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1

arXiv:cond-mat/0509364v1 [cond-mat.stat-mech] 14 Sep 2005

Canonical partition function for anomalous systems described by the κ-entropy

Antonio M. Scarfone1,?) and Tatsuaki Wada2, ??) Nazionale di Fisica della Materia (INFM) and Dipartimento di Fisica Unit? a del Politecnico di Torino, Corso Duca degli Abruzzi 24, I-10129 Torino, Italy

2 Department 1 Istituto

of Electrical and Electronic Engineering, Ibaraki University, Hitachi, Ibaraki, 316-8511, Japan

Starting from the κ-distribution function, obtained by applying the maximal entropy principle to the κ-entropy [G. Kaniadakis, Phys. Rev. E 66 (2002), 056125], we derive the expression of the canonical κ-partition function and discuss its main properties. It is shown that all important macroscopical quantities of the system can be expressed employing only the κ-partition function. The relationship between the associated κ-free energy and the κ-entropy is also discussed.

§1.

Introduction

Anomalous statistical systems which exhibit an asymptotic power law behavior in the probability distribution function (pdf) are ubiquitous in nature.1) – 3) Although not in equilibrium these distributions characterize a metastable con?guration in which the system remains for a very long period of time compared to the typical time scales of its underlying microscopical dynamics. The statistical properties of such systems can be investigated trough the introduction of a generalization of the Boltzmann-Gibbs (BG) entropy S B = ? i pi ln(pi ) with p ≡ {pi }i=1, ··· , N a discrete pdf, (throughout this paper we adopt units within kB = 1) from which, by means of the maximal entropy principle, the corresponding pdf can be derived. Recently, in4) – 6) it has been proposed a generalized statistical mechanics based on the deformed entropy Sκ = ?

i

pi ln{κ} (pi ) ,

(1.1)

which preserves the structure of the BG statistical mechanics. Eq. (1.1) mimics the well-known classical entropy by replacing the standard logarithm with the generalized version ln{κ} (x) = xκ ? x?κ , 2κ (1.2)

where ?1 < κ < 1, ln{κ} (x) = ln{?κ} (x) and ln{κ} (x) = ? ln{κ} (1/x). Eq. (1.2) reduces to the classical logarithm in the κ → 0 limit: ln{0} (x) = ln(x), as well as, in

?) ??)

antonio.scarfone@polito.it wada@ee.ibaraki.ac.jp

typeset using P T P TEX.cls Ver.0.9

2

A.M. Scarfone and T. Wada

the same limit, Eq. (1.1) reduces to the BG entropy. The κ-entropy (1.1) has many properties of the standard entropy like positivity, continuity, symmetry, expansibility, decisivity, maximality and concavity.8) Moreover, in9) it has been shown that the κ-entropy is also Lesche stable, an important property that must be ful?lled in order to represent a well de?ned physical observable. The maximization of Eq. (1.1) under the constraints on the normalization and on the mean energy pi = 1 , Ei pi = U , (1.3)

i i

leads to the following κ-distribution function6) 1 pi = α exp{κ} ? (γ + β Ei ) λ , (1.4)

where γ and β are the Lagrange multipliers associates to the constraints (1.3). The deformed exponential exp{κ} (x), the inverse function of ln{κ} (x), is de?ned as exp{κ} (x) = κ x + 1 + κ2 x2

1/κ

,

(1.5)

and reduces to the standard exponential in the κ → 0 limit: exp{0} (x) = exp(x). The constants λ and α in Eq. (1.4) are given by λ= 1 ? κ2 , α= 1?κ 1+κ

1/2 κ

,

(1.6)

respectively, and are related each other through the relation ln{κ} (α) = ?1/λ. In6), 10) it has been shown that, starting from the deformed logarithm (1.2) and the deformed exponential (1.5), the κ-algebra can be developed in a way that many algebraic properties of the standard logarithm and exponential can be reproduced κ in the deformed version. For instance, through the de?nition of the κ-sum x ⊕ y , given by κ x ⊕ y = x 1 + κ2 y 2 + y 1 + κ2 x2 , (1.7) it is easy to verify the following useful relation exp{κ} x ⊕ y = exp{κ} (x) · exp{κ} (x) .

κ

(1.8)

In6), 7) it has been shown that the κ-statistical mechanics emerges within the special relativity. The physical mechanism introducing the κ-deformation is originated from the Lorentz transformations and the κ → 0 limit, reproducing the ordinary statistical mechanics, corresponds to the classical limit c → ∞. The purpose of this work is to introduce the generalized canonical partition function Zκ in the framework of the κ-deformed statistical mechanics and to show that all the relevant relations, valid in the BG theory, still hold in the deformed version.

Canonical partition function in anomalous systems

3

We start by recalling the useful relations. Firstly, the κ-logarithm ful?lls the functional-di?erential equation6) – 8) d x [x Λ(x)] = λ Λ dx α , (1.9)

= 1. with the boundary conditions Λ(1) = 0 and (d/d x) Λ(x) x=1 Another solution of Eq. (1.9), which follows from the boundary conditions Λ(1) = 1 = 0, is given by11) and (d/d x)Λ(x)

x=1

u{κ} (x) =

xκ + x?κ . 2

(1.10)

It ful?lls the relations u{κ} (x) = u{?κ} (x), u{κ} (x) = u{κ} (1/x) and u{κ} (α) = 1/λ. From the de?nitions (1.2) and (1.10) we obtain the useful equations ln{κ} (x y ) = u{κ} (x) ln{κ} (y ) + ln{κ} (x) u{κ} (y ) , u{κ} (x y ) = u{κ} (x) u{κ} (y ) + κ2 ln{κ} (x) ln{κ} (y ) , (1.11) (1.12)

showing a deep link between both the functions ln{κ} (x) and u{κ} (x). Starting from Eq. (1.10), in analogy with Eq. (1.1), we introduce the function Iκ =

i

pi u{κ} (pi ) ,

(1.13)

which can also be de?ned as the mean value of u{κ} (x) according to the relation Iκ = u{κ} (p) . The κ-entropy Sκ = ln{κ} (p) as well, can be de?ned as the mean value of ln{κ} (x). It is worth to observe that, by using the de?nitions (1.1) and (1.13), we obtain from Eq. (1.11) the equation11) Sκ (A ∪ B) = Iκ (A) Sκ (B) + Sκ (A) Iκ (B) , (1.14)

stating the additivity rule of the κ-entropy for two statistically independent systems A and B, in the sense of pA∪B = pA · pB . In the κ → 0 limit u{0} (pi ) = 1 and I0 (p) = i pi = 1 so that Eq. (1.14) recovers the additivity rule of the BG entropy. Returning to the κ-distribution, we pose xi = γ + β Ei , so that Eq. (1.4) can be written in pi = α exp{κ} ? = exp{κ} xi 1 = exp{κ} ? λ λ x 1 κ , ⊕ ? i ? λ λ · exp{κ} ? xi λ (1.16) (1.15)

4

A.M. Scarfone and T. Wada

where Eq. (1.8) has been employed. By using the de?nition of the κ-sum the argument in Eq. (1.16) can be rewritten in ? 1 λ ⊕ ?

κ

xi λ

=?

1 λ

xi + λ

1+

κ2 2 x λ2 i

,

(1.17)

and Eq. (1.16) becomes pi = exp{κ} ? 1 λ xi + λ 1+ κ2 2 x λ2 i . (1.18)

On the other hand, by using Eq. (1.12) with x = α and y = exp{κ} (?xi /λ), and taking into account the relation u{κ} exp{κ} ? xi λ = 1+ κ2 2 x , λ2 i (1.19)

which follows from the de?nitions (1.5) and (1.10), we obtain u{κ} (pi ) = 1 λ κ2 x + λ i 1+ κ2 2 x λ2 i , (1.20)

so that Eq. (1.18) can be easily written as pi = exp{κ} ?u{κ} (pi ) ? γ ? β Ei , (1.21)

which is an alternative but equivalent expression of the κ-distribution (1.4).

§2.

Canonical partition function

We recall that in the classical theory the canonical partition function Z is an important quantity that encodes the statistical properties of a system. From the BG distribution 1 pi = e?1?γ e?β Ei = e?β Ei , (2.1) Z it follows that the partition function can be introduced as ln(Z ) = 1 + γ . (2.2)

It depends ?rstly on the Lagrange multiplier β and secondly on the microstate energies Ei which are determined by other macroscopical quantities like the volume or the number of particles. Remarkably, most of the thermodynamical functions of the system can be expressed in terms of the partition function or its derivatives. For instance, the entropy S BG = ln(Z ) + β U , the total energy U = ?d ln(Z )/d β and the free energy F = ? ln(Z )/β .

Canonical partition function in anomalous systems

5

We observe that, by using the expression (1.21), from the de?nition of the κ-entropy we obtain the relation Sκ = ?

i

pi ln{κ} (pi ) =

i

pi uκ (pi ) + γ + β Ei (2.3)

= Iκ + γ + β U ,

which reminds us the classical expression S = 1 + γ + β U , and it is recovered in the κ → 0 limit. According to Eq. (2.3), taking into account its limit for κ → 0 and the classical relationship between the BG entropy and the canonical partition function Z , we are guided to de?ne the canonical κ-partition function Zκ through ln{κ} (Zκ ) = Iκ + γ . (2.4)

Eq. (2.4) reduces to the de?nition (2.2) in the κ → 0 limit. In order to verify the consistency of the de?nition (2.4), we derive its main properties in the framework of the κ-statistical mechanics. Firstly, it is trivial to verify that the entropy (2.3) becomes Sκ = ln{κ} (Zκ ) + β U , (2.5)

which mimics the corresponding classical relationship. Successively, we compute the derivative of the entropy Sκ w.r.t the mean energy d Sκ =? dU =

i

i

d d pi = ?λ pi ln{κ} (pi ) d pi dU

ln{κ}

i

pi α

d pi dU (2.6)

d pi (γ + β Ei ) , dU

where we have taken into account Eqs. (1.4) and (1.9). Under the no-work condition i pi dEi = 0 (consequently dU = i Ei dpi ), and recalling that i d pi = 0, which follows from the normalization condition on the pdf, we obtain d Sκ =β . dU On the other hand, starting from Eq. (2.5) we have d Sκ d dβ = ln{κ} (Zκ ) + U +β , dU dU dU (2.8) (2.7)

and by comparing this relation with Eq. (2.7) it follows that, the following important property d ln (Z ) = ?U , (2.9) d β {κ} κ must hold. Eq. (2.9) can be proved directly from the de?nition (2.4). In fact, by using Eq. (1.9)

6 we obtain

A.M. Scarfone and T. Wada

d d ln{κ} (Zκ ) = (I + γ ) dβ dβ κ d dγ d pi = + p u (p ) d pi i {κ} i dβ dβ

i

=λ

i

u{κ}

p(xi ) α

dγ d p(xi ) d xi + , d xi d β dβ

(2.10)

where p(xi ) ≡ pi . Using in Eq. (2.10) the relation p(xi ) = ?λ u{κ} we ?nally obtain d ln (Z ) = ? d β {κ} κ pi

i

p(xi ) α

d p(xi ) , d xi

(2.11)

dγ + Ei dβ

+

dγ = ?U . dβ

(2.12)

Eq. (2.9) is the dual relation of Eq. (2.5). They state, from one hand that both β and U are canonically conjugate variables, and on the other hand that Sκ is a function of U whereas Zκ is a function of β , like in the BG theory. Accounting for the standard relationships between the free energy and the partition function, we introduce the κ-free energy Fκ = ? 1 ln (Z ) . β {κ} κ (2.13)

It is trivial to observe that the de?nition (2.13) can be obtained by means of a Legendre transformation on the mean energy Fκ = U ? dU 1 Sκ = U ? Sκ , d Sκ β (2.14)

as it follows through Eqs. (2.5) and (2.7). Finally, we observe that the κ-free energy is a function of 1/β as it follows from the relation d Fκ = ?Sκ , (2.15) d(1/β ) which mimics the classical relationship between the free energy and the BG entropy.

§3.

Mean value of an observable

Another interesting relation involving the partition function (2.4) can be obtained by evaluating the derivative of ln{κ} (Zκ ) w.r.t the energy levels Ei , for an equilibrium (meta-equilibrium) state with β ? const. We have d d ln{κ} (Zκ ) = (I + γ ) d Ei d Ei κ

Canonical partition function in anomalous systems =

j

7

d p u p d pj j {κ} j u{κ}

j

d pj dγ + d Ei d Ei d p(xj ) d xj dγ + d xj d Ei d Ei

=λ =?

j

p(xj ) α β δij +

pj

dγ d Ei

+

dγ d Ei

= ?β pi . Thus, if the partition function is known, we can compute the distribution function as 1 d pi = ? ln (Z ) . (3.1) β d Ei {κ} κ All the mean values of the macroscopical quantities associated with the system can be expressed employing the function Zκ : A =

i

pi Ai = ?

1 β

Ai

i

d ln (Z ) . d Ei {κ} κ

(3.2)

We recall that the microsate energies depend on other macroscopical variables like the volume or the number of particles. By following the standard literature,12) let us introduce the canonically conjugate variables through the relation Ai = ? Then, Eq. (3.2) assumes the expression A = 1 d ln (Z ) , β d A {κ} κ (3.4) d Ei . dA (3.3)

which, taking into account the de?nition of κ-free energy (2.13), becomes A =? d Fκ . dA (3.5)

Let us observe that if the energies Ei depend on a parameter ε as Ei = Ei(0) + ε Ai , then the mean value of A is given by A =? or equivalently A = d F (β, ε) . dε κ (3.8) 1 d Zκ (β, ε) , ln β d ε {κ} (3.7) (3.6)

8

A.M. Scarfone and T. Wada

This provides us with an alternative, useful trick for calculating the expected values of an observable. In fact, by adding the eigenvalues Ai of the observable A to the energy levels Ei we can calculate the new partition function Zκ (β, ε) and the mean value A according to Eq. (3.7), and then, we set ε to zero in the ?nal expression. Remark that this is analogous to the source ?eld method used in the path integral formulation of quantum ?eld theory.

§4.

Conclusions

In the framework of the generalized statistical mechanics based on κ-entropy we have derived the canonical partition function Zκ (β ) and studied its main properties. This function plays a relevant role in the formulation of κ-deformed thermostatistics theory based on the entropy Sκ (U ). It has been shown that most of the thermodynamic variables of the system, like the total energy U , the free energy Fκ (1/β ) and the entropy Sκ (U ) can be expressed in terms of the partition function Zκ (β ) or its derivatives. We have shown an algorithm to derive, starting from the expression of the partition function, the mean value of all macroscopical observables associated to the system.

References 1) S. Abe, “Nonextensive Statistical Mechanics and its Applications ”, Y. Okamoto editors, (Springer, 2001). 2) Special issue of Physica A 305, Nos. 1/2 (2002), edited by G. Kaniadakis, M. Lissia, and A. Rapisarda. 3) Special issue of Physica A 340 Nos. 1/3 (2004), edited by G. Kaniadakis, and M. Lissia. 4) G. Kaniadakis, Physica A 296 (2001), 405. 5) G. Kaniadakis, Phys. Lett. A 288 (2001), 282. 6) G. Kaniadakis, Phys. Rev. E 66 (2002), 056125. 7) G. Kaniadakis, Phys. Rev. E 72 (2005), 0261xy. 8) G. Kaniadakis, M. Lissia, and A.M. Scarfone, Phys. Rev. E 71 (2005), 046128. 9) G. Kaniadakis, and A.M. Scarfone, Physica A 340 (2004), 102. 10) G. Kaniadakis, and A.M. Scarfone, Physica A 305 (2002), 69. 11) A.M. Scarfone, and T. Wada, Phys. Rev. E (2005), in press. 12) H.B. Callen, Thermodynamics and an Introduction to Thermostatistics (Wiley, New York, 1985). 13) W. Greiner, L. Neiser, and H. St¨ ocker, Thermodynamics and Statistical Mechanics (Springer, New York, 1994).

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