Physics Letters A 375 (2011) 352–355 Contents lists available at ScienceDirect Physics Letters A www.elsevier.com/locate/pla 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]. 0375-9601/© 2010 Elsevier B.V. Open access under the Elsevier OA license. doi:10.1016/j.physleta.2010.11.045 A.P. Santos et al. / Physics Letters A 375 (2011) 352–355 353 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 coef- ficient 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.