当前位置：文档库 › Relativistic Nonextensive Thermodynamics

Relativistic Nonextensive Thermodynamics

a r X i v :c o n d -m a t /0207353v 1 [c o n d -m a t .s t a t -m e c h ] 15 J u l 2002Relativistic Nonextensive Thermodynamics

https://www.wendangku.net/doc/091539151.html,vagno Dipartimento di Fisica,Politecnico di Torino and INFN,Sezione di Torino C.so Duca degli Abruzzi 24,I-10129Torino,Italy 1Introduction Recently,there is an increasing evidence that the generalized nonextensive statistical mechanics,proposed by Tsallis [1],can be considered as the more appropriate basis of a theoretical framework to deal with physical phenom-ena where long-range interactions,long-range microscopic memories and/or fractal space-time constraints are present (cf.[1]for details).A considerable variety of physical applications involve a quantitative agreement between ex-perimental data and theoretical models based on Tsallis’thermostatistics [2].In particular there is a growing interest to high energy physics applications of nonextensive statistics.Several authors outline the possibility that experi-mental observations in relativistic heavy-ion collisions can re?ect nonextensive features during the early stage of the collisions and the thermalization evolu-tion of the system [3–7].

The basic aim of this letter is to study the nonextensive statistical mechanics formalism in the relativistic regime and to investigate,through an appropri-ate relativistic Boltzmann equation,the non-equilibrium and the equilibrium thermodynamics relations.

2Basic assumptions in nonextensive thermostatistics

Let us brie?y review some basic assumptions of the nonextensive thermostatis-tics that will be useful in view of the relativistic extension.

Starting point of the Tsallis’generalization of the Boltzmann-Gibbs statistical mechanics is the introduction of a q -deformed entropy functional de?ned,in a phase space system,as [1]

S q =?k B d ?p q ln q p ,(1)

where k B is the Boltzmann constant,p =p (x,v )is the phase space probability distribution,d ?stands for the corresponding phase space volume element and ln q x =(x 1?q ?1)/(1?q )is,for x >0,the q -deformed logarithmic function.For the real parameter q →1,Eq.(1)reduces to the standard Boltzmann-Gibbs entropy functional.

In the equilibrium canonical ensemble,under the constraints imposed by the probability normalization

d ?p =1,(2)

and the normalized q -mean expectation value of the energy [8]

E q = d ?p q H (x,v )

Z q

,(4)

where f (x,v )=[1?(1?q )β(H (x,v )? H q )]1/(1?q )

(5)and

Z q = d ?f (x,v ).(6)

Let us note that,depending from the extremization procedure,in Ref.[8]the above factor βis only proportional to the Lagrange multiplier,because of the

probability distribution is self-referential,while,in Ref.[9],βis actually the Lagrange multiplier associated to the energy constraint.

By using the de?nitions in Eqs.(4)and(6)into the relation Z1?q

q = d?p q

[9],the following identity holds[10]

d?f(x,v)≡ d?f q(x,v),(7) and the normalized q-mean expectation value for a physical observable A(x,v) can be expressed as

A q= d?f q A(x,v)Z q d?f q A(x,v).(8) Therefore,the probability distribution and the q-mean value of an observable have the same normalization factor Z q,as in the extensive statistical mechan-ics.Such a non-trivial property does not depend on the equilibrium frame but it comes from the normalization condition and holds at any time(if we require that the transport equation conserves the probability normalization or the number of particles).This observation will play a crucial role in the correct formulation of the relativistic Boltzmann equation and in the de?nition of the thermodynamic variables.

3Relativistic kinetic theory

On the basis of the above prescriptions,we are able to progress in the formula-tion of the relativistic nonextensive statistical mechanics.Let us start de?ning the basic macroscopic variables in the language of relativistic kinetic theory. Because we are going to describe a non-uniform system in the phase space,we introduce the particle four-?ow as

Nμ(x)=1

pμf(x,p),(9)

and the energy-momentum?ow as

Tμν(x)=1

pμpνf q(x,p),(10)

where we have set =c=1,x≡xμ=(t,x),p≡pμ=(p0,p)and p0=

√

the probability?ow j=j(x).The energy-momentum tensor contains the nor-malized q-mean expectation value of the energy density,as well as the energy ?ow,the momentum and the momentum?ow per particle.Its expression fol-lows directly from the de?nition(8);for this reason it is given in terms of f q(x,p).

In order to derive a relativistic Boltzmann equation for a dilute system in nonextensive statistical mechanics,we consider the?nite volume elements ?3x and?3p in the phase space.These volume elements are large enough to contain a very large number of particles but also small enough compared to the macroscopic dimension of the system.If,in the volume?3x?3p,is contained a representative sample of system,we can assume that Eq.(7)still holds in such phase space portion.Then,in the Lorentz frame,the particle fraction?N(x,p)in the volume?3x?3p can be written as

?N(x,p)=

Z q ?3σ

?3p

d3σμ

d3p

2 d3p1p′0d3p′1

p and p1,respectively.The factorization of h q in two single probability distri-butions(uncorrelated particles at the same spatial point)is the celebrated hy-pothesis of molecular chaos(Boltzmann’s Stosszahlansatz).Thus,the function h q de?nes implicitly a generalized nonextensive molecular chaos hypothesis. By assuming the conservation of the energy-momentum in the collisions(i.e. pμ+p′μ=pμ1+p′μ1)and requiring that the correlation function h q is symmetric and always positive(h q[f,f1]=h q[f1,f],h q[f,f1]>0),it is easy to show that collision term satis?es the following property

F[ψ]= d3p

?t d?f q(x,p)=0,(17) and this is,on account of Eq.(7),nothing else that the conservation of the probability normalization Z q.Otherwise,by settingψ=bμpμ,we have from Eq.(15)

?νTμν(x)=0,(18)

which implies the energy and the momentum conservation.

Let us remark that to have conservation of the probability normalization, energy and momentum,it is crucial that not only the collision term C q be explicitly deformed by means of the function h q,but also the streaming term pμ?μf q.This matter of fact is a directly consequence of the nonextensive sta-tistical prescription of the normalized q-mean expectation value(8)and is not taken into account in the non-relativistic formulation of Ref.[12].

4Local H-theorem

The relativistic local H-theorem states that the entropy productionσq(x)=?μSμq(x)at any space-time point is never negative.

Assuming the validity of the Tsallis entropy(1),it appears natural to introduce the nonextensive four-?ow entropy Sμq(x)as follows

Sμq(x)=?k

B d3p

ln q f pμ?μf q≡?k B F[ln q f],(20)

where the second identity follows from the Boltzmann equation(13)and the de?nition of F[ψ]in Eq.(15).After simple manipulations,Eq.(20)can be rewritten as

σq(x)=k

p01

d3p1

p′0

d3p1′

which satis?es the properties:e q(ln q x)=x and e q(x)·e q(y)=e q[x+y+ (1?q)xy].Let us note that a similar expression for the function h q was previ-ously introduced in Ref.s[12,13]and a rigorous justi?cation of the validity of the ansatz(23)can be found only by means of a microscopic analysis of the dynamics of correlations in nonextensive statistics.

5Equilibrium and equation of state

The condition that entropy production vanishes everywhere,together with re-quirement that the equilibrium probability distribution f eq must be a solution of the transport equation(13),uniquely de?nes the state of equilibrium.Tak-ing into account of Eqs.(15),(16)and(20),the conditionσq=0can only occur when ln q f eq=a+bμpμ.By imposing that f eq must satisfy Eq.(13)and after simple rede?nition of the coe?cients a and b,the equilibrium probability distribution can be written as a Tsallis-like distribution

f eq(p)=1

1/(1?q)

,(25)

where Uμis the hydrodynamic four-velocity[11]and f eq depends only on the

momentum in absence of an external?eld.At this stage,k

B T is a free parame-

ter and only in the derivation of the equation of state it will be identi?ed with the physical temperature.Moreover,it is easy to show that f eq is a solution of the transport equation(13).

We are able now to evaluate explicitly all other thermodynamic variables and provide a complete macroscopic description of a relativistic system at the equilibrium.Let us?rst calculate the probability density de?ned as

n=NμUμ=1

pμUμf eq(p).(26)

Since n is a scalar,it can be evaluated in the rest frame where Uμ=(1,0,0,0).

Settingτ=p0/k

B T and z=m/k

T,the above integral can be written as

4π

3Z q (k

T)3

∞

dτ(τ2?z2)3/2 e?τq q,(27)

where the last identity has been obtained by a partial integration.Let us introduce the q-modi?ed Bessel function of the second kind as follows

K n(q,z)=2n n!

z n

∞

dτ(τ2?z2)n?1/2 e?τq q,(28)

then,the particle density can be cast into the compact form

4π

3Tμν?μν=?

pμpν?μνf q eq(p),(30)

and can be expressed as

4π

Z q d3p

Z q

m4 3K2(q,z)z .(34)

Thus the energy per particle e=?/n is

e=3k

B T+m

K1(q,z)

T.(36)

Also in this case,no explicit q-dependence is detected.

6Conclusion

The physical motivation of this investigation lies in the strong relevance that nonextensive statistics could have in high energy physics.In this letter we have studied the thermostatistics of a relativistic system in nonextensive statistics; the obtained results can be easily extended to the case where an external force is present.

The prescription of the normalized q-mean expectation values implies a con-sistent nonextensive generalization of the macroscopic variables Nμand Tμν. On this basis,we have derived a generalized Boltzmann equation where both the streaming and the collision terms depend on the deformation parameter q. Such a transport equation conserves the probability normalization(or number of particles)and is consistent with the energy-momentum conservation laws. The collision term contains a generalized expression of the molecular chaos and for q>0implies the validity of a generalized H-theorem,if the nonextensive local four-density entropy(19)is assumed.At the equilibrium,the solution of the Boltzmann equation is a relativistic Tsallis-like distribution and the equa-tion of state of a classical relativistic gas in nonextensive statistical mechanics has the same form as in ordinary Boltzmann-Gibbs frame.

Finally,nonextensive statistical e?ects can be detected in connection to micro-scopic observables such as particle distribution,correlation functions,?uctu-ations of thermodynamical variables but not directly in connection to macro-scopic variables,such as temperature or pressure,because the equation of state and the macroscopic thermodynamical relations do not explicitly depend on

the deformation parameter q.

Acknowledgements

It is a pleasure to thank P.Quarati and C.Tsallis for useful discussions. References

[1] C.Tsallis,J.Stat.Phys.52,479(1988).See also:Nonextensive Statistical

Mechanics and Its Applications,Editors S.Abe,Y.Okamoto,Springer Verlag, 2001.

[2]See http://tsallis.cat.cbpf.br/biblio.htm for a regularly updated bibliography

on the subject.

[3]W.M.Alberico,https://www.wendangku.net/doc/091539151.html,vagno,P.Quarati,Eur.Phys.J.C12(2000)499;W.M.

Alberico,https://www.wendangku.net/doc/091539151.html,vagno,P.Quarati,Nucl.Phys.A680(2001)94;https://www.wendangku.net/doc/091539151.html,vagno, P.Quarati,Chaos,Solitons and Fractals13(2002)569;https://www.wendangku.net/doc/091539151.html,vagno,Physica A 305(2002)238.

[4]G.Wilk,Z.Wlodarczyk,Phys.Rev.Lett.84(2000)2770;O.V.Utyuzh,G.

Wilk,Z.Wlodarczyk,J.Phys.G26(2000)L39.

[5] D.B.Walton,J.Rafelski,Phys.Rev.Lett.84(2000)31.

[6]I.Bediaga,E.M.F.Curado,J.M.de Miranda,Physica A286(2000)156.

[7] C.Beck,Physica A286(2000)164.

[8] C.Tsallis,R.S.Mendes,A.R.Plastino,Physica A261(1998)534.

[9]S.Mart′inez,F.Nicol′a s,F.Pennini,A.Plastino,Physica A286(2000)489.

[10]S.Mart′inez,F.Pennini,A.Plastino,Phys.Lett.A278(2000)47.

[11]S.R.Groot,W.A.van Leeuwen,Ch.G.van Weert,Relativistic kinetic theory,

North-Holland,1980.

[12]J.A.S.Lima,R.Silva,A.R.Plastino,Phys.Rev.Lett.86(2001)2938.

[13]G.Kaniadakis,Physica A296(2001)405;Phys.Lett.A288(2001)283.

[14]S.Abe,S.Mart′inez,F.Pennini,A.Plastino,Phys.Lett.A281(2001)126.