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EPL, 89 (2010) 10004 www.epljournal.org
doi: 10.1209/0295-5075/89/10004
Nonextensive quantum H-theorem
R. Silva1,2(a), D. H. A. L. Anselmo1(b) and J. S. Alcaniz3(c)
1 Universidade Federal do Rio Grande do Norte, Departamento de F́ısica - Natal-RN, 59072-970, Brasil
2 Universidade do Estado do Rio Grande do Norte, Departamento de F́ısica - Mossoró-RN, 59610-210, Brasil
3 Departamento de Astronomia, Observatório Nacional - 20921-400, Rio de Janeiro-RJ, Brasil
received 4 August 2009; accepted in final form 7 December 2009
published online 18 January 2010
PACS 05.30.-d – Quantum statistical mechanics
PACS 03.65.Ta – Foundations of quantum mechanics; measurement theory
PACS 05.70.Ln – Nonequilibrium and irreversible thermodynamics
Abstract – A proof of the quantum H-theorem taking into account nonextensive effects on the
quantum entropy SQq is shown. The positiveness of the time variation of S
Q
q combined with a
duality transformation implies that the nonextensive parameter q lies in the interval [0,2]. It is
also shown that the stationary states are described by quantum q-power law extensions of the
Fermi-Dirac and Bose-Einstein distributions. Such results reduce to the standard ones in the
extensive limit, thereby showing that the nonextensive entropic framework can be harmonized
with the quantum distributions contained in the quantum statistics theory.
Copyright ©c EPLA, 2010
Introduction. – Boltzmann’s famous H-theorem, elements, multifractality of phase-space and long-range
which guarantees positive-definite entropy production interactions. In this regard, the nonextensive statistical
outside equilibrium, also describes the increase in the mechanics (NESM) framework proposed by Tsallis [8] is
entropy of an ideal gas in an irreversible process, by based on the nonadditive q-entropy
considering the Boltzmann equation. Roughly speaking, ∑W
1− pq
this theorem implies that in the thermodynamical equi- Sq = k
i=1 i , (1)
librium the distribution function of an ideal gas evolves q− 1
irreversibly towards Maxwellian equilibrium distribu- where k is a positive constant, W is the number of micro-
tion [1]. In the special relativistic domain, the very first scopic states, and pi is a normalized probability distribu-
derivation of this theorem was done by Marrot [2] and, in tion. In this approach, additivity for two probabilistically
the local form, by Ehlers [3], Tauber and Weinberg [4] and independent subsystems A and B is generalized by the
Chernikov [5]. As well known, the H-theorem furnishes following pseudo-additivity:
the Juttner distribution function for a relativistic gas Sq(A,B) Sq(A) Sq(B) Sq(A)Sq(B)
in equilibrium, which contains the number density, the = + +(1− q) . (2)
k k k k
temperature, and the local four-momentum as free para-
meters [6]. In the quantum domain, the first derivation For subsystems that have special probability correlations,
was done by Pauli [7], which showed that the change of extensivity may be no longer valid, so that a more realistic
entropy with time as a result of collisional equilibrium description may be provided by the Sq form with a
states are described by Bose-Einstein and Fermi-Dirac particular value of the index q 
= 1, called the q-entropic
distributions. parameter. In the limit q→∑1, not only the Boltzmann-W
Recently, a considerable effort has been done toward Gibbs (BG) entropy S1 = k i=1 pi ln pi is fully recovered,
the development of a generalization of thermodynamics but so is the additivity property for the subsystems A and
and statistical mechanics aiming at better understanding a B above, i.e., SBG(A,B) = SBG(A)+SBG(B).
number of physical systems that possess exotic properties, Several consequences of this generalized framework have
such as broken ergodicity, strong correlation between been investigated in the literature [9] and we refer the
reader to ref. [10] for a regularly updated bibliography.
(a)E-mail: raimundosilva@dfte.ufrn.br In particular, it is worth mentioning that the proofs
(b)E-mail: doryh@dfte.ufrn.br of both the nonrelativistic and relativistic nonextensive
(c)E-mail: alcaniz@on.br H-theorem have been discussed in refs. [11–13].
10004-p1
R. Silva et al.
The aim of this letter is twofold. First, to derive a particles are thrown from µ, ν to κ, λ instead of from κ, λ
proof of the quantum H-theorem by including nonexten- to µ, ν. This coefficient must have a value close to zero for
sive effects on the quantum entropy SQ in the Tsallis collisions which do not satisfy the energy partition:
formalism, as well as by considering statistical correlations
under a collisional term from the quantum Boltzmann µ + ν = κ + λ. (6)
equation1. Second, to obtain from this proof a natural
generalization of the quantum Bose-Einstein and Fermi- By taking into account the idea that the temporal evolu-
Dirac distributions. It is shown that the stationary states tion of the distribution nκ is affected by the nonextensive
4
are simply described by a q-power law extension of the effect , we may assume the following quantum q-transport
usual Fermi-Dirac and Bose-Einstein distributions. From equation:
the positiveness of the rate dSQq /dt we also discuss possi-
dnκ
=Cq(nκ), (7)
ble constraints on the dimensionless index q. dt
where Cq denotes the quantum q-collisional term. As CQuantum H-theorem. – Let us start by presenting q
must leads to a non-negative rate of change of quantum
the main results of the standard H-theorem in quantum-
entropy, its general form reads
statistical mechanics. The first one is a specific functional
form for the entropy2, which is expressed by the logarith- ∑
C (n ) = − A (g ±n )(g ±n )
mic measure [1] q κ κλ,µν µ µ ν ν
∑ λ,(µν)
SQ
n n
=− [nκ lnnκ ∓ (gκ ±nκ) ln(gκ ±nκ)± gκ ln gκ]. × (gκ ±nκ)(gλ ± κ λnλ)
∑ gκ ±
⊗q∗
n
κ κ
gλ ±nλ
(3) + Aµν,κλ(gκ ±nκ)(gλ ±nλ)
The second one is the well-known expression for quan-
λ,(µν)
tum distributions, which is the rule of counting of quantum nµ nν
states in the case of Bose-Einstein and Fermi-Dirac gases × (gµ ±nµ)(gν ±nν) ⊗q∗ ,gµ ±nµ gν ±nν
gκ (8)
nκ = . (4)
eα+βκ ∓ 1
where the sum above spans over all groups λ and also over
These two statistical expressions are the pillars of the all pairs of groups (µν). Also, we make a double inclu-
quantum H-theorem. As is well known, the evolution of sion of those terms in the summation for which λ= κ.
SQ with time as a result of molecular collisions leads to In the sum above, the standard product between the
the quantum distributions nκ. distributions (molecular chaos hypothesis) is replaced by
Proof of Hq-theorem. In order to study the influence the generalized form of the molecular chaos hypothesis,
of the NESM on the quantum H-theorem, let us now i.e, the q-product between the distributions (For similar
consider a spatially homogeneous gas of N particles arguments on the generalization of stosszahlansatz, see
(bosons or fermions) enclosed in a volume V . In this case, ref. [14]). Note that, in the limit q→ 1, the above expres-
the time derivative of the particle number nκ is given sion reduces to
by considering collisions of pairs of particles, where a ∑
pair of particles goes from a group κ, λ to another group C1(nκ) = − Aκλ,µνnκnλ(gµ ±nµ)(gν ±nν)
µ, ν. Here, the expected number of collisions per unit λ∑,(µν)
of time is given by3 Zκλ,µν =Aκλ,µνnκnλ(gµ ±nµ)(gν ± + Aµν,κλnµnν(gκ ±nκ)(gλ ±nλ), (9)
nν), where, as before, the upper sign refers to bosons, and λ,(µν)
the lower one to fermions. The coefficient Aκλ,µν must
satisfy the relation thereby showing that the molecular chaos hypothesis and
the standard dnκ/dt are readily recovered.
Aκλ,µν =Aµν,κλ, (5) Now, we introduce the generalized entropic measure
defined in5, i.e.,
which in turn determines the frequency of collisions that
are inverse to those considered, i.e., collisions in which ∑
SQq =− nq q qκ lnq nκ ∓ (gκ ±nκ) lnq(gκ ±nκ)± gκ lnq gκ,
1In nonextensive kinetic framework, this is equivalent to a κ
generalization of the molecular chaos hypothesis. (10)
2In this context, we assume a gas appropriately specified by
regarding the states of energy for a single particle in the container 4In nonextensive kinetic theory viewpoint, this effect corresponds
as divided up into groups of gκ neighboring states, and by stating to introduce statistical correlations in the collisional term of the
the number of particles nκ assigned to each such group κ. Boltzmann equation through the generalization of stosszahlansatz.
3In other words, the collisions in the sample of gas in a condition For details, see refs. [10,11] by adopting the Tsallis framework.
specified by taking nκ, nλ, nµ, nν , . . . as the numbers of particles in
5This is a q-quantum entropy which generalize the standart one.
different possible groups of gκ, gλ, gµ, gν , . . . , elementary states, are By considering the fermions and gκ = 1 this expression provides the
described quantitatively by Zκλ,µν . (For details see ref. [1].) equation of ref. [15].
10004-p2
Nonextensive quantum H -theorem
where we use the functionals Hq =−SQq /k. The general- where the summations include all groups κ and λ and all
ized q-logarithm is defined by [8] pairs of groups (µν).
1− − In order to rewrite dS
Q
q /dt in a more symmetrical formx q 1
ln (x) := − , (11) some elementary operations must be done in the aboveq 1 q expression. Following standard lines [1], we first notice
whose inverse function is given by the q-exponential that changing to a summation over all pairs of groups
function (κ, λ) does not affect the value of the sum. This happens
exp (x) := [1− (1− q)x]1/(1−q). (12) because the coefficients satisfies the equality for inverseq
→ collisions (see eq. (5)). By implementing these operationsNote that, when q 1, eq. (10) reduces to the standard
and symmetrizing the resulting expression, dSQq /dt can becase (3).
Q rewritten asBy taking the time derivative of Sq , we obtain
dSQq q ∑
dSQ ∑q − − ± dnκ = Aκλ,µν ñκñλ(gµ ±nµ)= q [lnq∗ nκ lnq∗(gκ nκ)] , (13) dt 2
dt dt (κλ),(µν)
κ ×[(gν ±nν)(gκ ±nκ)(gλ ±nλ)where we have used the transformation fq−1lnqf = ln ]q∗f
∗ − × nκ ⊗ nλ nµ nνwith q = 2 q. Now, we make use of the so-called ± q∗ ± − ± ⊗q∗g n g n g n g ±n
q-algebra, introduced in ref. [16], and define the [ κ κ λ λ µ µ ν ν
q-difference and the q-product, respectively, as × nκ nλlnq∗ + lnq∗
 x−
gκ ±nκ gλ ±nλ
y
x q∗ y := − ∗ , ∀ y=

1 ]
− ∗ , (14a) n n1+ (1 q )y 1 q − µ νlnq∗ − lnq∗
[ ] gµ ±nµ gν ±
. (20)
nν
1
⊗ ∗ ∗ 1−q∗x q∗ y := x1−q + y1−q − 1 , x, y > 0, (14b) Note that the summation in the above equation is
and the ln of a q-product and of a quotient never negative, because the terms ñκ, ñλ and gj ±nj withq
j = µ, ν, κ, λ are always positive and gj 
 nj on account for
lnq∗(x⊗q∗ y) := lnq∗(x)+ ln(q
∗()y), (15a) the Pauli exclusion principle. Note also that by defining
lnq∗(x) x
nκ nλ
q∗ lnq∗(y) := lnq∗ . (15b) X := ⊗q∗ , (21a)
y gκ ±nκ gλ ±nλ
From definitions (14a) and (15b), we can rewrite the nµ nν
term in sq(uare brac)kets in eq. (13) as
Y := ⊗q∗ , (21b)
gµ ±nµ gν ±nν
nκ lnq∗ nκ − lnq∗(gκ ±nκ) we can show that the function
lnq∗
gκ ± = , (16)nκ ñκ
ϕ(X,Y ) = (X −Y )(lnq∗ X − lnq∗ Y ), (22)
so that eq. (13) reads
Q ∑[ ( )] is also a positive quantity.dSq nκ · dnκ= q lnq∗ ± ñκ , (17) Finally, we note that, for positive values of q, anddt gκ nκ dtκ by considering the duality transformation q∗ = 2− q, i.e.,
q < 2 (as pointed out in ref. [17]), we obtain the quantum
where
Hq-theorem
6
∗
ñκ = 1+ (1− q∗) lnq∗(gκ ±nκ) = 2− (g ±n )q −1κ κ . (18) dSQq 
 0. (23)
dt
Substituting (7) into (17), we arrive at
∑ ∑ Note that, when q < 0 or q > 2, the quantum q-entropydSQq ± is a decreasing function of time. Consequently, it seems= q Aκλ,µν ñκ(gµ nµ)
dt that within the present context the parameter q should
κ λ,(µν)
be restricted to interval [0,2]. Notice also that the entropy
×(gν ±nν)(gκ ±nκ)(gλ ±nλ() ) does not change with time if q= 0. It should be emphasized
× nκ ⊗ nλ · nκ that, in quantum regime, the equivalent constraint on theq∗ lnq∗
gκ∑±n∑κ gλ ±nλ gκ ±nκ nonextensive parameter was also calculated based on the
− ± second law of thermodynamics, i.e., through Clausius’q Aµν,κλñκ(gκ nκ) inequality [18].
κ λ,(µν)
× ± ± ± In order to finalize the proof of the quantumH-theorem,(gλ nλ)(gµ nµ)(gν nν() ) let us now calculate the nonextensive Fermi-Dirac and
× nµ ⊗ nν nκ± q∗ ± · lnq∗ ± , (19)
6It is worth emphasizing that this same interval is also obtained
gµ nµ gν nν gκ nκ in both nonrelativistic and relativistic regimes. See, e.g., [11].
10004-p3
R. Silva et al.
Bose-Einstein distributions. As happens in the extensive equilibrium (steady state) is obtained by allowing this
case, dSQq /dt= 0 is a necessary and sufficient condition for system to evolve in time.
local and global equilibrium. From eq. (20), we note that
the following condition must occur, if and only if ∗ ∗ ∗
nκ nλ nµ nν
lnq∗ ± + lnq∗ ± = lnq∗ ± + lnq∗ ± ,gκ nκ gλ nλ gµ nµ gν nν RS would like to thank the hospitality of the Depar-
(24) tamento de Astronomia of Observatório Nacional/MCT
where for a null value of this rate of change, eq. (24) satis- where part of this work was developed. RS and JSA thank
fies the energy relation (6) for collisions with apprecia- CNPq - Brazil for the grants under which this work was
ble value of Aκλ,µν . Here, the above sum of q-logarithms carried out. DHALA acknowledges financial support from
remains constant during a collision, i.e., it is a summa- Fundação de Amparo à Pesquisa do Estado do Rio Grande
tional invariant. In the quantum regime, the solution of do Norte - FAPERN.
these equations is an expression of the form
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10004-p4
Nonextensive quantum H -theorem
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10004-p5