(20

s bl ea

te ff Scised

e 4

byAbstract

The original canonical ensemble formalism for the nonextensive entropy thermostatistics is reconsidered. It is shown that the unambiguousconnection of the statistical mechanics with the equilibrium thermodynamics is provided if the entropic parameter 1/(q1) is an extensive variableof the state. Based on a particular example of the perfect gas, it is proved that the Tsallis thermostatistics meets all the requirements of equilibriumthermodynamics in the thermodynamic limit. In particular, the entropy of the system is extensive and the temperature is intensive. However, forfinite systems both the Tsallis and BoltzmannGibbs entropies are nonextensive. The equivalence of the canonical and microcanonical ensemblesof Tsallis thermostatistics in the thermodynamic limit is established. The issue associated with physical interpretation of the entropic variable isdiscussed in detail. 2006 Elsevier B.V. All rights reserved.

PACS: 24.60.Ky; 25.70.Pq; 05.70.Jk

Keywords: Tsallis thermostatistics; Canonical ensemble; Equilibrium thermodynamics

1. Introduction

The main purpose of this Letter is to establish a clear way toimplement the equilibrium statistical mechanics based on thenonadditive entropy deferent from the usual BoltzmannGibbsstatistical one. For the first time this concept was formulated byTsallis in [1]. It is very well known that the conventional equi-librium statistical mechanics based on the BoltzmannGibbsentropy meets all the requirements of the equilibrium thermo-dynamics in the thermodynamic limit [2]. This is a necessarycondition for self-consistent definition of any equilibrium sta-tistical mechanics. In order to provide the connection of thestatistical mechanics with the thermodynamics, the statisticalentropy is usually used. From the mechanical and thermody-namical laws it allows one to determine a unique phase distri-bution function, or a statistical operator, which depends on two

E-mail address: parvan@theor.jinr.ru (A.S. Parvan).

different sets of variables: the first set specifies the dynamicstate of the microscopic system and the second one sets up thethermodynamic state of the macroscopic system. According tothe Liouville and von Neumann equations, the equilibrium dis-tribution function is a constant of motion which is expressedonly through the first additive integrals of motion of the system.The ensemble averages in the statistical mechanics correspondto the concrete functions of state from the thermodynamics anddepend only on the macroscopic variables of state. The sta-tistical entropy as a function of the variables of state in thethermodynamic limit must satisfy all properties of the thermo-dynamic entropy: concavity, extensivity and so one. Note thatthe thermodynamic potentials of the system are functions fix-ing the norm of the phase distributions. It is known that theequilibrium thermodynamics is the theory defined in the ther-modynamic limit. Therefore, the concept of the thermodynamiclimit plays a crucial role in comparing the equilibrium statisticalmechanics with thermodynamics. In this case, for thermody-namic systems the boundary effects must be neglected and onlyPhysics Letters A 360

Extensive statistical mechanicCanonica

A.S. PBogoliubov Laboratory of Theoretical Physics, Joint Institu

Institute of Applied Physics, Moldova Academy oReceived 9 February 2006; received in revi

Available onlin

Communicated0375-9601/$ see front matter 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.physleta.2006.07.05206) 2634www.elsevier.com/locate/pla

ased on nonadditive entropy:nsemble

rvan

or Nuclear Research, 141980 Dubna, Russian Federationences, MD-2028 Chisinau, Republic of Moldovaform 29 June 2006; accepted 18 July 2006

August 2006

C.R. Doering

etteA.S. Parvan / Physics L

the short-range interaction forces can be taken into account. Inthe thermodynamic limit the ensemble averages with the corre-sponding distribution function should provide performance ofthe zero, first, second, and third laws of thermodynamics, andthe principle of additivity that divide all variables into exten-sive and intensive ones. Moreover, the fundamental equation ofthermodynamics, the GibbsDuhem relation, and the Euler the-orem should be implemented.

In Ref. [1], the author tried to construct the equilibrium sta-tistical mechanics based on the nonextensive statistical entropy.The Tsallis generalized canonical distribution was derived fromGibbs microcanonical ensemble in [3]. The problems arisen inthe proof of the thermodynamical laws resulted in the occur-rence of the divers variants of the Tsallis thermostatistics [46].In these investigations not only the ensemble averages and thenorm equation for the distribution function but also the La-grange function were drastically redefined. Moreover, it wasfound that any of these variants do not satisfy the zeroth lawof thermodynamics if the parameter q must be a universal con-stant. See Ref. [7] for subsequent discussions of the critiqueof q-entropy for thermal statistics. To solve this problem, inAbe et al. [8] in the framework of the canonical ensemble thephysical temperature and pressure satisfying the zeroth law ofthermodynamics were introduced. However, as it was shown in[912] such transformations of the variables including the en-tropy lead to the transition from the Tsallis thermostatistics tothe extensive statistical mechanics of Gibbs or Rnyi one. Sosuch consideration does not pertain to the perception of the ze-roth law of thermodynamics. In [13], a stronger assumption wasevolved to make use of the nonextensive microscopic Hamil-tonian of special kind like the nonextensive entropy form. Inthis case, the Hamiltonian depends on the temperature of thesystem that entails the changes of the thermodynamic relationsand leads to the loss of self-consistency of the statistical me-chanics. Closely connected with the problem of the zeroth lawof thermodynamics is another one: the principle of additivityin the thermodynamic limit. In Abe [14], it was attempted todefine the thermodynamic limit on a particular example of theperfect gas in the canonical ensemble. However, this limit wascarried out incorrectly. The correct definition of the thermody-namic limit for the Tsallis thermostatistics in a particular casewas given in Botet et al. [11,15] and for the general case was de-veloped in [16]. Note that an important criterion of correctnessof the thermodynamic limit is the equivalence of all ensembles.Derivation of the fundamental equation of thermodynamics forthe Tsallis thermostatistics on the base of the canonical distrib-ution function was performed in [10,12]. In [16], it was provedthat the microcanonical ensemble of the Tsallis statistical me-chanics satisfies all requirements of the equilibrium thermody-namics if the entropic index 1/(q 1) is the extensive variableof state of the system. In the present Letter, we will show thatsimilar results are carried out also for the canonical ensemble.

The Letter is organized as follows. In Section 2, the canoni-cal ensemble and the derivation of the thermodynamic relations

are given. In Section 3, the performance of the thermodynamicprinciples in the thermodynamic limit on the example of theperfect gas is proved.rs A 360 (2006) 2634 27

2. Canonical ensemble

Let us consider the equilibrium statistical ensemble of theclassical dynamical systems of N particles at the constant tem-perature T , the volume V , and the thermodynamic coordinatez in a thermal contact with a heat bath. The system interactsweakly with its surroundings and only the energy can be trans-ferred in and out of it. In order to determine the equilibriumdistribution function, we consider the Tsallis equilibrium sta-tistical entropy which is a function of the parameter q and afunctional of the probing phase distribution function (x,p):

(1)S = k

q1 q d,

where d = dx dp is an infinitesimal element of phase space,k is the Boltzmann constant and q R is the real parametertaking values 0 < q < . The phase distribution function isnormalized to unity:

(2)

d = 1.In the classical statistical mechanics the expectation value ofthe Hamiltonian can be written as

(3)H =

H d.

The phase distribution function depends on the first additiveconstants of motion of the system. Nevertheless, the mechanicallaws are not sufficient to determine it unambiguously. For thisreason additional postulates of the equilibrium thermodynam-ics are required. To express the equilibrium phase distributionfunction from the macroscopic variables of state, we considerthe thermodynamic method explored in [16]. In the state of ther-mal equilibrium the macroscopic system is characterized by thefundamental equation of thermodynamics

(4)T dSth = dE + p dV + Xdz dN,where Sth(T ,V, z,N) is the thermodynamic entropy, z and Vare the thermodynamic coordinates; X and p are the associ-ated forces; is the chemical potential and E is the thermo-dynamic energy of the system. In the canonical ensemble thefundamental equation of thermodynamics at the fixed values ofT , V , z, N can be rewritten as

(5)(T dSth dE)T,V,z,N = 0.Then, to express the phase distribution function (x,p) throughthe variables of state (T ,V, z,N), let us replace in Eq. (5) theequilibrium thermodynamic entropy Sth and energy E of themacroscopic system with the statistical ones (1) and (3). Then,one finds

(6)T

S

qdq +

d

{[T

S

H

]d H

HdH

}= 0,where the symbol d before the functions H , and q is the totaldifferential in variables (T ,V, z,N). The statistical parametersq and must be expressed through the variables of state of the

ette28 A.S. Parvan / Physics L

system to provide the unambiguous conformity between statis-tical and thermodynamic entropies. Since dH = 0, dq = 0 andd = 0, we obtain

(7)T S

H

= ,

where is a certain constant. For the microcanonical ensembleit was stated in Ref. [16] that

(8)1q 1 = z,

where the parameter z takes the values < z < 1 for 0 0) asA(T ,V, z,N)|N,v,z=const

(27)= N[a(T , v, z) + O(N)] as= Na(T , v, z),whereas the intensive variables take the following form:

(T ,V, z,N)|N,v,z=const(28)= (T , v, z) + O(N) as= (T , v, z),

where v is the specific volume, z is the specific z, and a =A/Nis the specific A. It is important to note that the limiting sta-tistical procedures (V , z/V = const,N/V = const) and(z ,V/z = const,N/z = const) are equivalent with thelimit, N , given above. Note that in Abe [14], the thermo-dynamic limit for the Tsallis statistics is not correct because thelimits N and |z| are not coordinated among them-selves. Note that after applying the thermodynamic limit to thefunctions of state the BoltzmannGibbs limit, z , is pro-vided by expansion of these functions in powers of the smallparameter 1/z holding only the zero term of the power expan-sion.

In the canonical ensemble, how to prove from the generalpoint of view the principle of additivity (see Eqs. (25) and (26))and the zeroth law is not obvious. Therefore, we will illustrateexplicitly the implementation of these principles on the forego-ing example of the nonrelativistic ideal gas. Here, we investi-gate the Tsallis statistics from the point of view of the equilib-rium thermodynamics. The additive properties of the functionsof state in thermodynamics can be determined by division ofthe total system into two macroscopic dynamically indepen-dent (noninteracting) subsystems. Therefore, the perfect gasis the simplest physical system which can be applied for thisstudy. The principle of additivity in the microcanonical ensem-ble was proved both in the general form and for the ideal gas,in the thermodynamic limit [16]. The microcanonical ensembleis most convenient to analyze the fundamental questions of thestatistical mechanics. Therefore, for the canonical ensemble it isenough to conceder the perfect gas case if in the thermodynamiclimit the equivalence of ensembles is implemented. Moreover,

in the framework of the nonrelativistic ideal gas of N identicalparticles in the canonical ensemble the functions of state can beexplicitly expressed from the variables of state. Hence, the ther-

ette30 A.S. Parvan / Physics L

modynamic properties of the Tsallis statistics can thoroughlybe investigated. In order to evaluate the expectation values ofthe dynamical variables, we use the method based on the inte-gral representation of the Euler gamma function [12,17]. Let usinvestigate the thermodynamic properties of the nonrelativisticperfect gas of N identical particles in the thermodynamic limit(N , v = const, z = const). The exact relations for it canbe found in Appendix B. In the thermodynamic limit one easilyconfirms that the canonical partition function of the ideal gasfor Gibbs statistics is simply [18]

(29)Z1/NG (T ,V,N) = (gve)(

mkT

2h2

)3/2 ZG(T , v),

where ZG is the one-particle partition function, m is the particlemass and g is the spin degeneracy factor. Then in the thermo-dynamic limit and in the limit of BoltzmannGibbs statistics,z , Eqs. (B.1) and (B.2) can be written as(T ,V, z,N)

(30)= kT zN[

1 (

1 + 32z

)(ZGe

3/2) 1z+ 32],

(31)|z = kT lnZG FG,where z < 3/2 or z > 0 with respect to the conditions of theintegration method used. Note that the norm function is ex-tensive. The energy (B.3) takes the following form:

(32)E(T ,V, z,N)N

= 32kT

(ZGe

3/2) 1z+ 32 = (T , v, z),where is the specific energy depending only on intensive vari-ables and it is intensive. Consequently, the energy E is exten-sive (cf. Eqs. (27) and (28)). Note that in the BoltzmannGibbslimit the energy (32) is reduced to E|z = (3/2)kT N . Thefree energy (B.4) is found to be

(33)F(T ,V, z,N) = (T ,V, z,N) = Nf (T , v, z),where f is the specific free energy which is intensive. In theBoltzmannGibbs limit the free energy (33) takes the usualform F |z = kT lnZG. Then in the thermodynamic limitand in the limit of BoltzmannGibbs statistics the entropy (B.5)can be written as

(34)S(T ,V, z,N)N

= kz[1 (ZGe3/2)1

z+ 32]= s(T , v, z),

(35)S|z = 32kN + k lnZG = SG,where s is the specific entropy and SG is the entropy of Gibbsstatistics. The function s, however, is an intensive function of(T , v, z). Thus, the Tsallis entropy (34) in the thermodynamiclimit is extensive. It is important to note that the Tsallis en-tropy (B.5) for the finite values of N and z does not satisfythe homogeneous condition (25), for instance, in the case of

= 1/N , and it is nonextensive. However, the Gibbs entropy,SG, is also nonextensive for the nonrelativistic perfect gas ofN identical particles in the canonical ensemble, because thers A 360 (2006) 2634

canonical partition function ZG (N !)1. See Appendix B. Inthe thermodynamic limit the Gibbs entropy (35) of the perfectgas in contrast with its exact value is extensive, because Eq. (29)has been implemented. Note that the values of the tempera-ture and the specific volume are restricted, T T0 and v v0,by the physical conditions S 0 and SG 0, where the pa-rameters (T0, v0) are determined from the following equationZG(T0, v0)e3/2 = 1.

Let us deduce the pressure p, the chemical potential , andthe variable X in the thermodynamic limit. The pressure (B.6)takes the form

(36)p(T ,V, z,N) = kTv

(ZGe

3/2) 1z+ 32 = p(T , v, z),where p is the intensive function and in the Gibbs limit we havep|z = kT /v. The function of state (B.7) can be written as

X(T ,V, z,N)

= kT [1 (ZGe3/2)1

z+ 32(1 ln(ZGe3/2)

1z+ 32

)](37)= X(T ,v, z)

and in the Gibbs limit it is reduced to X|z = 0. The chem-ical potential (B.8) is now

(T ,V, z,N)

= kT (ZGe3/2)1

z+ 32[

52

+ z ln(ZGe3/2)1

z+ 32]

(38)= (T , v, z),where is an intensive one and in the Gibbs limit we find that|z = kT (1 ln ZG). Then, Eqs. (32), (34), and (36)(38)in the thermodynamic limit yield the Euler theorem

(39)T S = E + pV + Xz N.This equation enables us to interpret X as the force observ-able of the system conjugate to the position variable z. More-over, Eq. (39) allows us to write

(40)F = E T S = pV Xz + N.Its differential leads to the GibbsDuhem relation

(41)S dT = V dp + z dX N d,which shows that the variables T , p, X and are not inde-pendent. The heat capacity of the perfect gas of N identicalparticles is deduced from Eq. (24). Using Eq. (32) we obtain

(42)Cvz(T , v, z) = 32kN(

1 + 32z

)1(ZGe

3/2) 1z+ 32 .In the BoltzmannGibbs limit the heat capacity takes its usualform Cvz|z = (3/2)kN .To better understand the thermodynamic properties of theperfect gas, it is necessary to study the equilibrium distributionof the momenta pi of the gas particles. The N -particle distrib-

2 the effective interaction energy which can be written as

ntumexp( p /2mkT ). It is clearly seen that the effective mass (44)is meff m for z < 3/2 and meff m for z > 0. At z 3/2we have meff . Let us investigate the single-particle aver-ages. The particle mean kinetic energy with the single-particledistribution function (43) can be written as

(45) p2

2m

= 3

2kT

meffm

= .

So the average kinetic energy is equivalent with the specificenergy per particle (32). The average momentum of the particleand the highest probability momentum of the distribution f (p)can be written as

(46)p =

8meffkT

, php =

2meffkT ,

where p = | p|. Their ratio is p/php = 2/ 1.13 as inthe Gibbs statistics for which the average momentum and thehighest probability momentum are pG = (8mkT/)1/2 and(php)G = (2mkT )1/2. The variance and the relative statisticalfluctuations for the distribution (43) are

Fig. 1. The dependence of the single-particle distribution function on the mome

degrees of freedom in the non-relativistic approximation for the different values ofT = 100 MeV and the specific volume v = 0.25/0. The ratio of the effective quarkof T = 100 MeV (dashed line) and T = 200 MeV (solid one). The dotted lines corre(49) p2

2meff

= 3

2kT , = 3

2kT

[meffm

1].

So the effective interaction energy is positive > 0, and theforces are repulsive for z < 3/2 and they are attractive < 0for z > 0. Note that from Eq. (49) follows the physical interpre-tation for the temperature of the system T as the average kineticenergy of the quasiparticles with the effective mass meff.

Fig. 1 shows the dependence of the single-particle distribu-tion function (left panel) on the momentum p for the perfect gasof free quarks in the nonrelativistic approximation. The calcu-lations are carried out for the system of quarks with two flavorand three color degrees of freedom, and the constituent massm = 300 MeV at temperature T = 100 MeV and the specificvolume v = 0.25/0, where 0 = 0.168 fm3. Note that thequarks at short distances are treated as almost free pointlikenoninteracting particles because of the property of asymptoticfreedom. It is remarkable that the single-particle distributionfunction at small values of the variable |z| considerably dif-fers from the limiting MaxwellBoltzmann distribution. Such

p (left) for the classical perfect gas of quarks with two flavor and three colorA.S. Parvan / Physics Letters A 360 (2006) 2634 31

ution function is defined in Appendix B. In the thermodynamiclimit the single-particle distribution function (B.11) takes theform

(43)f ( p) =(

12meffkT

)3/2e p22meffkT ,

where meff is the effective particle mass written as

meff = m(ZGe

3/2) 1z+ 32(44)=

{m

[gv

(kT e5/3

2h2

)3/2]1/z}1/(1+ 1z

32 )

.

In the BoltzmannGibbs limit, meff|z = m, and the single-particle distribution function is reduced to the MaxwellBoltzmann distribution f ( p)|z = (2mkT )3/2

(47)(p)2= p2 p2 = kTmeff(

3 8

),

(48)p =(p)2

p =

38

1

or p 0.424. So from the investigation of the distributionfunction (43) and its averages we arrive at the conclusion that inthe Tsallis statistics the mean kinetic energy of the particles andthe momentum are larger than their values in the Gibbs statis-tics, G and p pG, for z < 3/2 and smaller, Gand p pG, for z > 0. Now we can give a physical in-terpretation for the variable of state z in the framework of theTsallis thermostatistics if we consider the system of noninter-acting particles as the system of the interacting quasiparticleswith the effective mass meff. Then the total energy of the qua-siparticle, , is equal to the sum of the mean kinetic energy andz = 2, 3, 3 and 2 (the curves 1,2,3 and 4, respectively) at the temperaturemass to the constituent one as a function of the specific z (right) for the valuesspond to the conventional BoltzmannGibbs thermostatistics.

ette32 A.S. Parvan / Physics L

a behavior has really been caused by the sharp changes of theeffective mass of quasiparticles in the dependence on z. Thisdependence can be seen even better in the right panel of Fig. 1which shows the ratio of the effective mass to the constituentone vs. the variable z for two values of the temperature T . Fig. 1clearly shows that the variable z is the order parameter andthe system is physically unstable in the region 3/2 < z < 0.For the microcanonical ensemble similar results were obtainedin [16].

Our exploration of the perfect gas has brought the followingpoints to the foreground: for the Tsallis statistics in the thermo-dynamic limit the principle of additivity and the zeroth law ofthermodynamics are valid. The entropy (34) is a homogeneousfunction of the first order, it is an extensive variable satisfyingthe relation (25) with = 1/N provided that the temperatureT must be the intensive variable of state. It should, however,be noted that in the thermodynamic limit the equivalence of thecanonical and microcanonical ensembles is implemented. Giv-ing constants of motion exactly or only as averages leads tothe same results. For instance, if we express the temperature Tthrough the variables (E,V, z,N) from Eq. (32) as

(50)kT = 23w1/z, w = gv

(me5/3

3h2

)3/2

and substitute it into Eqs. (34) and (36)(38), then we obtain theresults of the microcanonical ensemble derived in Ref. [16]. Itis important to note that in terms of the z variable and our ther-modynamic limit the results for the perfect gas in the canonicalensemble of Abe et al. [14,19] are the same as here. Note thatthe problem of extensivity of the Tsallis entropy has been re-cently investigated in [22] using the discrete and continuoussets of probabilities {pi} which do not maximize the entropy.In this case the equilibrium thermodynamics cannot be applied,therefore, the extensivity property of the statistical entropy hasno sense et all.

4. Conclusions

In this Letter, the canonical ensemble of the nonextensivethermostatistics introduced by Tsallis has been reconsidered.It is shown that the equilibrium statistical mechanics based onthe nonadditive statistical entropy completely satisfies all re-quirements of the equilibrium thermodynamics in the thermo-dynamic limit. The unique non-Gibbs phase distribution func-tion corresponding to the Tsallis entropy is obtained from theconstraints imposed by the equilibrium thermodynamics laws.The microscopic foundation of the equilibrium statistical me-chanics proceeds on the Gibbs idea of the statistical ensemblesfor the quantum and classical mechanics. The phase distributionfunction and the statistical operator depend only on the first ad-ditive constants of motion of the system. Here they were derivedwithin a formalism based on the fundamental equation of ther-modynamics and statistical definition of the functions of state.

It allows us to avoid introduction of the controversial Lagrangemultiplies. Nevertheless, it is shown that the distribution func-tion derived from the Jaynes principle exactly coincides withrs A 360 (2006) 2634

ours if the Lagrange parameters are expressed through a set ofindependent variables of state of the system. The unambiguousconnection of the statistical mechanics with thermodynamicsis established. The equilibrium distribution function satisfiesthe fundamental equation of thermodynamics, the first and thesecond principles preserving the Legendre transformation. Allthermodynamic relations relative to the Helmholtz free energyfor the thermodynamic system in the thermostat are carried out.The heat capacity of the system was derived from the first andthe second laws of thermodynamics. Note that in the funda-mental equation of thermodynamics the new term related to thework of the conjugate force X at changes of the variable of statez appeared. In the limit z the conventional Gibbs statis-tics is recovered.

It is well known that the statistical mechanics should sat-isfy all requirements of the equilibrium thermodynamics onlyin the thermodynamic limit. Based on a particular example ofthe ideal gas we obviously proved the fulfillment of the prin-ciple of additivity and the zero law of thermodynamics forthe Tsallis statistics in the thermodynamic limit. It was shownthat all functions of state of the system are the homogeneousfunctions of the first degree, extensive, or the homogeneousfunctions of the zero degree, intensive. In particular, the temper-ature is an intensive variable and thus provides implementationof the zero law of thermodynamics. It should be marked thatfor the finite values of the number of particles N and the pa-rameter z of the system both the Tsallis entropy and the Gibbsentropy are nonextensive functions of state, while in the ther-modynamic limit they become extensive variables. The homo-geneous properties of the functions of state allow us to findthe Euler theorem and the GibbsDuhem relation. After ap-plying the thermodynamic limit the expressions of the Gibbsstatistics are obtained by the limiting procedure z . Theone-particle distribution function in the thermodynamic limitleads to the MaxwellBoltzmann distribution function with theeffective mass of particles meff. This distribution allows us tofind the physical interpretation for the variable of state z as theorder parameter of the interacting system of quasiparticles withmass meff and the physical interpretation for the temperature Tas the average kinetic energy of quasiparticles. The numericalexample for the ideal gas of quarks with two flavor and threecolor degrees of freedom in the nonrelativistic approximationshows that the dense system of quarks in the dependence of val-ues of the parameter z can pass from the strong coupled state tothe repulsive state of quarks. In the framework of the ideal gasof identical particles the equivalence of the canonical and mi-crocanonical ensembles in the thermodynamic limit is proved.This property is the key to identifying the self-consistency ofthe statistical mechanics and thermodynamics.

Acknowledgements

This work has been supported by the MTA-JINR Grant. We

acknowledge valuable remarks and fruitful discussions withT.S. Bir, R. Botet, K.K. Gudima, M. Poszajczak, V.D. Toneev,and P. Van.

etteA.S. Parvan / Physics L

Appendix A. Phase distribution function

Here, to derive the distribution function in the canonical en-semble we use the Jaynes principle [20]. In this respect, theLagrange function can be written as

[] = S[]

k

( d 1

)

(A.1) (

H d H ),

where is the probing distribution function. After maximizingthe Lagrange function (A.1), |= = 0, and using Eq. (1) toeliminate the parameter , we arrive at the following expressionfor the equilibrium phase distribution function

(A.2) =[

1 + (q 1)q( H)

] 1q1

,

where = H qS/k . Differentiating the function andEq. (1) with respect to , and using the distribution func-tion (A.2), one finds

(A.3)S

= k H

.

The parameter can be related to the temperature

(A.4)1T

SE

= S/H / = k, =

1kT

.

Then the distribution function (A.2) takes the form

(A.5) =[

1 + (q 1) HkT q

] 1q1 =

[1 + 1

z + 1 HkT

]z,

where is determined from Eq. (2)

(A.6) [

1 + (q 1) HkT q

] 1q1

d = 1.

Note that Eqs. (A.5) and (A.6) are identical with (9) and (10).So the form of the distribution function in terms of the variablesof state is independent of the method of derivation.

Let us show that the distribution function expressed throughthe variables of state (T ,V, z,N ) in [1] is equivalent toEq. (A.5). So in [1] the Lagrange function was written as

(A.7)[] = S[]

k+

d (q 1)

H d.

After maximizing (A.7) we obtain

(A.8) = 1Z

[1 (q 1)H ] 1q1 ,

(A.9)Z = [

1 (q 1)H ] 1q1 d.To express the Lagrange parameter through the variables ofstate T , V , z, N , we use the method described in [12]. Finally,for the Lagrange parameter we get(A.10) = Zq1

kT q.rs A 360 (2006) 2634 33

Substituting Eq. (A.10) into Eqs. (A.8) and (A.9) and introduc-ing the new function in the following form:

(A.11)Z1q 1 + (q 1) kT q

,

we obtain the phase distribution function (A.5) with the normal-ization condition (A.6). So the form of the Lagrange functiondoes not disturb the distribution function in terms of the vari-ables of state.

Appendix B. The finite perfect gas

Following the arguments given in Ref. [12] we easily derivethe norm function from Eq. (10) in the case of z < 1:

1 + 1z + 1

kT=[ZG

(z 32N)(z 1) 32 N (z)

] 1z+ 32 N ,

(B.1)z < 1,where ZG = ((gV )N/N !)(mkT /2h2)3N/2 is the partitionfunction of the conventional ideal gas of the BoltzmannGibbsstatistics [18,21] and z 32N > 0. In the case of z > 0, weobtain

1 + 1z + 1

kT=[ZG

(z + 1) 32 N (z + 1) (z + 1 + 32N)

] 1z+ 32 N ,

(B.2)z > 0.In order to determine the energy of system, we insert the Hamil-ton function A(x,p) = H(x,p) into Eq. (11) and after per-forming integration, we obtain

(B.3)H = 32kT N

1 + 1z+1

kT

1 + 1z+1

32N

,

where the function is determined from Eqs. (B.1) and (B.2).The norm function and the energy H allow us to calcu-late the thermodynamic potential of the canonical ensemble, thefree energy F . Substituting Eqs. (B.1)(B.3) into Eq. (13), wefind

(B.4)F = kT z[

1 (

1 + 1z

32N

) 1 + 1z+1

kT

1 + 1z+1

32N

].

Then, the entropy of the system can be easily obtained fromEqs. (12) or (13):

(B.5)S = kz[

1 1 +1

z+1kT

1 + 1z+1

32N

].

Let us calculate the pressure p, the chemical potential ,and the variable X. Taking into account Eqs. (21) and (B.4),one finds

(B.6)p = NV

kT1 + 1

z+1kT

1 + 1z+1

32N

= 23E

V.Differentiating Eq. (B.4) with respect to z in conformity withEq. (22), we obtain

34 A.S. Parvan / Physics Letters A 360 (2006) 2634

X = kT[

1 (1 + 1

z+1kT

)(1 + z(z+1)2

32N)

(1 + 1z+1

32N)

2

]

+ kT 1 +1

z+1kT

1 + 1z+1

32N

[ln(

1 + 1z + 1

kT

)

(B.7) (a + 3

2N

)+ (a)

],

where (y) is the psi-function which depends on argumentsa = z, = 1 for z < 1 and a = z + 1, = 1 for z > 0.Taking the derivative of (B.4) with respect to N in conformitywith Eq. (22), we get

= 32kT

1 + 1z+1

kT

1 + 1z+1

32N

[ 1z+1

1 + 1z+1

32N

ln( (z + 1)

(1 + 1

))]

(B.11)f ( p) = C[

1 1z + 1

p22mkT (1 + 1

z+1kT

)

]z+ 32 (N1),

C = 1( (z + 1)2mkT [1 + 1

z+1kT

])3/2

(B.12)[

(a + 32N) (a + 32 (N 1))

].

References

[1] C. Tsallis, J. Stat. Phys. 52 (1988) 479.[2] R. Balescu, Equilibrium and Nonequilibrium Statistical Mechanics, Wi-

ley, New York, 1975.[3] A.R. Plastino, A. Plastino, Phys. Lett. A 193 (1994) 140.[4] C. Tsallis, R.S. Mendes, A.R. Plastino, Physica A 261 (1998) 534.[5] S. Martnez, F. Nicols, F. Pennini, A. Plastino, Physica A 286 (2000) 489.z + 1 kT

kT 1 +1

z+1kT

1 + 1z+1

32N

[ln(V

(mkT

2h2

)3/2)

(B.8) 32

(a + 3

2N

) (N + 1)

].

The N -particle distribution function of the classical ideal gasin the canonical ensemble for the Tsallis statistics can be writtenas

(B.9)f ( p1, . . . , pN) = (gV )N

N !h3N[

1 + 1z + 1

Ni=1 p2i2mkT

]z,

which is normalized to unity

(B.10)

d3p1 d3pN f ( p1, . . . , pN) = 1.Then the reduced single-particle distribution function can beeasily obtained by directly performing the integral over the mo-menta ( p2, . . . , pN):[6] Q.A. Wang, Chaos Solitons Fractals 12 (2001) 1431.[7] M. Nauenberg, Phys. Rev. E 67 (2003) 036114;

C. Tsallis, Phys. Rev. E 69 (2004) 038101;M. Nauenberg, Phys. Rev. E 69 (2004) 038102.

[8] S. Abe, S. Martnez, F. Pennini, A. Plastino, Phys. Lett. A 281 (2001) 126.[9] R. Toral, cond-mat/0106060.

[10] E. Vives, A. Planes, Phys. Rev. Lett. 88 (2002) 020601.[11] R. Botet, M. Poszajczak, K.K. Gudima, A.S. Parvan, V.D. Toneev, Phys-

ica A 344 (2004) 403.[12] A.S. Parvan, T.S. Bir, Phys. Lett. A 340 (2005) 375.[13] Q.A. Wang, Eur. Phys. J. B 26 (2002) 357.[14] S. Abe, Phys. Lett. A 263 (1999) 424;

S. Abe, Phys. Lett. A 267 (2000) 456, Erratum.[15] R. Botet, M. Poszajczak, J.A. Gonzlez, Phys. Rev. E 65 (2002)

015103(R).[16] A.S. Parvan, Phys. Lett. A 350 (2006) 331.[17] D. Prato, Phys. Lett. A 203 (1995) 165.[18] K. Huang, Statistical Mechanics, Wiley, New York, 1963.[19] S. Abe, Physica A 269 (1999) 403.[20] E.T. Jaynes, Phys. Rev. 106 (1957) 620.[21] A.S. Parvan, V.D. Toneev, M. Poszajczak, Nucl. Phys. A 676 (2000) 409.[22] C. Tsallis, M. Gell-Mann, Y. Sato, Proc. Natl. Acad. Sci. USA 102 (2005)

15377.

Extensive statistical mechanics based on nonadditive entropy: Canonical ensembleIntroductionCanonical ensembleThe thermodynamic limit. The perfect gasConclusionsAcknowledgementsPhase distribution functionThe finite perfect gasReferences