The $H$-theorem in $κ$-statistics: influence on the molecular chaos hypothesis

0.13 MB·English

by  R. Silva

6 The H-theorem in κ-statistics: influence on the molecular chaos hypothesis
0
0 R. Silva∗†
2
Observat´orio Nacional, Rua Gal. Jos´e Cristino 77, 20921-400 Rio de Janeiro - RJ, Brasil

n (Dated: February 2, 2008)
a
J We rediscuss recent derivations of kinetic equations based on the Kaniadakis’ entropy concept.
Ourprimary objective hereisto derivea kineticalversion ofthesecond law of thermodynamycsin
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suchaκ-framework. Tothisend,weassumeaslightmodificationofthemolecularchaoshypothesis.
2
For the Hκ-theorem, it is shown that the collisional equilibrium states (null entropy source term)
] aredescribedbyaκ-powerlawextensionoftheexponentialdistributionand,asshouldbeexpected,
h all these results reduce to thestandard one in thelimit κ→0.
c
e PACSnumbers: 05.90.+m;05.20.-y;05.45.+b
m
-
t I. INTRODUCTION The κ-statistic is defined by the κ-deformed functions,
a
given by
t
s
t. Over the last few years, a great deal of attention exp (f)=( 1+κ2f2+κf)1/κ, (4)
a has been paid to nonextensive statistic mechanics based κ
m on the deviations of Boltzmann-Gibbs-Shannonentropic p
- measure. Basically, in this extended framework, the key
d point is to substitute the exponential behaviour of the fκ−f−κ
n entropybyapower-lawone(see,e.q. [1]). Recently,sim- lnκ(f)= 2κ , (5)
o
ilar motivations also led at least to two new examples,
c and
[ namely, Abe [2, 3] and Kaniadakis entropies [4, 5]. In
thislatterworks,byusingκ-exponentialandκ-logarithm exp (ln f)=ln (exp (f))=f. (6)
2 functions (see Eqs. (4) and (5)) and the kinetic interac- κ κ κ κ
v
8 tion principle (KIP), Kaniadakis proposed a statistical Asonemaycheck,theabovefunctionsreducetothestan-
2 framework based on κ-entropy [4, 5] dard exponential and logarithm when κ → 0. In partic-
2 ular, this κ-framework leads to a class of one parameter
1 S (f)=− d3vf[a fκ+a f−κ+b ] (1) deformedstructureswithinterestingmathematicalprop-
1 κ Z κ −κ κ erties [6]. Recently, a connection with the generalized
5
Smoluchowski equation was investigated [7], and a fun-
0 where a and b are coefficients so that in limit κ → 0,
κ κ damentaltest,i.e.,theso-calledLeschestabilitywasalso
/
t Eq. (1) reduces to the standard entropy Sκ=0. Expres- checked in the κ-framework [8]. More recently, it was
a
sion (1) is also the most general one that leads to the
m shown that it is possible to obtain a consistent form for
κ-framework.
the entropy (linked with a two-parameter deformations
- Previous works have already discussed some specific
d oflogarithmfunction),whichgeneralizestheTsallis,Abe
n choices for the constants aκ and bκ. For instance, for and Kaniadakis logarithm behaviours [9]. In the experi-
o the pair [aκ = 1/2κ, bκ = 0], it is possible to write the mentalviewpoint,thereexistsomeevidencerelatedwith
c Kaniadakis entropy as [3, 4]
the κ-statistic, namely, cosmic rays flux, rain events in
:
v meteorology[6], quark-gluonplasma [10], kinetic models
Xi Sκ =− d3vflnκf =−hlnκ(f)i, (2) describing a gas of interacting atoms and photons [11],
Z
fracture propagationphenomena [12], and income distri-
r
a bution [13], as well as constructfinancial models [14]. In
whichhasaperfectanalogywiththestandardformalism.
thetheoreticalfront,somestudiesonthecanonicalquan-
The second and third choices are, respectively, [a =
κ tization of a classical system has also been investigated
1/2κ(1+κ), b = 0] [4] and [a = 1/2κ(1+κ), b =
κ κ κ [15].
−a −a ] [3] while the fourth one is given by [a =
κ −κ κ In this letter, we aim at rediscussing the H-theorem
Zκ/2κ(1+κ), b = 0] [4]. In particular, in this latter
κ in the context of Kaniadakis entropy framework, Eq.
choice, the κ-entropy is given by [4, 6]
(3). However, instead of using the KIP introduced in
zκ z−κ Ref. [6], we propose a different route to the κ-statistic
S =− d3v f1+κ− f1−κ . (3) which follows similar arguments of Ref. [16]. In real-
κ
Z (cid:18)2κ(1+κ) 2κ(1−κ) (cid:19)
ity, the main result is to obtain the equilibrium veloc-
ity κ-distribution of a slight modification of the kinetic
BoltzmannH-theorem,wherethecentralideafollowsby
modifing the molecular chaos hypothesis and generaliza-
∗OnleaveofabsencefromUniversidadedoEstadodoRioGrande
doNorte,59610-210, Mossoro´,RN,Brasil tion of the local entropy formula in accordance with a
†Electronicaddress: [email protected],[email protected] κ-statistic.
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II. THE MOLECULAR CHAOS HYPOTHESIS volume V. In BKT the state of a non-relativistic gas
is characterized by the one-particle distribution func-
As is widely known, the Boltzmann’s kinetic theory tion f(r,v,t), which is defined in such a way that
(BKT) relies on two statistical ingredients, namely [17, f(r,v,t)d3rd3v gives at a time t, the number of parti-
18]: cles in the volume element d3rd3v around the particle
position r and velocity v. Let us consider that distribu-
• The local entropy which is expressed by Boltz- tion function is a solution of the κ-Boltzmann equation,
mann’slogarithmicmeasure(Boltzmann’sconstant given by
is an unity)
∂f ∂f F ∂f
+v· + · =C (f), (10)
∂t ∂r m ∂v κ
H[f] = − f(r,v,t) lnf(r,v,t)d3v. (7) where Cκ defines the κ-collisional term. In expression
Z
(10), the left-hand-side (LHS) is just the total time
derivative of the distribution function, thus it is reason-
• The hypothesis of molecular chaos
able to consider that the unique possible modification
(“Stosszahlansatz”), i.e., the two point corre-
describing κ-statistic must be associated with the colli-
lation function of the colliding particles can be
sional term, which is clearly a hypothesis of work. Basi-
factorized
cally,C (f)maybecalculatedthroughthelawsofelastic
κ
collisions,where the standardassumptions arealso valid
[17, 18].
f(r1,v1,r2,v2,t) = f(r1,v1,t)f(r2,v2,t). (8)
As in the canonical H-theorem, our main goal here is
toshowthatC (f)leadsto anonnegativeexpressionfor
Thereissomecontroversyassociatedwiththe secondas- κ
the time derivativeofthe κ-entropy,anddoesnotvanish
sumption, Eq. (8). In particular, Burbury [19], was the
unless the distribution function assumes the equilibrium
first to point out that this hypothesis provides the fun-
form for a κ-distribution which has been recently pro-
damental role within the BKT. Physically, the equation
posed [6]. Here, we define
(8) represents that colliding molecules are uncorrelated,
i.e., the velocities and positions of pairs of molecules are s2
statistically independents. The irreversibility associated Cκ(f)= |V·e|Rκdωd3v1, (11)
2 Z
withtheBoltzmann’sequationcanbetracedbacktothis
assumption. Indeed,asthemoleculeswhichhasbeenas- whered3v1 isanarbitraryvolumeelementinthevelocity
sumed to be uncorrelated before a collision, become cor- space, V denotes the relative velocity before collision,
related after the collision, this clearly represents a time V = v1 −v, e denotes an arbitrary unit vector, dω is
asymmetric hypothesis [20]. In particular, this hypothe- an elementary solid angle such that s2dω is the area of
sismaynotalwaysbe validforrealgases(see[21,22]for the “collision cylinder” (for details see Refs. [17, 18]),
details). finallyR (f,f′)isadiferenceoftwocorrelationfunctions
κ
In this investigation, we introduce a consistent gener- (just before and after collision). In this approach, such
alizationofthis hypothesis,expression(8), within the κ- expression is assumed to satisfy a κ-generalized form of
statistic proposed by Kaniadakis. We remark that equa- molecular chaos hypothesis, which is given by
tion (8) implies that the logarithm of the joint distri-
bution f(r1,v1,r2,v2,t) describing coliding molecules is Rκ =expκ(lnκz′f′+lnκz′1f1′)
given by
lnf(r1,v1,r2,v2,t) = lnf(r1,v1,t)+lnf(r2,v2,t) (9) −expκ(lnκzf +lnκz1f1), (12)
where each term involve only the marginal distribution whereprimesrefertothedistributionfunctionaftercolli-
associated with one of the colliding molecules. The new sion,z,z′arearbitraryconstantandexp (f),ln (f),are
κ κ
hypothesis assumed here is to consider that a power of defined by Eqs. (2) and (3). Note that κ→0 the above
the joint distribution (instead of the logarithm) be equal expression reduces to R0 = (z′f′)(z1′f′1) − (zf)(z1f1),
to the sumoftwo terms,eachonedepending onjust one whereisonestandardtothemolecularchaoshypothesis.
of the coliding molecules. Considering the κ-logarithm In the present framework, we adopt Kaniadakis for-
function,theconditionabovecanbeformulatedinaway mula for local entropy
that extend the standard hypothesis, Eq. (8).
zκ z−κ
H =− d3v f1+κ− f1−κ .
κ
Z (cid:18)2κ(1+κ) 2κ(1−κ) (cid:19)
III. H-THEOREM AND κ-STATISTICS (13)
Here, in order to obtain the source term, we need the
partial time derivative of H
Let us now introduce a spatially homogeneous gas of κ
N hard-sphere particles of mass m and diameter s, un- ∂H ∂f
κ =− d3vln fz , (14)
der the action of an external force F, and enclosed in a ∂t Z κ ∂t
3
and combining with the κ-Boltzmann equation (10) and Now, in order to make the H-theorem and the κ-
(11), one may rewrite the expression in the form of a statistic compatible,we need to recoverthe relatedequi-
balance equation librium distribution previously obtained by an extrem-
izationofκ-entropyandKIP[6]. TheH -theoremstates
κ
∂H
κ +∇·S =G (r,t), (15) that Gκ = 0 is a necessary and sufficient condition for
k κ
∂t equilibrium. Since the integrandof (18) cannot be nega-
tive, this occur if and only if
wherethe κ-entropyfluxvectorrelatedtoH isgivenby
κ
lnκz′f′+lnκz1′f1′ =lnκzf +lnκz1f1 (22)
zκ z−κ
Sκ =− d3vv f1+κ− f1−κ , Therefore,theabovesumofκ-logarithmsremainscon-
Z (cid:18)2κ(1+κ) 2κ(1−κ) (cid:19)
stant during a collision, or equivalently, it is a summa-
(16)
tional invariant. Indeed, the unique quantities satisfying
and the source term reads
(18) are the particle masses, and the expressionsfor mo-
−s2 mentumandenergyconservationlaws,whichleadto the
Gκ = 2 Z |V·e|lnκfz Rκdωd3v1d3v. (17) following expression
At this point, it is convenient to rewrite Gκ in a more lnκzf =ao+a1.v+a2v2, (23)
symmetrical form by using some elementary symmetry
operations which also take into account the inverse col- where ao and a2 are constants and a1 is an arbitrary
lisions. First we notice that by interchanging v and v1 constantvector. Byintroducingthe barycentricvelocity,
the value of the integral is preserved. This happens be- u, we may rewrite (23) as
cause the magnitude of the relative velocity vector and
2
the scattering cross section are invariants [18]. In addi- ln zf =α−γ(v−u) , (24)
κ
tion, the value of G is not altered if we integrate with
κ
respect to the variables v′ and v′. Actually, although with a different set of constants. Thus, we obtain a κ-
1
changing the sign of R in this step (inverse collision), distribution
κ
the quantity d3vd3v1 is also a collisional invariant [18]. 1
As one may check, such considerations imply that the f = exp [α−γ(v−u)2], (25)
z κ
κ-entropy source term can be written as
whereγ andumaybefunctionsofthetemperature. The
s2
G (r,t)= |V·e|(ln z′f′ +ln z′f′− expressionaboveisthe generalformofthe κ-Maxwellian
κ 8 Z κ 1 1 κ distribution function.
−lnκz1f1−lnκzf)[expκ(lnκz′f′+lnκz′1f1′)− IV. FINAL REMARKS
−expκ(lnκzf +lnκz1f1)]dωd3v1d3v. (18) Summing up, we have discussed a κ-generalization of
Boltzmann’s kinetic equation along the lines defined by
Notethattheintegrandin(18)isnevernegative,because
κ-statistic. The main results follow from a slight modifi-
the expressions
cationofthemainstatisticalhypothesisunderlyingBoltz-
mann’s approach:
(lnκz′f′+lnκz1′f1′ −lnκzf −lnκz1f1) (19)
1. The κ-statistic has been explicitly introduced
and
through a new functional formula for the local en-
expκ(lnκz′f′+lnκz1′f′1)−expκ(lnκzf +lnκz1f1)(20) tropy;
2. A nonfactorizable expression for the molecular
alwayshavethesamesigns. Therefore,forvaluesofκon
chaos hypothesis has been adopted.
the interval [−1;1], we obtain
∂H Both ingredients are shown to be consistent with the
κ +∇·Sκ =Gκ ≥0, (21) standard laws describing the microscopic dynamics, and
∂t
reduce to the familiar Boltzmann assumptions in the
which is the mathematical expression for the Hκ- limit (κ=0).
theorem. Thisinequalitystatesthattheκ-entropysource Acknowledgments: The author is grateful to J. S. Al-
must be positive or zero, thereby furnishing a kinetic caniz,I.QueirozandJ.A.S.Limaforhelpfuldiscussions.
derivation of the second law of thermodynamics in the This work was supported by Conselho Nacional de De-
κ-ststistic. senvolvimento Cient´ifico e Tecnol´ogico - CNPq (Brasil).
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