Open access peer-reviewed chapter

Foundation of Equilibrium Statistical Mechanics Based on Generalized Entropy

By A.S. Parvan

Submitted: November 21st 2014Reviewed: June 8th 2015Published: December 21st 2015

DOI: 10.5772/60997

Downloaded: 737

Abstract

The general mathematical formulation of the equilibrium statistical mechanics based on the generalized statistical entropy for the first and second thermodynamic potentials was given. The Tsallis and Boltzmann-Gibbs statistical entropies in the canonical and microcanonical ensembles were investigated as an example. It was shown that the statistical mechanics based on the Tsallis statistical entropy satisfies the requirements of equilibrium thermodynamics in the thermodynamic limit if the entropic index z=1/(q-1) is an extensive variable of state of the system.

Keywords

  • Equilibrium statistical mechanics
  • Tsallis nonextensive statistics

1. Introduction

In modern physics, there exist alternative theories for the equilibrium statistical mechanics [1, 2] based on the generalized statistical entropy [3-12]. They are compatible with the second part of the second law of thermodynamics, i.e., the maximum entropy principle [13-14], which leads to uncertainty in the definition of the statistical entropy and consequently the equilibrium probability density functions. This means that the equilibrium statistical mechanics is in a crisis. Thus, the requirements of the equilibrium thermodynamics shall have an exclusive role in selection of the right theory for the equilibrium statistical mechanics. The main difficulty in foundation of the statistical mechanics based on the generalized statistical entropy, i.e., the deformed Boltzmann-Gibbs entropy, is the problem of its connection with the equilibrium thermodynamics. The proof of the zero law of thermodynamics and the principle of additivity in general serves as a primarily problem. The equilibrium thermodynamics is a phenomenological theory defined on the class of homogeneous functions of the zero and first order [15].

The formalism of the statistical mechanics agrees with the requirements of the equilibrium thermodynamics if the thermodynamic potential, which contains all information about the physical system, in the thermodynamic limit is a homogeneous function of the first order with respect to the extensive variables of state of the system [14, 6-7]. It was proved that for the Tsallis and Boltzmann-Gibbs statistics [6, 7], the Renyi statistics [10], and the incomplete nonextensive statistics [12], this property of thermodynamic potential provides the zeroth law of thermodynamics, the principle of additivity, the Euler theorem, and the Gibbs-Duhem relation if the entropic index zis an extensive variable of state. The scaling properties of the entropic index zand its relation to the thermodynamic limit for the Tsallis statistics were first discussed in the papers [16, 17].

The aims of this study are to establish the connection between the Tsallis statistics, i.e., the statistical mechanics based on the Tsallis statistical entropy, and the equilibrium thermodynamics and to prove the zero law of thermodynamics.

The structure of the chapter is as follows. In Section 2, we review the basic postulates of the equilibrium thermodynamics. The equilibrium statistical mechanics based on generalized entropy is formulated in a general form in Section 3. In Section 4, we describe the Tsallis statistics and analyze its possible connection with the equilibrium thermodynamics. The main conclusions are summarized in the final section.

2. Equilibrium thermodynamics

2.1. Thermodynamic potentials

In the equilibrium thermodynamics, the physical properties of the system are fully identified by the fundamental thermodynamic potential f=f(x1,,xn)as a real-valued function of nreal variables, which are called the variables of state. The macroscopic state of the system is fixed by the set of independent variables of state x=(x1,,xn). Each variable of state xi, which is related to the certain thermodynamic quantity, describes some individual property of the system. The first and the second partial derivatives of the thermodynamic potential with respect to the variables of state define the thermodynamic quantities (observables) of the system, which describe other individual properties of this system. The first differential and the first partial derivatives of the fundamental thermodynamic potential with respect to the variables of state can be written as

df=i=1nuidxi,     ui=fxi,E1

where the vector u=(u1,,un). Equation (1) is the fundamental equation of thermodynamics and expresses the first law of thermodynamics. The second differential of the fundamental thermodynamic potential is written as a quadratic form

d2f=i=1nj=1naijdxidxj,     aij=2fxixj,E2

where aijis the element of the symmetric matrix Aof the dimension (n×n). The symmetry conditions for the matrix elements, aij=aji, lead to the equalities (Maxwell relations)

ujxi=uixj        (i,j=1,,n).E3

If the function f(x1,,xn)is convex (concave) on snvariables of state, then the quadratic form (Eq. (2)) in svariables is positive definite (negative definite). The quadratic form (Eq. (2)) in svariables for which aij=ajiis positive definite (negative definite) if [18]

a11ξ>0, |a11a12a21a22|ξ2>0, , |a11a12a1sa21a22a2sas1as2ass|ξs>0E4

for every nonzero vector x, where ξ=1for the positive definite quadratic form and ξ=1for the negative definite quadratic form. Note that the fundamental thermodynamic potential f, the set of variables of state x, and the vector uconstitute the complete set of 2n+1variables, which completely define the given thermodynamic system.

The first thermodynamic potential g=g(y) is a function of a new set of independent variables of state y=(u1,,um,xm+1,,xn), which is obtained by the Legendre transform from the fundamental thermodynamic potential f(x1,,xn)changing mnvariables of state (x1,,xm)with their conjugate variables (u1,,um). The set of unknown variables x1,,xmis a solution of a system of mdifferential equations [19]:

fxi=ui     (i=1,,m).E5

Solving this system of equations, we obtain mfunctions of the variables of state,

xi=xi(u1,,um,xm+1,,xn)     (i=1,,m).E6

Substituting Eq. (6) into the fundamental thermodynamic potential fand using the Legendre transform, we obtain [19]

g=fi=1mfxixi=fi=1muixi.E7

This Legendre transform is always well defined when the fundamental thermodynamic potential f(x1,,xn)is a convex function of the variables (x1,,xm), i.e., the quadratic form i,j=1maijdxidxjis positive definite [19]. To obtain this, it is necessary and sufficient to satisfy the relations (4) for s=m[18].

The first differential and the first partial derivatives of the first thermodynamic potential gcan be written as

dg=i=1nvidyi,       vi=gyi,vi=gui=xi(i=1,,m),vi=gxi=ui  (i=m+1,,n).E8

The second differential and the second partial derivatives of the first thermodynamic potential gare

d2g =i=1nj=1nbijdyidyj,    bij=2gyiyj.E9

The symmetry conditions for the matrix elements, bij=bji, impose the following equalities

xjui=xiuj          (i,j=1,,m),ujui=xixj       (i=1,,m,j=m+1,,n),ujxi=uixj          (i,j=m+1,,n).E10

If the function g(u1,,um,xm+1,,xn)is convex (concave) on snvariables of state, then the quadratic form (Eq. (9)) in svariables is positive definite (negative definite). The quadratic form (Eq. (9)) in svariables for which bij=bjiis positive definite (negative definite) if [18]

b11ξ>0, |b11b12b21b22|ξ2>0, , |b11b12b1sb21b22b2sbs1bs2bss|ξs>0E11

for every nonzero vector y, where ξ=1for the positive definite quadratic form and ξ=1for the negative definite quadratic form.

The Legendre transform (Eq. (7)) is involutive [19], i.e., if under the Legendre transformation fis taken to g, then the Legendre transform of gwill again be f. The fundamental thermodynamic potential f(x1,,xn)can be obtained from the first thermodynamic potential g(u1,,um,xm+1,,xn)by the Legendre back-transformation

f=gi=1mguiui=g+i=1mxiui,E12

where mfunctions ui=ui(x1,,xn),i=1,,mare the solutions of the system of mdifferential equations

gui=xi    (i=1,,m).E13

This Legendre transform is well defined when the function g(u1,,um,xm+1,,xn)is a convex function of the variables (u1,,um), i.e., the quadratic form i,j=1mbijduidujis positive definite [19]. To obtain this, it is necessary and sufficient to satisfy the relations (Eq. (11)) for s=m[18].

The second thermodynamic potential h=h(r)is obtained from the fundamental thermodynamic potential f=f(x1,,xn)by expressing the variable xkthrough the set of independent variables r=(x1,,xk1,f,xk+1,,xn):

h=xk(x1,,xk1,f,xk+1,,xn),E14

where, now, xkis the second thermodynamic potential and fis a variable of state. The condition of independence of the variables of state rcan be written as

fxi+fhhxi=0.E15

Then the first differential and the first partial derivatives of the second thermodynamic potential hcan be written as

dh=i=1nwidri,    wi=hri,E16
wi=xkf=1uk       (i=k),wi=xkxi=uiuk    (i=1,,n,ik).E17

The second differential and the second partial derivatives of the second thermodynamic potential hcan be written as

d2h=i=1nj=1ncijdridrj,cij=2hrirj,E18
cii=2xkf2=1uk2ukf    (i=k).E19

The symmetry conditions for the matrix elements, cij=cji, impose the following equalities

uiuk2ukf1ukuif=1uk2ukxi             (ik),ujuk2ukxi1ukujxi=uiuk2ukxj1ukuixj     (i,jk).E20

2.2. Principle of additivity

In the equilibrium thermodynamics, all thermodynamic quantities belong to the class of homogeneous functions of zero and first order, which imposes the additional constraints on the thermodynamic system. The homogeneous function of kth order satisfies the relation [14, 15, 20]

f(λx1,,λxm,xm+1,,xn)=λkf(x1,,xm,xm+1,,xn)E21

and the Euler theorem[1] -

i=1mfxixi=kf,E22

where (x1,,xm)are extensive variables and (xm+1,,xn)are intensive variables. Note that the function fis extensive if k=1, and it is intensive if k=0.

Let us divide the system into two subsystems: system(1+2) = system(1) + system(2).

Then m-extensive and nm-intensive variables of state satisfy the additivity relations [6, 15]

xi1+2=xi1+xi2(i=1,,m),xi1+2 =xi1=xi2(i=m+1,,n).E23

The homogeneous function of the first degree (k=1), which is extensive, is an additive function of the first order [15]

f1+2(x11+2,,xn1+2)=f1(x11,,xn1)+f2(x12,,xn2)E24

and the homogeneous function of the zero degree (k=0), which is intensive, is an additive function of zero order [15]

f1+2(x11+2,,xn1+2)=f1(x11,,xn1)=f2(x12,,xn2).E25

Note that the zero law of thermodynamics is expressed by Eqs. (21) and (25) when the temperature Tis a function or the second equation of Eq. (23) when temperature Tis a variable of state.

3. Equilibrium statistical mechanics

In comparison with the equilibrium thermodynamics, the system in the equilibrium statistical mechanics is described by two additional elements: the microstates of the system and the probabilities of these microstates. As in the equilibrium thermodynamics, the macrostates of the system are fixed by the set of independent variables of state. The thermodynamic potential is a universal function that depends not only on the macroscopic state variables of the system but also on the microstates of the system and their probabilities. The extensive thermodynamic quantities are calculated as averages over the ensemble of microstates. However, the intensive thermodynamic quantities are defined in terms of the first derivatives of the thermodynamic potential with respect to the extensive variables of state.

Let us formulate the main statements of the equilibrium statistical mechanics. Let the thermodynamic potential be a function Y=Y(p1,,pW;X1,,Xn)of W-independent variables (p1,,pW)and nvariables of state (X1,,Xn). All arguments of the function Yare independent.[1] -

The first thermodynamic potential Y=g(p1,,pW;u1,,um,xm+1,,xn)is a function of mintensive variables of state Xj=uj (j=1,,m)conjugated to the variables (x1,,xm)and nmextensive variables of state Xj=xj (j=m+1,,n). The first thermodynamic potential Yis related to the fundamental thermodynamic potential f=f(p1,,pW;x1,,xn)by the Legendre transform (7) [19]

Y=fj=1mujxj,     uj=fxj.E26

Here and in the following, the first thermodynamic potential will be considered only for the statistical ensembles for which x1=S, X1=u1=T, and f=E, where Sis the entropy, Tis the temperature, and Eis the energy.

The second thermodynamic potential Y=h=xk(p1,,pW;x1,,xk1,f,xk+1,,xn)is a function of n1variables of state Xj=xj (j=1,,n,jk)and one variable Xk=ffor 1kn. In the following, the second thermodynamic potential will be associated only with the microcanonical ensemble (k=1)for which Y=x1=Sand X1=f=E.

Let Yiand xi1,,xinbe the values of the dynamical extensive variables Yand x1,,xn, respectively, in the ith microscopic state of the system. Moreover, let us impose an additional constraint on the variables (p1,,pW)[18],

φ(p1,,pW;X1,,Xn)=iδXm+1,Xim+1δXn,Xinpi1=0,E27

where δx,x'is the Kronecker delta for the integer x,x'and the Dirac delta function for the real x,x'. In Eq. (27), the variables Xij=xij (j=m+1,,n)are for the first thermodynamic potential, and m=0, X1=f=E, Xi1=fi=Ei, and Xij=xij (j=2,,n)are for the second thermodynamic potential.

The ensemble averages for the extensive dynamical variables Acan be written as

A(p1,,pW;X1,,Xn)=iδXm+1,Xim+1δXn,XinpiAi,E28

where Aiis the value of the variable Ain the ith microscopic state of the system.

The first and the second thermodynamic potentials, which are the extensive functions of the variables of state, can also be written as (28)

Y(p1,,pW;X1,,Xn)=iδXm+1,Xim+1δXn,XinpiYi,E29
Yi=fij=1mujxij       for Y=g,E30
Yi=xi1=Si        for Y=x1=S,E31

where Siand fiare the values of the entropy Sand the fundamental thermodynamic potential f, respectively, in the ith microstate of the system, which are both determined by Eq. (28).

In the equilibrium statistical mechanics, the unknown probabilities of microstates {pi}are found from the second part of the second law of thermodynamics, i.e., from the constrained extremum of the thermodynamic potential (Eq. (29)) as a function of the variables (p1,,pW)under the condition that the variables (p1,,pW)satisfy Eq. (27). Moreover, it is supposed that the value of the entropy in the ith microstate of the system is a function of the probability piof this microstate, i.e., xi1=Si=Si(pi). Then to determine the unknown probabilities {pi}at which the function Yattains the constrained local extrema, the method of Lagrange multipliers [18] can be used

Φ(p1,,pW;X)=Y(p1,,pW;X)λφ(p1,,pW;X),E32
Φ(p1,,pW;X)pi=0     (i=1,,W),E33

where λis an arbitrary constant and the vector X=(X1,,Xn). Substituting Eqs. (27) and (29) into Eq. (32), we obtain

Yi+piYipiλ=0    (i=1,,W).E34

Substituting Eq. (30) into Eq. (34) and using Eq. (27), we obtain the probabilities related to the first thermodynamic potential

pi=ψ(1u1(Λfi+j=2mujxij)),E35
iδxm+1,xim+1δxn,xinψ(1u1(Λfi+j=2mujxij))=1,E36

where Λϕ(λ)=Λ(u1,,um,xm+1,,xn)is the solution of Eq. (36) and ψ(x)is a function related to the given function xi1=Si(pi).

Substituting Eq. (31) into Eq. (34) and using Eq. (27), we obtain the probabilities related to the second thermodynamic potential

pi=1W(f,x2,,xn),E37
W(f,x2,,xn)=iδf,fiδx2,xi2δxn,xin,E38

where pi=ψ(λ)is a constant the same for all microstates of the system. Note that in these derivations, the conditions fi/pi=0, uj/pi=0 (j=1,,m), and xij/pi=0 (j=2,,m)were used.

Let us consider the first thermodynamic potential. Substituting Eq. (35) into Eq. (26) and using Eq. (36), we obtain the expression for the first thermodynamic potential as

Y(u1,,um,xm+1,,xn)=f(u1,,um,xm+1,,xn)                                       j=1mujxj(u1,,um,xm+1,,xn).E39

Then the partial derivatives of the first thermodynamic potential (Eq. (39)) with respect to the variables of state can be written as

xk=Yuk=iδxm+1,xim+1δxn,xinpixik           (k=1,,m),E40
uk=Yxk=iδxm+1,xim+1δxn,xinpifixk              +iδxm+1,xim+1δxn,xin(δxk,xikxk)piYi     (k=m+1,,n).E41

Here we used the conditions fi/uk=0, xij/uk=0 (j=2,,n)and Eqs. (34) and (36).

The fundamental thermodynamic potential can be written as

f=Yj=1mujYuj=iδxm+1,xim+1δxn,xinpifi.E42

Let us consider the second thermodynamic potential. Substituting Eqs. (37), (38), and (31) into Eq. (29), we obtain the expression for the second thermodynamic potential

Y(f,x2,,xn)=Si(pi),E43

where piis determined from Eq. (37) and Y=S. The partial derivatives (Eq. (17)) of the second thermodynamic potential (Eq. (43)) with respect to the variables of state can be written as

1u1=Yf=γlnWf,                         γpiSi(pi)pi,E44
uju1=Yxj=γlnWxjSixj|pi=const    (j=2,,n),E45

where Wis determined from Eq. (38).

Finally, it should be mentioned that the equilibrium statistical mechanics is thermodynamically self-consistent if the statistical variables (x1,,xn), the potentials (f,g,), and the variables (u1,,un)are homogeneous variables of the first- or zero-order satisfying Eqs. (21)-(25).

4. Tsallis statistical mechanics

The Tsallis statistical mechanics is based on the generalized entropy which is a function of the entropic parameter qand probing probabilities pi[3, 4]:

S=ipiSi,Si=kBz(1pi1/z),E46

where z=1/(q1), kBis the Boltzmann constant, and qis a real parameter, 0<q<. In the limit q1 (z±), the entropy (Eq. (46)) recovers the usual Boltzmann-Gibbs entropy S=ipiSi, where Si=kBlnpi[3]. Note that throughout the work, we use the system of natural units =c=kB=1.

4.1. Canonical ensemble

The thermodynamic potential of the canonical ensemble, the Helmholtz free energy, is the first thermodynamic potential g=F, which is a function of the variables of state u1=T, x2=V, x3=N, and x4=z. It is obtained from the fundamental thermodynamic potential f=E(the energy) by the Legendre transform (Eq. (7)), exchanging the variable of state x1=Sof the fundamental thermodynamic potential with its conjugate variable u1=T. In the canonical ensemble, the first partial derivatives (Eq. (1)) of the fundamental thermodynamic potential are defined as u2=p, u3=μ, and u4=Ξ. The entropy (Eq. (46)) for the Tsallis and Boltzmann-Gibbs statistics in the canonical ensemble can be rewritten as

S=iδV,ViδN,Niδz,zipiSi,E47
Si=zi(1pi1/zi)          for |z|<,E48
Si=lnpi                 for |z|=.E49

The first thermodynamic potential (Eqs. (26) and (29)), Y=F, for the Tsallis and Boltzmann-Gibbs statistics can be rewritten as

F=ETS=iδV,ViδN,Niδz,zipiFi,FiEiTSi,E50
Fi=EiTzi(1pi1/zi)          for |z|<,E51
Fi=Ei+Tlnpi                    for |z|=.E52

Here the constraint (Eq. (27)) in the canonical ensemble is in the form

φ=iδV,ViδN,Niδz,zipi1=0.E53

Applying the method of Lagrange multipliers (Eqs. (32)-(34)) with the Lagrange function Φ=Fλφto Eqs. (50)-(53), we can write down Eqs. (34) and (35) for the Tsallis and Boltzmann-Gibbs statistics immediately as

Fi+piFipiλ=0E54

and [7]

pi=[1+1zi+1ΛEiT]zi    for |z|<,E55
pi=eΛEiT                           for |z|=,E56

where ΛλTand Ei/pi=0. Then the constraint (Eq. (53)) for the probabilities (Eqs. (55) and (56)) of the Tsallis and Boltzmann-Gibbs statistics can be written as [7]

iδV,ViδN,Niδz,zi[1+1zi+1ΛEiT]zi=1    for |z|<,E57
iδV,ViδN,Niδz,zieΛEiT=1                            for |z|=,E58

where Λ=Λ(T,V,N,z)is the solution of Eq. (57) for the Tsallis statistics and Λ=TlnZGis the solution of Eq. (58) for the Boltzmann-Gibbs statistics, where ZG=iδV,ViδN,Niδz,zieEi/Tis the partition function. Substitution of the probabilities given in Eqs. (55) and (56) into Eqs. (50)-(53) gives the Helmholtz free energy [7]

F=zz+1(Λ+Ez)           for |z|<,E59
F=Λ=TlnZG             for |z|=,E60

where Eis the energy (Eq. (42)), which can be written in terms of the canonical ensemble as

E=FTFT=iδV,ViδN,Niδz,zipiEi.E61

Making use of Eqs. (40), (50), and (54), we can write the entropy of the system as

S=FT=iδV,ViδN,Niδz,zipiSi.E62

Here we have used the conditions that the derivative of the constraint (Eq. (53)) with respect to Tis zero, Ei/T=0, Ei/pi=0, and T/pi=0. Substituting Eqs. (48), (49), (55), and (56) into Eq. (62) and using Eqs. (57) and (58), we obtain [7]

S=zz+1ΛET          for |z|<,E63
S=ΛET                  for |z|=.E64

Using Eqs. (41), (50), and (54), we obtain the pressure, u2=p, and the chemical potential, u3=μ:

p=FV=iδV,ViδN,Niδz,zipiEiV+iδN,Niδz,zi(δV,ViV)piFi,E65
μ=FN=iδV,ViδN,Niδz,zipiEiN+iδV,Viδz,zi(δN,NiN)piFi.E66

Here we have used the conditions that the derivatives of the constraint (Eq. (53)) with respect to the variables of state Nand Vare zero, Ei/pi=0and T/pi=0.

Substituting Eqs. (50) and(54) into Eq. (41), we obtain the variable Ξ:

Ξ=Fz=TiδV,ViδN,Niδz,zipi[1pi1/zi(1lnpi1/zi)]              +iδV,ViδN,Ni(δz,ziz)piFi                            for |z|<,E67
Ξ=Fz=0                                                                for |z|=,E68

where we have used the conditions that the derivative of the constraint (53) with respect to zis zero, Ei/z=0, Ei/pi=0, and T/pi=0.

Thus, from the results given in Eqs. (62) and (65)-(67), we see that the differential of the thermodynamic potential (Eq. (50)) satisfies [7, 15]

dF=SdTpdV+μdNΞdz.E69

Using Eqs. (50) and (69), we obtain the fundamental equation of thermodynamics [7, 14, 15]

TdS=dE+pdVμdN+Ξdz.E70

To prove the homogeneity properties of the thermodynamic quantities and the Euler theorem for the Tsallis statistics in the canonical ensemble, we will consider, as an example, the exact analytical results for the nonrelativistic ideal gas.

4.1.1. Nonrelativistic ideal gas: canonical ensemble

Let us investigate the nonrelativistic ideal gas of identical particles governed by the classical Maxwell-Boltzmann statistics in the framework of the Tsallis and Boltzmann-Gibbs statistical mechanics.

It is convenient to obtain the exact results for the ideal gas in the Tsallis statistics by means of the integral representation for the Gamma function (see [9] and reference therein):

xy=1Γ(y)0dtty1etx,                   Rex>0, Rey>0,E71
xy1=Γ(y)i2πCdt(t)yetx,          Rex>0, |y|<.E72

Thus, solving Eqs. (57) and (58) for the ideal gas in the canonical ensemble, the norm function Λis [7]

1+1z+1ΛT=[ZGΓ(z3N/2)(z1)3N/2Γ(z)]1z+3N/2     for z<1,E73
1+1z+1ΛT=[ZG(z+1)3N/2Γ(z+1)Γ(z+1+3N/2)]1z+3N/2     for z>0,E74
ZG=(gV)NN!(mT2π)32NE75

and Λ=TlnZGfor |z|=. Here mis the particle mass, and Eq. (73) is restricted by the condition z3N/2>0.

The energy (Eq. (61)) and the thermodynamic potentials (Eqs. (59) and (60)) for the ideal gas in the canonical ensemble for the Tsallis and Boltzmann-Gibbs statistics can be written as [7]

E=32TNη,   η=(1+1z+1ΛT)(1+1z+132N)1    for |z|<,E76
F=T[z(z+32N)η]                                       for |z|<E77

and E=3TN/2and F=Λ=TlnZGfor |z|=.

The entropies (Eqs. (63) and (64)) and the pressure (Eq. (65)) for the ideal gas in the canonical ensemble for the Tsallis and Boltzmann-Gibbs statistics can be written as [7]

S=z(1η)                        for |z|<,E78
p=NVTη=23EV               for |z|<E79

and S=lnZG+3N/2, p=NT/Vfor |z|=.

The chemical potential (Eq. (66)) and the variable (Eqs. (67) and (68)) for the ideal gas in the canonical ensemble for the Tsallis and Boltzmann-Gibbs statistics become [7]

μ=Tη[ln(gV(mT2πγ1(z+1)(1+1z+1ΛT))3/2)321z+1+3N/2]  +Tη[ψ(N+1)+32ψ(a1+γ132N)]                                               for |z|<,E80
Ξ=T[1η(1+z(z+1)232N)(1+1z+132N)1]   +Tη[ln(1+1z+1ΛT)+ψ(a1)ψ(a1+γ132N)]                        for |z|<E81

and μ=T[ln(gV(mT/2π)3/2)ψ(N+1)], Ξ=0for |z|=, where ψ(y)is the psi-function, a1=z, γ1=1for z<1and a1=z+1, γ1=1for z>0.

4.1.2. Nonrelativistic ideal gas in the thermodynamic limit: canonical ensemble

Let us try to express the thermodynamic quantities of the nonrelativistic ideal gas directly in terms of the thermodynamic limit when the entropic parameter zis considered as an extensive variable of state

V, N, |z|, v=VN=const, z˜=zN=const.E82

Note first that the canonical partition function (Eq. (75)) for the nonrelativistic ideal gas for the Boltzmann-Gibbs statistics can be rewritten as

ZG=Z˜GN,  Z˜Ggve(mT2π)3/2.E83

The norm functions (Eqs. (73) and (74)) for the ideal gas in the thermodynamic limit in the Tsallis and Boltzmann-Gibbs statistics can be rewritten as [7]

Λ=TN[z˜(z˜+32)(Z˜Ge3/2)1z˜+32]    for |z˜|<,E84
Λ=TNlnZ˜G                                     for |z˜|=,E85

where Eq. (84) is restricted by the conditions z˜<3/2and z˜>0.

In the thermodynamic limit (Eq. (82)), the energy of the system (Eq. (76)) and the thermodynamic potential (Eq. (77)) for the ideal gas in the canonical ensemble for the Tsallis and Boltzmann-Gibbs statistics become [7]

E=32TN(Z˜Ge3/2)1z˜+32           for |z˜|<,E86
E=32TN                            for |z˜|=E87

and [7]

F=Λ=TN[z˜(z˜+32)(Z˜Ge3/2)1z˜+32]    for |z˜|<,E88
F=Λ=TNlnZ˜G                                   for |z˜|=,E89

respectively. The entropy (78) and the pressure (79) for the ideal gas in the Tsallis and Boltzmann-Gibbs statistics in the thermodynamic limit can be written as [7]

S=Nz˜[1(Z˜Ge3/2)1z˜+32]       for |z˜|<,E90
S=N(lnZ˜G+3/2)             for |z˜|=E91

and [7]

p=Tv(Z˜Ge3/2)1z˜+32=23εv         for |z˜|<,E92
p=Tv=23εv                            for |z˜|=,E93

where ε=E/Nis the specific energy given in Eqs. (86) and (87).

In the thermodynamic limit (82), the chemical potential (80) and the variable (81) for the ideal gas in the canonical ensemble for the Tsallis and Boltzmann-Gibbs statistics are [7]

μ=T(Z˜Ge3/2)1z˜+32[52+z˜ln(Z˜Ge3/2)1z˜+32]   for |z˜|<,E94
μ=T(1lnZ˜G)                                         for |z˜|=E95

and [7]

Ξ=T[1(Z˜Ge3/2)1z˜+32(1ln(Z˜Ge3/2)1z˜+32)]     for |z˜|<,E96
Ξ=0                                                                for |z˜|=.E97

Thus, from the results for the Tsallis statistics given in Eqs. (86), (90), (92), (94), and (96), we see that the Euler theorem (Eq. (22)) is satisfied [7]

TS=E+pVμN+Ξz.E98

Moreover, the thermodynamic quantities (86), (88), (90), (92), (94) and (96) satisfy the relation for the thermodynamic potential

F=ETS=pV+μNΞz.E99

Next we shall verify that, when the entropic parameter zis an extensive variable of state in the thermodynamic limit, the ideal gas is in accordance with the principle of additivity. Suppose that the system is divided into two subsystems (1 and 2). Then the extensive variables of state of the canonical ensemble are additive

V1+2=V1+V2,     N1+2=N1+N2,     z1+2=z1+z2.E100

However, the temperature and the specific variables of state (Eq. (82)) are the same in each subsystem

T1+2=T1=T2,    v1+2=v1=v2,    z˜1+2=z˜1=z˜2.E101

Considering Eqs. (83), (100), and (101), we can verify that the Tsallis thermodynamic potential (Eq. (88)) and the entropy (Eq. (90)) of the canonical ensemble are homogeneous functions of the first order, i.e., F(T,V,N,z)/N=F(T,v,z˜)and S(T,V,N,z)/N=S(T,v,z˜), respectively, and they are additive (extensive)

F1+2(T1+2,V1+2,N1+2,z1+2)=F1(T1,V1,N1,z1)+F2(T2,V2,N2,z2),E102
S1+2(T1+2,V1+2,N1+2,z1+2)=S1(T1,V1,N1,z1)+S2(T2,V2,N2,z2).E103

The Tsallis pressure (Eq. (92)), the chemical potential (Eq. (94)), and the variable (Eq. (96)) are the homogeneous functions of the zero order, i.e., p(T,V,N,z)=p(T,v,z˜), μ(T,V,N,z)=μ(T,v,z˜), and Ξ(T,V,N,z)=Ξ(T,v,z˜), respectively, and they are the same in each subsystem (intensive)

p1+2(T1+2,V1+2,N1+2,z1+2)=p1(T1,V1,N1,z1)=p2(T2,V2,N2,z2),E104
μ1+2(T1+2,V1+2,N1+2,z1+2)=μ1(T1,V1,N1,z1)=μ2(T2,V2,N2,z2),E105
Ξ1+2(T1+2,V1+2,N1+2,z1+2)=Ξ1(T1,V1,N1,z1)=Ξ2(T2,V2,N2,z2).E106

Thus, the principle of additivity (Eqs. (21), (24), and (25)) is totally satisfied by the Tsallis statistics. Equations (101) and (103) prove the zero law of thermodynamics for the canonical ensemble.

4.2. Microcanonical ensemble

The thermodynamic potential of the microcanonical ensemble, the entropy, is the second thermodynamic potential h=x1=Sdefined in Eq. (14), which is a function of the variables of state f=E, x2=V, x3=Nand x4=z. It is obtained from the fundamental thermodynamic potential fby exchanging the variable of state x1with variable f. In the microcanonical ensemble, the first partial derivatives of the fundamental thermodynamic potential (1) are defined as u1=T, u2=p, u3=μ, and u4=Ξ.

The entropy Sfor the Tsallis and Boltzmann-Gibbs statistics in the microcanonical ensemble can be written as

S=iδE,EiδV,ViδN,Niδz,zipiSi,E107

where Siis defined in Eqs. (48) and (49). The set of probabilities {pi}is constrained by Eq. (27):

φ=iδE,EiδV,ViδN,Niδz,zipi1=0.E108

Applying the method of Lagrange multipliers (Eqs. (32)-(34)) with the Lagrange function Φ=Sλφto Eqs. (107), (108), (48), and (49), we can write down Eqs. (34), (37), and (38) for the Tsallis and Boltzmann-Gibbs statistics immediately as [6]

Si+piSipiλ=0,E109
pi=1W,E110
W=iδE,EiδV,ViδN,Niδz,zi,E111

where W=W(E,V,N)is the statistical weight for the Tsallis and Boltzmann-Gibbs statistics and zi=zfor all microstates. Substituting Eqs. (110) and (111) into Eqs. (107), (48), and (49) and using Eq. (108), we can express the second thermodynamic potential (Eq. (43)) as [6]

S=z(1W1/z)        for |z|<,E112
S=lnW                    for |z|=.E113

Then the first derivative (Eq. (44)) of the thermodynamic potential Swith respect to the variable of state E, i.e., the temperature T, can be rewritten as [6]

1T=SE=W1/zlnWE        for |z|<,E114
1T=SE=lnWE                 for |z|=.E115

The first derivative (Eq. (45)) of the thermodynamic potential Swith respect to the variable of state V, i.e., the pressure p, becomes [6]

p=TSV=TW1/zlnWV        for |z|<,E116
p=TSV=TlnWV                 for |z|=.E117

The first derivative (Eq. (45)) of the thermodynamic potential Swith respect to the variable of state N, i.e., the chemical potential μ, is [6]

μ=TSN=TW1/zlnWN        for |z|<,E118
μ=TSN=TlnWN                for |z|=.E119

The first derivative (Eq. (45)) of the thermodynamic potential Swith respect to the variable of state z, i.e., the variable Ξ, can be rewritten as [6]

Ξ=TSz=T[1W1/z(1lnW1/z)]        for |z|<,E120
Ξ=0                                                            for |z|=,E121

where W/z=0. Then the differential of the thermodynamic potential (107) satisfies the fundamental equation of thermodynamics (70).

4.2.1. Nonrelativistic ideal gas: microcanonical ensemble

Let us consider the nonrelativistic ideal gas of Nidentical particles governed by the classical Maxwell-Boltzmann statistics in the framework of the Tsallis and Boltzmann-Gibbs statistics in the microcanonical ensemble. For this special model, the statistical weight (111) can be written as (see [6] and reference therein)

W=(gV)NN!(m2π)32NE32N1Γ(32N),E122

where mis the particle mass. Then the entropies of the ideal gas for the Tsallis and Boltzmann-Gibbs statistics are calculated by Eqs. (112), (113), and (122).

The temperatures (Eqs. (114) and (115)) for the ideal gas for the Tsallis and Boltzmann-Gibbs statistics in the microcanonical ensemble correspond to [6]

T=EW1/z3N/21            for |z|<,E123
T=E3N/21             for |z|=.E124

The pressures (Eqs. (116) and (117)) and the chemical potentials (Eqs. (118) and (119)) for the ideal gas for the Tsallis and Boltzmann-Gibbs statistics in the microcanonical ensemble can be written as [6]

p=NVE3N/21                                 for |z|< and |z|=, E125
μ=E3N/21[ln(gV(mE2π)3/2)ψ(N+1)32ψ(32N)]                                                           for |z|< and |z|=.E126

The variable (Eq. (120)) for the ideal gas for the Tsallis statistics in the microcanonical ensemble is [6]

Ξ=E3N/21[1W1/z+lnW1/z]        for |z|<.E127

However, the variable (Eq. (121)) for the Boltzmann-Gibbs statistics vanishes, Ξ=0.

4.2.2. Nonrelativistic ideal gas in the thermodynamic limit: microcanonical ensemble

Let us rewrite the thermodynamic quantities of the nonrelativistic ideal gas in the microcanonical ensemble in the terms of the thermodynamic limit when the entropic parameter zis considered to be an extensive variable of state

E, V, N, |z|,ε=EN=const, v=VN=const, z˜=zN=const.  E128

Then in the thermodynamic limit (Eq. (128)), the statistical weight (Eq. (122)) for the nonrelativistic ideal gas can be rewritten as [6]

W=wN,         wgv(mεe5/33π)3/2.E129

Substituting Eq. (129) into Eqs. (112) and (113) and using Eq. (128), we obtain the entropy as [6]

S=Nz˜(1w1/z˜)             for |z˜|<,E130
S=Nlnw                        for |z˜|=.E131

The temperatures (Eqs. (123) and (124)) for the nonrelativistic ideal gas in the thermodynamic limit (128) can be rewritten as [6]

T=23εw1/z˜            for |z˜|<,E132
T=23ε                  for |z˜|=.E133

The pressure (125) and the chemical potential (126) for the nonrelativistic ideal gas in the thermodynamic limit (128) become [6]

p=23εv                        for |z˜|< and |z˜|=, E134
μ=23ε(52lnw)      for |z˜|< and |z˜|=.E135

The variable (Eq. (127)) for the Tsallis statistics in the thermodynamic limit (Eq. (128)) corresponds to [6]

Ξ=23ε[1w1/z˜+lnw1/z˜]        for |z˜|<.E136

For the Boltzmann-Gibbs statistics, we have Ξ=0. Using the results so far obtained for the Tsallis statistics given in Eqs. (130), (132), and (134)-(136), we can verify that the Euler theorem defined in Eqs. (22) and (98) is satisfied [6], i.e., TS=E+pVμN+Ξz.

Let us verify the principle of additivity for the nonrelativistic ideal gas in the microcanonical ensemble in the thermodynamic limit when the entropic parameter zis an extensive variable of state. Suppose that the system is divided into two subsystems (1 and 2). Then the extensive variables of state of the microcanonical ensemble are additive [6]

E1+2=E1+E2,    V1+2=V1+V2,    N1+2=N1+N2,    z1+2=z1+z2.E137

However, the specific variables of state (Eq. (128)) are the same in each subsystem (intensive)

ε1+2=ε1=ε2,     v1+2=v1=v2,     z˜1+2=z˜1=z˜2.E138

Considering Eqs. (129), (137), and (138), we can verify that the Tsallis thermodynamic potential (Eq. (130)) of the microcanonical ensemble is a homogeneous function of the first order, i.e., S(E,V,N,z)/N=S(ε,v,z˜), and it is additive (extensive) [6]

S1+2(E1+2,V1+2,N1+2,z1+2)=S1(E1,V1,N1,z1)+S2(E2,V2,N2,z2).E139

Now, considering Eqs. (129), (137), and (138), we find that the Tsallis temperature (Eq. (132)), the pressure (Eq. (134)), the chemical potential (Eq. (135)), and the variable (Eq. (136)) are the homogeneous functions of the zero order, i.e., T(E,V,N,z)=T(ε,v,z˜), p(E,V,N,z)=p(ε,v,z˜), μ(E,V,N,z)=μ(ε,v,z˜), and Ξ(E,V,N,z)=Ξ(ε,v,z˜), respectively, and they are the same in each subsystem (intensive) [6]

T1+2(E1+2,V1+2,N1+2,z1+2)=T1(E1,V1,N1,z1)=T2(E2,V2,N2,z2),E140
p1+2(E1+2,V1+2,N1+2,z1+2)=p1(E1,V1,N1,z1)=p2(E2,V2,N2,z2),E141
μ1+2(E1+2,V1+2,N1+2,z1+2)=μ1(E1,V1,N1,z1)=μ2(E2,V2,N2,z2),E142
Ξ1+2(E1+2,V1+2,N1+2,z1+2)=Ξ1(E1,V1,N1,z1)=Ξ2(E2,V2,N2,z2).E143

Thus, the principle of additivity (Eqs. (21), (24), and (25)) is totally satisfied by the Tsallis statistics in the microcanonical ensemble. Equation (140) proves the zero law of thermodynamics for the microcanonical ensemble [6].

4.3. Equivalence of canonical and microcanonical ensembles

We can now easily prove the equivalence of the canonical and microcanonical ensembles for the Tsallis statistics in the thermodynamic limits (Eqs. (82) and (128)). Using Eqs. (83) and (129), it is easy to verify that Eq. (132) for the temperature of the microcanonical ensemble and Eq. (86) for the energy of canonical ensemble are identical. Comparing Eqs. (83) and (129) and using Eq. (86), we have

w=(Z˜Ge3/2)z˜z˜+32.E144

Substituting Eq. (144) into Eq. (130) for the entropy of the microcanonical ensemble, we obtain the entropy of the canonical ensemble (Eq. (90)). Equation (134) for the pressure of the microcanonical ensemble is identical to Eq. (92) for the pressure of the canonical ensemble. Substituting Eqs. (144) and (86) into Eq. (135) for the chemical potential of the microcanonical ensemble, we obtain Eq. (94) for the chemical potential of the canonical ensemble. Moreover, substituting Eqs. (144) and (86) into Eq. (136) for the variable Ξof the microcanonical ensemble, we obtain Eq. (96) for the variable Ξof the canonical ensemble. Thus, for the Tsallis statistics, the canonical and microcanonical ensembles are equivalent in the thermodynamic limit when the entropic parameter zis considered to be an extensive variable of state.

5. Conclusions

In conclusion, let us summarize the main principles of the equilibrium statistical mechanics based on the generalized statistical entropy. The basic idea is that in the thermodynamic equilibrium, there exists a universal function called thermodynamic potential that completely describes the properties and states of the thermodynamic system. The fundamental thermodynamic potential, its arguments (variables of state), and its first partial derivatives with respect to the variables of state determine the complete set of physical quantities characterizing the properties of the thermodynamic system. The physical system can be prepared in many ways given by the different sets of the variables of state and their appropriate thermodynamic potentials. The first thermodynamic potential is obtained from the fundamental thermodynamic potential by the Legendre transform. The second thermodynamic potential is obtained by the substitution of one variable of state with the fundamental thermodynamic potential. Then the complete set of physical quantities and the appropriate thermodynamic potential determine the physical properties of the given system and their dependences. In the equilibrium thermodynamics, the thermodynamic potential of the physical system is given a priori, and it is a multivariate function of several variables of state. However, in the equilibrium statistical mechanics, the thermodynamic potential is a composed function that can depend on the set of independent variables of state explicitly and implicitly through the probabilities of microstates. The probabilities of microstates are determined from the second part of the second law of thermodynamics, i.e., the maximum entropy principle. The equilibrium probability distributions are found from the constrained extremum of the thermodynamic potential as a function of a multidimensional set of probabilities considering that the statistical entropy is defined. The equilibrium thermodynamics and statistical mechanics are defined only on the class of homogeneous functions, i.e., all thermodynamic quantities describing the thermodynamic system should belong to the class of homogeneous functions of the first or zero orders.

In the present work, the general mathematical scheme of construction of the equilibrium statistical mechanics on the basis of an arbitrary definition of statistical entropy for two types of thermodynamic potential, the first and the second thermodynamic potentials, was proposed. As an example, we investigated the Tsallis and Boltzmann-Gibbs statistical entropies in the canonical and microcanonical ensembles. On the example of a nonrelativistic ideal gas, it was proven that the statistical mechanics based on the Tsallis entropy satisfies the requirements of the equilibrium thermodynamics only in the thermodynamic limit when the entropic index zis an extensive variable of state of the system. In this case the thermodynamic quantities of the Tsallis statistics belong to one of the classes of homogeneous functions of the first or zero orders.

Acknowledgments

This work was supported in part by the joint research project of JINR and IFIN-HH, protocol N 4342.

Notes

  • In this subsection, the symbol f denotes any function not only the fundamental thermodynamic potential.
  • In this section, the thermodynamic quantities are numbered by the index at the top. The index at the bottom of the variable denotes the microstate of the system.

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A.S. Parvan (December 21st 2015). Foundation of Equilibrium Statistical Mechanics Based on Generalized Entropy, Recent Advances in Thermo and Fluid Dynamics, Mofid Gorji-Bandpy, IntechOpen, DOI: 10.5772/60997. Available from:

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