Kaniadakis Statistics and the Quantum H-Theorem ∗ A.P

Physics Letters A 375 (2011) 352–355

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Kaniadakis statistics and the quantum H-theorem ∗ A.P. Santos a,R.Silva a,b, ,J.S.Alcaniz c, D.H.A.L. Anselmo a a Universidade Federal do Rio Grande do Norte, Departamento de Física, Natal-RN, 59072-970, Brazil b Departamento de Física, Universidade do Estado do Rio Grande do Norte, Mossoró-RN, 59610-210, Brazil c Departamento de Astronomia, Observatório Nacional, Rio de Janeiro-RJ, 20921-400, Brazil article i nfo a bstract

Q Article history: A proof of the quantum H-theorem in the context of Kaniadakis’ entropy concept Sκ and a generaliza- Received 16 August 2010 tion of stosszahlansatz are presented, showing that there exists a quantum version of the second law Received in revised form 13 October 2010 of thermodynamics consistent with the Kaniadakis statistics. It is also shown that the marginal equi- Accepted 16 November 2010 librium states are described by quantum κ-power law extensions of the Fermi–Dirac and Bose–Einstein Available online 19 November 2010 distributions. Communicated by C.R. Doering © 2010 Elsevier B.V. Open access under the Elsevier OA license. Keywords: Statistical mechanics Fermion Boson H-theorem

Non-Gaussian statistics are generalizations of the standard ponential and logarithmic functions ∀κ = 0. We refer the reader Boltzmann–Gibbs statistics aiming at describing a number of phys- to [4] for some studies on the above approach in both theoretical ical systems that present some restrictions to the applicability of and experimental contexts. A discussion on the thermodynamical the usual statistical mechanics. foundations of this formalism can be found in [5] and a study in In this concern, the Kaniadakis statistics has been proposed the perspective of quantum statistics has been presented in [6,7]. as a real possibility in this direction [1] (see also [2] for other Indeed, in order to introduce the effects of the Kaniadakis frame- approaches). Recently, efforts on the kinetic foundations of the Ka- work in quantum statistics, it has been considered the power law niadakis statistics have led to a power-law distribution function generalization of Boltzmann factor which is quantified by parame- and the κ-entropy which emerges in the context of the so-called ter κ [7]. Physically, the deformation of the Boltzmann factor can kinetic interaction principle (see Ref. [3] for a review on this sub- be associated with the statistical correlations (for Tsallis statistics, ject). see [14]). From the mathematical viewpoint, the κ-framework is based on In this Letter, we are particularly interested in thermodynami- κ-exponential and κ-logarithm functions, defined as cal aspects, especially in the so-called quantum H-theorem.1 The aim of this Letter is to derive a proof of the quantum H-theorem 1/κ by including the effects of Kaniadakis statistics on the quantum = + 2 2 + expκ( f ) 1 κ f κ f , (1) entropy S Q , as well as by introducing statistical correlations on a − collisional term of Boltzmann equation, i.e., the κ-generalization of f κ − f κ lnκ ( f ) = , (2) stosszahlansatz. Additionally, we propose that the stationary states 2κ of a quantum gas are simply described by a κ-power law extension of the usual Fermi–Dirac and Bose–Einstein distributions. with exp [lnκ ( f )]= f .The κ-parameter lies in the interval 0 κ We start with the main results of the standard H-theorem in κ < 1 and, as the real index κ → 0, the above expressions re- quantum statistical mechanics, namely, the specific functional form produce the usual exponential and logarithmic properties, so that Eqs. (1) and (2) constitute direct generalizations of the usual ex-

1 In the quantum domain, the ﬁrst derivation of H-theorem was done by Pauli [10], who showed that the change of entropy with time (as a result of molec- * Corresponding author at: Universidade Federal do Rio Grande do Norte, Depar- ular collision) provides the equilibrium states which are described by Bose–Einstein tamento de Física, Natal-RN, 59072-970, Brazil. and Fermi–Dirac distributions. Some proofs of the H-theorem, taking into account E-mail addresses: [email protected] (A.P. Santos), non-extensive effects under the chaos molecular hypothesis and entropy, have been [email protected] (R. Silva), [email protected] (J.S. Alcaniz), discussed in the non-relativistic and relativistic domains in Refs. [11–13],as well as [email protected] (D.H.A.L. Anselmo). in the quantum domain [14].

2 for the entropy and the well-known expression for occupation Cκ (nα) number, which is the rule of counting quantum states in the case (gμ ± nμ)(gν ± nν)(gα ± nα)(gλ ± nλ) of Bose–Einstein and Fermi–Dirac [8] =− Aαλ,μν nα λ,(μν) Q S =− n ln(n ) ∓ (g ± n ) ln(g ± n ) ± g ln(g ) , κ α α α α α α α α × nα ⊗ nλ α gα ± nα gλ ± nλ (3a) (gα ± nα)(gλ ± nλ)(gμ ± nμ)(gν ± nν) gα + Aμν,αλ nα = , (3b) nα γ +βα ∓ λ,(μν) e 1 nμ κ nν where gα is the number of states, β inverse of thermal energy and × ⊗ . (9) γ a constant, the upper sign refers to bosons, and the lower one gμ ± nμ gν ± nν to fermions. These two expressions are the main statistical ingre- The sum above spans over all groups λ and also over all pairs of dients of the quantum H-theorem. As is well known, the evolution groups ( ). Also, we make a double inclusion of those terms Q μν of S with time as a result of molecular collisions leads to the oc- in the summation for which λ = α. In the sum above, the cupation number nα, i.e., this quantity increases with time towards standard product between the distributions is replaced by the an equilibrium value as a result of molecular collision: κ-generalization from the molecular chaos hypothesis and nα,de- dSQ fined in Eq. (16), is introduced by mathematical convenience. Note 0. (4) that, in the limit κ → 0, the above expression reduces to the stan- dt dard case [8] Let us now consider a spatially homogeneous gas of N parti- =− ± ± cles (bosons or fermions) enclosed in a volume V . The state of a C0(nα) Aαλ,μνnαnλ(gμ nμ)(gν nν) λ,(μν) quantum gas is characterized by the occupation number nα.Inthis case, the time derivative of the occupation number nα is given by + Aμν,αλnμnν(gα ± nα)(gλ ± nλ), (10) considering collisions of pairs of particles, where a pair of particles λ,(μν) goes from a group α,λ to another group μ, ν. The standard the- ory [8] shows us that the expected number of collisions per unit with the molecular chaos hypothesis and the standard dnα/dt of time is given by readily recovered. Let us now assume the generalized entropic measure,4 = ± ± Zαλ,μν Aαλ,μνnαnλ(gμ nμ)(gν nν). (5) Q nα gα S =− nα lnκ ± gα lnκ , (11) The collisions in the sample of gas in a condition specified by tak- κ g ± n g ± n α α α α α ing nα,nλ,nμ,nν as the numbers of particles in different possible Q Q groups of gα, gλ, gν, gμ, are described by Zαλ,μν . The coefficient where we use the functionals Hκ =−Sκ /kB with kB = 1. Note Aαλ,μν must satisfy the relation once again that, in the limit κ → 0, the expression above reduces to the standard case (3). A = A , (6) αλ,μν μν,αλ Before proceeding with our proof of H-theorem, let us consider which determines the frequency of collisions that are inverse to some properties of the so-called κ-algebra, i.e., [1] those considered, in other words, collisions in which particles are κ x ⊕ y := x 1 + κ2 y2 + y 1 + κ2x2, (12a) thrown from μ, ν to α,λ instead of from α,λ to μ, ν. This coefficient must have a value close to zero for collisions which do not κ κ ⊕ − = := + 2 2 − + 2 2 satisfy the energy partition: x ( y) x y x 1 κ y y 1 κ x , (12b) + = + so that Eq. (11) is rewritten as μ ν α λ. (7) Q =− + 2 ± 2 By considering that the temporal evolution of the distribution nα Sκ nα lnκ (nα) 1 κ lnκ (gα nα) is affected by κ-statistical correlations introduced in the collisional α term trough the generalization of stosszahlansatz, we may assume − ± + 2[ ]2 3 lnκ (gα nα) 1 κ lnκ nα the following quantum κ-transport equation 2 dnα ∓ gα lnκ (gα) 1 + κ2 lnκ (gα ± nα) = Cκ (nα). (8) α dt Here, Cκ defines the quantum κ-collisional term. Our main goal − lnκ (gα ± nα) 1 + κ2[lnκ gα]2 . (13) is to show that a generalized collisional term Cκ (nα) leads to a non-negative expression for the time derivative of the κ-entropy Now, by combining the κ-properties discussed in Ref. [1], i.e., (see Eq. (11)), and that it does not vanish unless the distribution κ ln (x ⊗ y) := ln (x) + ln (y), (14a) function assumes the stationary form associated with quantum κ κ κ κ-distributions [1]. κ x lnκ (x) lnκ (y) := lnκ (14b) Let us now introduce the κ-collisional term that must lead to a y non-negative rate of change of quantum κ-entropy: and the expression (13), we obtain the temporal evolution for the κ-entropy: 2 We assume a gas appropriately specified by regarding the states of energy for a single particle in the container as divided up into groups of gα neighboring states, 4 and by stating the number of particles nα assigned to each such group α. By following, the results of the paper [9], is possible to show the physical vi- 3 For similar arguments in the so-called non-extensive Tsallis framework, see ability between the combinatorial structure of W±,κ (κ-generalization of Stirling Q = ∞ Refs. [8,9]. approximation) and the quantum entropy give by Eq. (11). i.e., Sκ lnκ (W±,κ ). 354 A.P. Santos et al. / Physics Letters A 375 (2011) 352–355

Q dSκ κ dnα By taking the arithmetical mean of this expression with the = nα lnκ (nα) lnκ (gα ± nα) dt dt equivalent result obtained by interchanging the pair of indices α (α,λ) with the pair (μ, ν), we obtain nα dnα = nα lnκ , (15) Q g ± n dt dSκ 1 α α α = Aαλ,μν(gμ ± nμ)(gν ± nν)(gα ± nα)(gλ ± nλ) dt 2 (αλ),(μν) where − 1 × nα + nλ = + nα + − lnκ lnκ nα 1 lnκ (Aα Bα Dα) (16) gα ± nα gλ ± nλ gα ± nα nμ nν and − lnκ − lnκ g ± n g ± n κ + −κ μ μ ν ν (nα nα ) 2 Aα = 1 + κ2 lnκ (gα − nα) , (17a) nα κ nλ nμ κ nν 2 × ⊗ − ⊗ . (20) g ± n g ± n g ± n g ± n (g ln (g ) − n ln (n )) ln (g − n ) α α λ λ μ μ ν ν = 2 α κ α α κ α 2κ α α Bα κ The summation in the above equation is never negative because (gα − nα) 1 + κ2[lnκ (gα − nα)]2 the terms g j ± n j with j = μ, ν, α,λ are always positive and ln2κ (nα) lnκ (gα − nα) g n accounts for the Pauli exclusion principle. Note also that − , (17b) j j 1 + κ2(ln (n ))2 by defining κ α − κ + − −κ nα κ nλ (gα nα) (gα nα) X := ⊗ (21a) Dα = ± ± 2(g − n ) gα nα gλ nλ α α and × + 2 2 gα 1 κ lnκ (gα) n κ n := μ ⊗ ν Y , (21b) 2 gμ ± nμ gν ± nν − nα 1 + κ2 lnκ (nα) . (17c) we can show that the function In particular, when κ → 0, we have nα = 1 and the standard cal- Φ(X, Y ) = (X − Y ) · lnκ (X) − lnκ (Y ) (22) culation is fully recovered. Now, by substituting (9) into (15),wefind is also a positive quantity. Therefore, the quantum κ-entropy is an increasing function of time, i.e., dSQ κ = ± ± ± ± Q Aαλ,μν(gμ nμ)(gν nν)(gα nα)(gλ nλ) dSκ dt 0 (23) α λ,(μν) . dt nα κ nλ nα Therefore, this inequality states that the quantum κ-entropy must × ⊗ lnκ gα ± nα gλ ± nλ gα ± nα be positive or zero, thereby furnishing a quantum derivation of the second law of thermodynamics in the κ-statistic. − ± ± Aμν,αλ(gα nα)(gλ nλ) In order to make the proof of the quantum H-theorem con- α λ,(μν) sistent with the κ-statistics, let us now calculate the generalized × (gμ ± nμ)(gν ± nν) Fermi–Dirac and Bose–Einstein distributions, which recover the stationary distribution previously obtained by a maximization of nμ κ nν nα Q × ⊗ κ-entropy [4]. As happens in the standard case, dS /dt = 0isa ± ± lnκ ± (18) κ gμ nμ gν nν gα nα necessary and sufficient condition for local and global equilibrium. with the summations including all groups α and λ and all pairs of From Eq. (20), we note that the condition must be satisfied, if and groups (μν). only if Q Following standard lines [8],werewritedSκ /dt in a more sym- nα nλ metrical form. First, note that changing to a summation over all lnκ + lnκ gα ± nα gλ ± nλ pairsofgroups(α,λ) does not affect the value of the sum. This happens because the coefficients satisfy the equality for inverse nμ nν = lnκ + lnκ . (24) collisions [see Eq. (6)]. By implementing these operations we have gμ ± nμ gν ± nν Q Here, for a null value of this rate of change, the expression Eq. (24) dSκ = Aαλ,μν(gμ ± nμ)(gν ± nν)(gα ± nα)(gλ ± nλ) satisfies the energy relation (7) for collisions with appreciable dt (αλ),(μν) value of A . As a matter of fact, the above sum of κ-logarithms αλ,μν remains constant during a collision, i.e., it is a summational invari- nα κ nλ × ⊗ ant. In the quantum regime, the solution of these equations is an g ± n g ± n α α λ λ expression of the form

× nα + nλ lnκ lnκ nα + + = gα ± nα gλ ± nλ lnκ γ βα 0, (25) gα ± nα − ± ± Aμν,αλ(gμ nμ)(gν nν) where γ and β are constants independent of α. After some alge- (αλ),(μν) bra, we may rewrite Eq. (25) as the quantum κ-distribution nμ κ nν gα × (gα ± nα)(gλ ± nλ) ⊗ nα = , (26) g ± n g ± n exp (γ + β ) ± 1 μ μ ν ν κ α n n where exp (x) is the κ-exponential function defined in Eq. (1).The × α + λ κ lnκ lnκ . (19) above expression, which coincides, for g = 1, with the κ-occupa- gα ± nα gλ ± nλ α A.P. Santos et al. / Physics Letters A 375 (2011) 352–355 355 tion number derived in Ref. [1,6], seems to be the most general G. Kaniadakis, Phys. Rev. E 66 (2002) 056125; expression which leads to a vanishing rate of change, and clearly G. Kaniadakis, Phys. Rev. E 72 (2005) 036108. reduces to Fermi–Dirac and Bose–Einstein occupation number in [2] M. Gell-Mann, C. Tsallis (Eds.), Nonextensive Entropy – Interdisciplinary Appli- → cations, Oxford Univ. Press, New York, 2004; the limit κ 0. C. Beck, E.G.D. Cohen, Physica A 322 (2003) 267; Summingup,wehaveinvestigatedaκ-generalization of the V. Badescu, P.T. Landsberg, J. Phys. A 35 (2002) L591; quantum H-theorem based on the Kaniadakis statistics. We have V. Badescu, P.T. Landsberg, Complexity 15 (3) (2002) 19. shown that the Kaniadakis statistics can be extended in order to [3] G. Kaniadakis, Eur. Phys. J. B 70 (2009) (special number). [4] S. Abe, G. Kaniadakis, A.M. Scarfone, J. Phys. A: Math. Gen. 37 (2004) 10513; achieve the occupation numbers concepts of the quantum statis- = G. Kaniadakis, Phys. Lett. A 288 (2001) 283; tical mechanics. We notice, for gα 1, that the quantum κ-dis- G. Kaniadakis, A.M. Scarfone, Physica A 305 (2002) 69; tributions [Eq. (26)] reproduces the result originally obtained in G. Kaniadakis, P. Quarati, A.M. Scarfone, Physica A 305 (2002) 76; Ref. [1,6]. It is worth mentioning that a different expression for G. Kaniadakis, M. Lissia, A.M. Scarfone, Physica A 340 (2004) 41; Q n G. Kaniadakis, M. Lissia, A.M. Scarfone, Phys. Rev. E 71 (2005) 046128; the quantum entropy, given by Sκ = d v σκ ( f ), where σκ ( f ) = − + A. Rossani, A.M. Scarfone, J. Phys. A: Math. Gen. 37 (2004) 4955; df lnκ ( f /( f 1)) has been studied in Ref. [6] in the context of M. Cravero, G. Iabichino, G. Kaniadakis, E. Miraldi, A.M. Scarfone, Physica A 340 the maximal entropy principle. In their analysis, the authors calcu- (2004) 410; lated the κ-generalization of Bose–Einstein stationary occupation D. Rajaonarison, D. Bolduc, H. Jayet, Econ. Lett. 86 (2005) 13; A.M. Scarfone, Phys. Rev. E 71 (2005) 051103; number, which is similar to our expression (26) for gα = 1. J.M. Silva, R. Silva, J.A.S. Lima, Phys. Lett. A 372 (2008) 5754; Finally, it is worth emphasizing that this work seems to com- J.C. Carvalho, R. Silva, J.D. do Nascimento Jr., J.R. De Medeiros, Europhys. Lett. 84 plement a series of studies on the compatibility between Kani- (2008) 59001; adakis statistics and the Boltzmann H-theorem and shows, to- J.C. Carvalho, J.D. do Nascimento Jr., R. Silva, J.R. De Medeiros, Astrophys. J. gether with Refs. [1,15], that an Hκ -theorem can be derived in Lett. 696 (2009) L48. non-relativistic, relativistic and also in a quantum regimes. Also, [5] J. Naudts, Physica A 316 (2002) 323; Y. Yamano, Phys. Lett. A 308 (2003) 364; our formalism is very general, since our assumptions can be ap- T. Wada, Physica A 340 (2004) 126; Q plied to any ensemble whose quantity Sκ can be defined and cal- G. Kaniadakis, A.M. Scarfone, Physica A 340 (2004) 102. culated within the framework of Kaniadakis statistics, and whose [6] A. Aliano, G. Kaniadakis, E. Miraldi, Physica B 325 (2003) 35; value of equilibrium (steady state) is obtained by letting this sys- A.M. Teweldeberhan, H.G. Miller, R. Tegen, Int. J. Mod. Phys. E 12 (2003) 699. [7] F.I.M. Pereira, R. Silva, J.S. 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