is a 2-dimensional continuity equation. The derivation of equation (15) implies that there exist two random variables satisfying the Langevin equations dqi = Ki (qα1 , qα2 )dt + Gi dwi , where i = 1, 2, wi is white noise satisfying the conditions hdwi i = 0 and hdwi dw j i = δi j dt. Let us discuss a stationary solution of equation (15). In this case ∂∂tφ = 0 and after substitution of the variables qαk = γk , from the equation (15) we obtain n=2

∂j

∑ ∂ γkk = 0.

(16)

k=1 2

1 ,γ2 ) . Let us introduce the notation Here jk = jk (γ1 , γ2 ) = Kk φ (γ1 , γ2 ) − 21 ∑ Qkl ∂ φ (γ ∂ γl

l=1

2

jk (γ1 , γ2 ) =

∂ hkl , l=1 ∂ γl

∑

(17)

where hkl (γ1 , γ2 ) = h(γ1 , γ2 )φ (γ1 , γ2 )εkl , εkl is the Levi – Civita tensor and consequently ε11 = ε22 = 0 and ε12 = −ε21 = 1. Then from equation (17) we have jk (γ1 , γ2 ) =

∂ h(γ1 , γ2 )φ (γ1 , γ2 )εk1 ∂ h(γ1 , γ2 )φ (γ1 , γ2 )εk2 + ∂ γ1 ∂ γ2

(18)

and hence j1 (γ1 , γ2 ) =

∂ h(γ1 , γ2 )φ (γ1 , γ2 ) ∂ h(γ1 , γ2 )φ (γ1 , γ2 ) , j2 (γ1 , γ2 ) = − . ∂ γ2 ∂ γ1 n

1 ,γ2 ) As jk = Kk φ (γ1 , γ2 ) − 21 ∑ Qkl ∂ φ (γ we obtain ∂ γl

l=1

∂ h(γ1 , γ2 )φ (γ1 , γ2 ) 1 n ∂ φ (γ1 , γ2 ) = K1 (γ1 , γ2 )φ (γ1 , γ2 ) − ∑ Q1l , ∂ γ2 2 l=1 ∂ γl −

∂ h(γ1 , γ2 )φ (γ1 , γ2 ) 1 n ∂ φ (γ1 , γ2 ) = K2 (γ1 , γ2 )φ (γ1 , γ2 ) − ∑ Q2l . ∂ γ1 2 l=1 ∂ γl

(19)

(20)

Further we introduce the notations γ1 = v and γ2 = β . Suppose that v is a fast variable whereas β is a slow one. In this instance an adiabatic approximation allows to suppose that K2 (γ1 , γ2 ) = K2 (γ2 ) that is K1 = K1 (γ1 , γ2 ) and K2 = K2 (γ2 ). We will find a solution of the equations in the form φ (γ1 , γ2 ) = φ˜ (β )e−β ε where ε = 21 v2 . After some simplifications the equations for definition of h(γ1 , γ2 ) and φ (γ1 , γ2 ) ) ˜ (β )e−β 12 v2 = −vβ φ in accord with (19) and (20) and taking into account that ∂ φ∂(v,β v equations (19) and (20) take the form     1 2 1 2 1 ∂ ˜ ∂h 1 Q12 + h(v, β ) φ (β ) = K1 (v, β ) − + Q11 β v + v Q12 + v h(v, β ) φ˜ (β ), 2 ∂β ∂β 2 4 2 (21)   1 ∂ ˜ ∂h 1 1 2 (22) Q22 φ (β ) = K2 (β ) + − vβ h(v, β ) + Q21 β v + v Q22 φ˜ (β ). 2 ∂β ∂v 2 4 √ β φ (β ) ∂ ˜ We require that ∂ β φ (β ) = Ξ(β )φ˜ (β ). The function φ˜ (β ) = √2π can be defined from this equation. An immediate substitution into equation (22) leads to the equation for definition of φ (β )   1 ∂ φ (β ) = Ξ(β ) − φ (β ). (23) ∂β 2β Equations (21) and (22) for definition of f˜(β ) and h(v, β ) functions take the form     ∂h 1 1 2 1 2 1 Q12 + h(v, β ) Ξ(β ) = K1 (v, β ) − + Q11 β v + v Q12 + v h(v, β ) , (24) 2 ∂β 2 4 2   1 ∂h 1 1 2 Q22 Ξ(β ) = K2 (β ) + − vβ h(v, β ) + Q21 β v + v Q22 . (25) 2 ∂v 2 4 After differentiation of equation (25) with respect to v we obtain   1 1 ∂ ∂h − vβ h(v, β ) + Q21 β + vQ22 = 0. ∂v ∂v 2 4

(26)

The solution of this equation can be found in the form h = a + cv.

(27)

An immediate substitution of (27) into equation (26) and then from the comparison of the coefficients of the same powers v we obtain 1 1 h = Q21 + Q22 v. 2 8β p− 1

(28)

Consider some special case. Let Ξ(β ) = − β1 + β 2 where β0 and p are positive 0 constants. Then equation (26) can be written in the following way

    p−1 ∂ φ (β ) 1 1 = Ξ(β ) − φ (β ) = − + φ (β ). ∂β 2β β0 β − ββ

0

β p−1 . From the normalization

β p−1 ,

(30)

The solution of this equation has the form φ (β ) = ce condition we finally find − ββ

√

φ (β ) = Γ−1 (p)e

0

(29)

β φ (β )

or φ (γ1 , γ2 ) = √2π e−β ε , qαk = γk , γ1 = v and γ2 = β . Thus after integration over β we arrive at distribution function (14). Note that at α = 1 we obtain the results of [8]. It should be stressed that equation (1) represents the Fokker – Planck equation for homogeneous substructures. However in conformity with (2) averaging over β leads to q-exponential distribution (3). On the other hand within the framework of the maximum entropy approach and applying the Tsallis entropy characteristic of a multifractal structure we also arrive at q-exponential distribution (3). It means that due to the distribution β in the structure in accord with a gamma-distribution the entire system manifests itself as a multifractal.

CONCLUSION We have introduced the equation of Langevin type for two random variables for the fractal medium. The joint evolution of these parameters is described by the Fokker – Planck equation as (13). We have demonstrated that in case of a special choice of the parameters of the Fokker – Planck equation the distribution of the intensive parameter of the system is described by a gamma-distribution. In this instance the behavior of the entire system is described by the q-exponential distribution. The other implementations of the intensive parameter may lead to more complicated distributions. We have shown that generalization of the Boltzmann – Gibbs entropy for fractal structures leads us to the Shafee entropy. For a multifractal structure an appropriate entropy of the system depends on the distribution of the dimensions of fractal substructures and specification of this function may lead to various generalized entropies.

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