# Logarithmic Generalization of the LambertW Function and Its Applications to Adiabatic Thermostatistics of the Three-Parameter Entropy

Authors: Cristina B. Corcino; Roberto B. Corcino
Journal: Advances in Mathematical Physics
 (2021)

---

## Abstract

A generalization of the LambertW function called the logarithmic Lambert function is introduced and is found to be a solution to the thermostatistics of the three-parameter entropy of classical ideal gas in adiabatic ensembles. The derivative, integral, Taylor series, approximation formula, and branches of the function are obtained. The heat functions and specific heats are computed using the “unphysical” temperature and expressed in terms of the logarithmic Lambert function.


---

## Body

## 1. Introduction

The first law of thermodynamics [1] states that the total energy of a system remains constant, even if it is converted from one form to another. For example, kinetic energy—the energy that an object possesses when it moves—is converted to heat energy when a driver presses the brakes on the car to slow it down. The first law of thermodynamics relates the various forms of kinetic and potential energy in a system to the work, which a system can perform, and to the transfer of heat. This law is sometimes taken as the definition of internal energy and also introduces an additional state variable, enthalpy. The first law of thermodynamics allows for many possible states of a system to exist. However, experience indicates that only certain states occur. This eventually leads to the second law of thermodynamics and the definition of another state variable called entropy.Entropy is a measure of the number of specific ways in which a thermodynamic system may be arranged, commonly understood as a measure of disorder. According to the second law of thermodynamics, the entropy of an isolated system never decreases; such a system will spontaneously proceed towards thermodynamic equilibrium, the configuration with maximum entropy [2].A system in thermodynamic equilibrium with its surroundings can be described using three macroscopic variables corresponding to the thermal, mechanical, and chemical equilibrium. For each fixed value of these macroscopic variables (macrostates), there are many possible microscopic configurations (microstates). A collection of systems existing in the various possible microstates, but characterized by the same macroscopic variables, is called an ensemble. The adiabatic class has the heat function as its thermal equilibrium variable. The specific form of each of the four adiabatic ensembles and its heat function and corresponding entropy are listed in Table1 (see [3]).Table 1
Adiabatic ensembles.
EnsembleHeat functionEntropyMicrocanonicalN,V,EInternal energyESN,V,EIsoenthalpic-isobaricN,P,HEnthalpyH=E+PVSN,P,HThird adiabatic ensembleμ,V,LHill energyL=E−μNSμ,V,LFourth adiabatic ensembleμ,P,RRay energyR=E+PV−μNSμ,P,RIt is known that some physical systems cannot be described by Boltzmann-Gibbs (BG) statistical mechanics [4, 5]. Among these physical systems are diffusion [6], turbulence [7], transverse momentum distribution of hadron jets in e+e− collisions [8], thermalization of heavy quarks in a collisional process [9], astrophysics [10], and solar neutrinos [11]. To overcome some difficulties in dealing with these systems, Tsallis [12] introduced a generalized entropic form, the q-entropy:(1)Sq=k∑i=1ωpilnq1pi,where k is a positive constant and ω is the total number of microscopic states.Forq>0, lnqx called the q-logarithm is defined as(2)lnqx=x1−q−11−q,ln1x=lnx.The inverse function of theq-logarithm is called q-exponential and is given by(3)expqx=1+1−qx1/1−q,exp1x=expx.In the case of equiprobability, BG is recovered in the limitq⟶1.A two-parameter entropySq,q′ that recovered the q-entropy Sq in the limit q′⟶1 was defined in [13] as(4)Sq,q′x≡∑i=1ωpilnq,q′1pi=11−q′∑i=1ωpiexp1−q′1−qpiq−1−1−1.Applications ofSqto a class of energy-based ensembles were done in [14] while applications of Sq,q′ to adiabatic ensembles were done in [3]. Results in the applications of Sq,q′ involved the well-known Lambert W function.A three-parameter entropySq,q′,r that recovers Sq,q′ in the limit r⟶1 was defined in [15] as(5)Sq,q′,r≡k∑i=1ωpilnq,q′,r1pi,where k is a positive constant and(6)lnq,q′,rx≡11−rexp1−r1−q′e1−q′lnqx−1−1.This three-parameter entropy was shown to be nonextensive (see [15]).One of the interesting properties of entropy is the Lesche stability [16]. The Lesche stability of the κ-entropy was shown in [17].It was shown in [15] that the three-parameter logarithm lnq,q′,rx is differentiable. Hence, the function(7)fx=xlnq,q′,r1x,x≠0is also differentiable. Since differentiability implies continuity, for every ϵ>0, there exists δi>0 such that if ρi−λi<δi, then fρi−fλi<ϵlnq,q′,rω/ω. Now, given(8)ρ≡ρi:i=1,2,⋯,ω,λ≡λi:i=1,2,⋯,ω,ρ−λ1=∑i=1ωρi−λi<∑i=1ωδiω≤δ,where δ=minδ1,δ2,⋯,δω. This implies that(9)Sq,q′,rρ−Sq,q′,rλ=k∑i=1ωρilnq,q′,r1ρi−k∑i=1ωλilnq,q′,r1λi≤k∑i=1ωρilnq,q′,r1ρi−λilnq,q′,r1λi<k∑i=1ωϵlnq,q′,rωω=kϵlnq,q′,rω.Note that the maximum value ofSq,q′,rρ corresponding to the uniform distribution ρ≡ρi=1/ω:i=1,2,⋯,ω is given by (see [15])(10)Sq,q′,rmax=klnq,q′,rω.Thus, we have(11)Sq,q′,rρ−Sq,q′,rλSq,q′,rmax<ϵ.Therefore, the three-parameter entropy is Lesche-stable. Moreover, it was also shown that the three-parameter entropy is concave and convex in specified ranges of the parameters (see [15]).The goal of this paper is to introduce a generalized LambertW function and derive its applications to the adiabatic thermostatistics of the three-parameter entropy of classical ideal gas. This generalized Lambert W function will be called the logarithmic Lambert function. As the logarithmic Lambert function is new, it is imperative that we study its analytic properties, namely, the derivative, integral, Taylor series, approximation, and branches of the function. These calculations are motivated by the hope to obtain results which are similar to those obtained for the quadratic Lambert function [18] which have applications to resolving the Einstein-Maxwell field equation with plane symmetry.The properties of the logarithmic Lambert function have implications in the applications to the adiabatic thermostatistics of the three-parameter entropy of classical ideal gas. In particular, the derivative is useful to determine the branches of the function and the specific heat functions in the applications. In the computation of the heat functions and the specific heats, we used the inverse of the Lagrange multiplier for the temperatureT which is referred to in [19] as the “unphysical” temperature.The integral is being derived because when the derivative exists, the natural property that should be considered is the integral of the function. Taylor series and approximation formulas are useful in the computations when a parameter involved becomes large.The analytic properties of the logarithmic Lambert function are presented in Section2. Its applications to the adiabatic thermostatistics of the three-parameter entropy of classical ideal gas are derived in Section 3. Relationships among the specific heats are derived in Section 4 with some important remarks given. Finally, a conclusion is presented in Section 5.


## 2. Logarithmic Generalization of the LambertW Function

A generalization of the LambertW function will be called the logarithmic Lambert function denoted by WLx. Its formal definition is given below, and fundamental properties of this function are proven.Definition 1.
For any real numberx and constant B, the logarithmic Lambert function WLx is defined to be the solution to the equation(12)ylnByey=x.Observe thaty cannot be zero. Moreover, By must be positive. By Definition 1, y=WLx. The derivatives of WLx with respect tox can be readily determined as the following theorem shows.Theorem 2.
The derivative of the logarithmic Lambert function is given by(13)dWLxdx=e−WLxWLx+1lnBWLx+1.Proof.
Taking the derivative of both sides of (12) gives(14)lnByyeydydx+lnBy+1eydydx=1,from which(15)dydx=1ylnBy+lnBy+1ey.
Withy=WLx, (15) reduces to (13).
The integral of the logarithmic Lambert function is given in the next theorem.Theorem 3.
The integral ofWLx is(16)∫WLxdx=eWLx1+WL2x−WLx+1lnWLx−2EiWLx+C,where Eix is the exponential integral given by(17)Eix=∫exxdx.Proof.
From (12),(18)dx=ylnBy+lnBy+1eydy.
Thus,(19)∫ydx=∫yylnBy+lnBy+1eydy=∫y2eylnBydy+∫yeylnBydy+∫yeydy.
These integrals can be computed using integration by parts to obtain(20)∫yeydy=y−1ey+C1,(21)∫yeylnBydy=eyy−1lnBy−1+Eiy+C2,(22)∫y2eylnBydy=eyy2−2y+2lnBy−y+3−2Eiy+C3,where C1,C2,C3 are constants. Substitution of (20), (22), and (21) to (19) with C=C1+C2+C3 and writing WLx for y will give (16).
The next theorem contains the Taylor series expansion ofWLx.Theorem 4.
Few terms of the Taylor series ofWLx about 0 are given below:(23)WLx=1B+e−1/Bx+2+B2!e−2/Bx2+4B2+9B+93!e−B/3x3+⋯.Proof.
Being the inverse of the function defined byx=ylnByey, the Lagrange inversion theorem is the key to obtain the Taylor series of the function WLx.
Letfy=ylnByey. The function f is analytic for By>0. Moreover, f′y=y+1lnBy+1ey,f′1/B=e1/B≠0, and for finite B, f1/B=0. By the Lagrange inversion theorem (taking a=1/B),(24)WLx=1B+∑n=1∞gnxnn!,where(25)gn=limy⟶1Bdn−1dyn−1y−1/Bfyn.
The values ofgn for n=1,2,3 are(26)g1=e−1/B,g2=2+Be−2/B,g3=4B2+9B+9e−3/B.
Substituting these values to (24) will yield (23).
An approximation formula forWLx expressed in terms of the classical Lambert W function is proven in the next theorem.Theorem 5.
For largex,(27)WLx∼Wx−lnlnBWx,where Wx denotes the Lambert W function.Proof.
From (12), y=WLxsatisfies(28)x=ylnByey∼yey.
Then,(29)y=Wx+ux,where ux is a function to be determined. Substituting (29) to (12) yields(30)Wx1+uxWxlnBWx1+uxWxeWx·eux=x.
Withux<<Wx, (30) becomes(31)WxeWxlnBWxeux=x.
By definition ofWx, WxeWx=x. Hence, (31) gives(32)lnBWxeux=1,from which(33)ux=−lnlnBWx.
Thus,(34)WLx∼Wx−lnlnBWx.
Table2 illustrates the accuracy of the approximation formula in (27).
The next theorem describes the branches of the logarithmic Lambert function.Theorem 6.
Letx=fy=ylnByey. Then, the branches of the logarithmic Lambert function y=WLx can be described as follows:(1)
WhenB>0, the branches are as follows:(i)
WL0x:fδ,+∞⟶δ,+∞ is strictly increasing(ii)
WL1x:fδ,0⟶0,δ is strictly decreasingwhere δ is the unique solution to(35)y+1lnBy=−1.(2)
WhenB<0, the branches are as follows:(i)
WL,<0x:0,fδ2⟶δ2,+∞ is strictly decreasing(ii)
WL,<1x:fδ1,fδ2⟶δ1,δ2 is strictly increasing(iii)
WL,<2x:fδ1,0⟶−∞,δ1 is strictly decreasingwhere δ1 and δ2 are the two solutions to (35) with δ1<1/B<δ2<0.Proof.
Consider the case whenB>0. Let x=fy=ylnByey. From equation (13), the derivative of y=WLx is not defined when y satisfies (35). The solution y=δ (35) can be viewed as the intersection of the functions:(36)gy=−1,hy=y+1lnBy.
Clearly, the solution is unique. Thus, the derivativedWLx/dx is not defined for x=fδ=δlnBδeδ. The value of fδ can then be used to determine the branches of WLx. To explicitly identify the said branches, the following information is important:(1)
The value ofy must always be positive; otherwise, lnBy is undefined(2)
The functiony=WLx has only one y-intercept, i.e., y=1/B(3)
Ify<δ, y+1lnBy+1<0 which gives dy/dx<0(4)
Ify>δ, y+1lnBy+1>0 which gives dy/dx>0(5)
Ify=δ, y+1lnBy+1=0 and dy/dx does not exist
These imply that(1)
wheny>δ, the function y=WLx is increasing in the domain fδ,+∞ with range δ,+∞ and the function crosses the y-axis only at y=1/B(2)
wheny<δ, the function y=WLx is decreasing, the domain is fδ,0, and the range is δ,0 because this part of the graph does not cross the x-axis and y-axis(3)
wheny=δ, the line tangent to the curve at the point fδ,δ is a vertical line
These proved the case whenB>0. For the case B<0, the solution to (35) can be viewed as the intersection of the functions:(37)gy=−1y+1,hy=lnBy.
These graphs intersect at two pointsδ1 and δ2. Thus, the derivative dWLx/dx is not defined for(38)x1=fδ1=δ1lnBδ1eδ1,x2=fδ2=δ2lnBδ2eδ2.
Note that(1)
the value ofy must always be negative; otherwise, lnBy is undefined(2)
the functiony=WLx has only one y-intercept, i.e., y=1/B(3)
gy is not defined at y=−1
The desired branches are completely determined as follows:(1)
Ifδ2<y<0, then y+1lnBy+1<0. This gives dy/dx<0. Thus, the function y=WLx is a decreasing function with domain 0,fδ2 and range δ2,0(2)
Ifδ1≤y≤δ2, then y+1lnBy+1>0. This gives dy/dx>0. Thus, the function y=WLx is an increasing function with domain fδ1,fδ2 and range δ1,δ2(3)
If−∞<y<δ1, then y+1lnBy+1<0. This gives dy/dx<0. Thus, y=WLx is a decreasing function with domain fδ1,0 and range −∞,δ1
These complete the proof of the theorem.Table 2xWLxApproximate valueRelative error302.756443.89142.71438×10−21194.308854.87662.46807×10−24337.084265.87562.07321×10−214937.647176.87921.72518×10−249589.822987.88441.44500×10−2160238.656498.88991.22306×10−2507178.1179109.89531.04662 × 10−2Figure1 depicts the graphs of the logarithmic Lambert function (red color) when B=1 and B=−1. The y-coordinates of the points of intersection of the blue- and gray-colored graphs correspond to the value of δ,δ1,δ2.Figure 1
Graphs of the logarithmic Lambert function withB=1 and B=−1. The graphs with red, blue, and gray colors are the graphs of x=fy, x=gy, and x=hy, respectively.


## 3. Applications to Classical Ideal Gas

As discussed in [3], the microstate of a system of N particles can be represented by a single point in the 2DN-dimensional phase space. Corresponding to a particular value of the heat function which is a macrostate is a huge number of microstates. The total number of microstates has to be computed as it is a measure of entropy S. The points denoting the microstates of the system lie so close to each other that the surface area of the constant heat function curve in the phase space is regarded as a measure of the total number of microstates.In an adiabatic ensemble, theoretically the density of states in the regionE and E+∆E should be calculated, where ∆E is very small. But it is not possible to calculate the number of states in this region. So the volume density of the states is calculated. Since the number of particles is very large, most of the volume of the region enclosed by a constant energy curve lies in its surface which is the surface density of states. So the volume density of states is equal to the surface density of states. Hence, in the calculation, the volume density of states is used.In this section, applications of the logarithmic Lambert function to classical ideal gas in the four adiabatic ensembles are derived. In what follows,m denotes the mass of the system;P, pressure;V, volume; h, Planck’s constant (see [20], p. 119); and μ, the chemical potential of the system (see [21]).


### 3.1. Microcanonical EnsembleN,V,E

The Hamiltonian of a nonrelativistic classical ideal gas inD dimensions is(39)H=∑iPi22m,Pi=pi,where Pifori=1,2,⋯,N represent the D-dimensional momentum of the gas molecules. This classical nonrelativistic ideal gas is studied in the microcanonical ensemble. In order to compute the entropy of the system, the phase space volume enclosed by the constant energy curve is computed and is given by (see [3])(40)∑N,V,E=VNN!MNΓDN/2+1EDN/2,where(41)M=2πmh2D/2.The three-parameter entropy of the system is(42)Sq,q′,r=klnq,q′,r∑N,V,E=k1−rexp1−r1−q′exp1−q′1−q∑N,V,E1−q−1−1−1.Computing the inner exponential,(43)exp1−q′1−q∑N,V,E1−q−1=exp1−q′1−qVNMNEDN/2N!ΓDN/2+11−q−1=exp1−q′1−qξmc1−q·EDN/21−q−1,where(44)ξmc=VNMNN!ΓDN/2+1.Let(45)u=1−q′1−qξmc1−qEDN/21−q.Then,(46)Sq,q′,r=k1−rexp1−r1−q′euAe−1−r/1−q′−1,where(47)A=e−1−q′1−q.With(48)z=eu,(49)Sq,q′,r=k1−re1−r/1−q′zAe−1−r/1−q′−1.From the definition of temperature,(50)T=∂Sq,q′,r∂E−1.Hence,(51)1T=∂Sq,q′,r∂E=k1−re−1−r/1−q′e1−r/1−q′zA1−r1−q′Adzdu·dudE,where(52)dz=eu=z,dudE=1−q′ξmc1−qDN2EDN/21−q−1.Thus,(53)1T=ke−1−r/1−q′e1−r/1−q′zAzAξmc1−qDN2EDN/21−q−1.Substituting(54)β=1kT,the preceding equation becomes(55)βe1−r/1−q′DN/2A=e1−r/1−q′euAeuξmc1−qEDN/21−q−1.Let(56)y=1−r1−q′Aeu.Then,(57)1−qξmc1−q1−q′lny1−q′1−rA=EDN/21−q.For largeN,(58)DN21−q−1≈DN21−q.Then, (55) is approximated by(59)2βe1−r/1−q′1−r1−qDN=yeylnBy,where(60)B=1−q′1−rA.Let(61)x=2βe1−r/1−q′1−r1−qDN.It follows from (59) that(62)y=WLx.From (57),(63)EDN/21−q=1−qξmc1−q1−q′lnBWL2βe1−r/1−q′1−r1−qDN,which gives(64)E=1−qξmc1−q1−q′lnBWL2βe1−r/1−q′1−r1−qDN2/DN1−q.The specific heat at constant volume is(65)CV=∂E∂T.Let(66)a=1−q1−q′ξmc1−q′,c=21−re1−r/1−q′1−qDN.Then, (64) can be written(67)E=alnBWLckT2/1−qDN.Taking the partial derivative ofE with respect to T,(68)CV=∂E∂T=2a2/DN1−qlnBWLc/kT2/DN1−q−1WLc/kTddtWLckT,where(69)ddTWLckT=−c/kT2e−WLc/kTWLc/kT+1lnBWLc/kT+1.Let us consider the following regions depending on the values of the deformation parametersq,q′, and r. (i) When r>1 and q>1, the argument of WL is positive. If q′>1, then B>0 and WL must be the principal branch WL0. If q′<1, then B<0. With the argument of WL being positive, we shall take WL to be the branch WL,<0. (ii) If r<1 and q<1, then the argument of WL is positive. If q′<1, then B>0 and we take WL to be the principal branch WL0. If q′>1, then B<0 and again we take WL to be the branch WL,<0. (iii) If r<1,q>1, then the argument of WL is negative. If q′<1, thenB is positive and we take WL to be the principal branch WL0. If q′>1, then B is negative. Here, we have two choices for WL, either WL,<1 or WL,<2. Since the heat function must be a continuous function of the deformation parameters, we must in this case restrict WL to WL,<1.The specific heat at constant pressure is either positive or negative depending on the values of the deformation parametersq,q′ and r.The molar heat capacity must be positive in a stable system [22]. In order to find the molar heat capacity of a compound or element, simply multiply the specific heat by the molar mass [23]. For the molar mass to be positive, the specific heat must be positive. Thus, stability of the system requires that specific heat is positive. Looking at (69), stability is guaranteed if BWLc/kT>1 with B and WLc/kT being dependent on the deformation parameters. The analysis above which is based on the deformation parameters will help determine the stability of the system. On the other hand, negative specific heat appears in astrophysical physics [24].For some values of the parameters, the specific heatCV is depicted in Figures 2(a) and 2(b).Figure 2
(a) Graph ofCV when q′,r vary and q=0.4, T=2, V=3, m=2, and h=D=N=k=A=1. (b) Graph of CV when q,q′ vary and r=0.6, T=3, V=4, m=2, h=1, D=1, and N=k=A=1.
(a)(b)In the applications to the three other adiabatic ensembles, the definition ofM, A, z,β, y, and B will be the same as those in (41), (47), (48), (54), (56), and (60), respectively. However, since definition of u differs in every ensemble, z will have different values in every ensemble.


### 3.2. The Isoenthalpic-Isobaric EnsembleN,P,H

A system which exchanges energy and volume with its surroundings in such a way that its enthalpy remains constant is described by the isoenthalpic-isobaric ensemble. In order to calculate the thermodynamic quantities, the phase space volume enclosed by the constant enthalpy curve was computed in [25] and was given by(70)∑N,P,H=MNN!1ΓD+N/2+1∑VVNH−PVDN/2.Since the volume states are very closely spread, the summation is replaced by an integration. But an integral over the volume leads to overcounting of the eigenstates. To overcome this, the shell particle method of counting volume states was employed (see [2, 15]). In this method, only the minimum volume needed to confine in a particular configuration is taken into account. The minimum volume needed to confine a particular configuration is found by imposing a condition that requires at least one particle to lie on the boundary of the system. All the equivalent ways of choosing a minimum volume for a particular configuration are treated as the volume eigenstate and are considered only once. Using this shell particle technique to reject the redundant volume states, the following expression for the phase space volume was obtained (see [15]):(71)∑N,P,H=MN1PNHDN/2+NΓDN/2+N+1.The three-parameter entropy of the system is(72)Sq,q′,r=klnq,q′,r∑N,P,H=k1−re1−r/1−q′zAe−1−r/1−q′−1,where α=DN/2+N,(73)u=1−q′1−qξie1−qHα1−q,ξie=MN1/PNΓDN/2+N+1.From the definition of temperature,(74)1T=∂Sq,q′,r∂H=ke−1−r/1−q′e1−r/1−q′zA·1−q′AdzdududH,where(75)dzdu=eu=z,dudH=α1−q′ξie1−qHα1−q−1.For largeN, approximately(76)Hα1−q−1≈Hα1−q.Then,(77)dudH≈α1−q′ξie1−qHα1−q,(78)1T=ke−1−r/1−q′e1−r/1−q′zAAzαξie1−qHα1−q.Substitutingβ=1/kT, (78) becomes(79)βe1−r/1−q′Aαξie1−q=e1−r/1−q′zAzHα1−q,from which(80)Hα1−q=1−qξie1−q1−q′lny1−q′1−rA,where y is defined in (56). Then, (79) can be written(81)βe1−r/1−q′1−rα1−q=yeylnBy.It follows from (81) that(82)y=WLβe1−r/1−q′1−rα1−q.From (80),(83)H=1−qξie1−q1−q′ln1−q′1−rAWLβe1−r/1−q′1−rα1−q1/α1−q.Let(84)a=1−qξie1−q1−q′,c=e1−r/1−q′1−rα1−q.Then,(85)H=alnBWLckT1/α1−q.The specific heat at constant pressure is(86)CP=∂H∂T.Taking the partial derivative of (85),(87)∂H∂T=1α1−qalnBWLckT1/α1−q−1·ad/dtWLc/kTWLc/kT.Using the derivative ofWLx with respect to x given in Section 2,(88)CP=−c/α1−qkT2a1/α1−qe−WLc/kTlnBWLc/kT1/α1−q−1WLc/kTWLc/kT+1lnBWLc/kT+1.The specific heat at constant pressure is either positive or negative depending on the values of the deformation parametersq,q′ and r. For some values of the parameters, the specific heat CP is depicted in Figures 3(a) and 3(b).Figure 3
(a) Graph ofCP when q′,r vary and q=0.4, T=2, P=3, M=2,D=1, and N=k=A=1. (b) Graph of CP when q,q′ vary and r=0.2, T=3, P=4,M=1, D=1, and N=k=A=1.
(a)(b)


### 3.3. Theμ,V,L Ensemble

The Hill energyL is the heat function corresponding to the μ,V,L ensemble. For large N, an approximate expression of the phase space volume obtained in [3] is(89)∑μ,V,L=expD2LμexpVμD/2eD/2MD/2D/2.This is a first-order approximation of the exact value:(90)∑μ,V,L=∑N=0∞VNN!MNΓDN/2+1L+μNDN/2.The 3-parameter entropy of the classical ideal gas in this adiabatic ensemble is(91)Sq,q′,r=k1−rexp1−r1−q′zAe−1−r/1−q′−1,where z=eu,(92)ξhe=expVμeD/2MD/2D/2,u=1−q′1−qξhe1−qeDL/2μ1−q.From the definition of temperature,(93)1T=∂Sq,q′,r∂H=ke−1−r/1−q′e1−r/1−q′zA·A1−q′AdzdududL,where(94)dudL=1−q′D2μξhe1−qeDL/2μ1−q.Then, (93) becomes(95)2βμexp1−r/1−q′ADξhe1−q=exp1−r1−q′zAzeDL/2μ1−q,and further(96)1−q1−q′ξie1−qlny1−q′1−rA=eDL/2μ1−q,from which(97)L=2μD1−qln1−q1−q′ξie1−qlny1−q′1−rA.To solve fory, rewrite (95) into(98)β1−re1−r/1−q′D/2μ1−q=yeylny1−q′1−rA.Thus,(99)y=WL2βμe1−r/1−q′1−rD1−q.Therefore,(100)L=2μD1−qln1−q1−q′ξhe1−qlnBWL2βμe1−r/1−q′1−rD1−q.The specific heat at constant volume is(101)CV∗=∂L∂T.Let(102)a=1−q1−q′ξie1−q,c=2μ1−re1−r/1−q′D1−q.Then,(103)L=2μD1−qlnalnBWLcβ,∂L∂T=2μD1−q·d/dtlnBWLcβlnBWLcβ,where(104)ddTlnBWLcβ=−c/kT2e−WLc/kTWLc/kTWLckT+1lnBWLckT+1.Thus,(105)CV∗=2μD1−q−c/kT2e−WLc/kTlnBWLc/kT−1WLc/kTWLc/kT+1lnBWLc/kT+1.The specific heat at constant volume is either positive or negative depending on the values of the deformation parametersq,q′ and r. For some values of the parameters, the specific heat CV∗ is depicted in Figures 4(a) and 4(b).Figure 4
(a) Graph ofCV∗ when q′,r vary and q=0.4,T=2, P=3, D=1, and k=A=1. (b) Graph of CV∗ when q,q′ vary and r=0.8, T=3, P=4, D=1, and k=A=1.
(a)(b)


### 3.4. Theμ,P,R Ensemble

The adiabatic ensemble with both the number and volume fluctuations is illustrated here using the classical ideal gas.In the largeN limit, the expression of the phase space volume [3] is(106)∑μ,P,R=expDRμ1−MPμααexpα−1,where α=DN/2+N.The three-parameter entropy is(107)Sq,q′,r=k1−re1−r/1−q′zA·e−1−r/1−q′−1,where z=eu,(108)ξre=1−MPμααeα−1,u=1−q′1−qξre1−qeDR/μ1−q.From the definition of temperature,(109)1T=∂Sq,q′,r∂R=ke−1−r/1−q′e1−r/1−q′zA·A1−q′·dzdu·dudR,where(110)dudR=1−q′Dμξre1−qeDR/μ1−q.Then,(111)1T=ke−1−r/1−q′e1−r/1−q′zAAzDμξre1−qeDR/μ1−q,from which(112)μβe1−r/1−q′ADξre1−q=e1−r/1−q′zAzeDR/μ1−q.Then,(113)R=μD1−qln1−q1−q′ξie1−qlny1−q′1−rA.From (112),(114)μβ1−re1−r/1−q′D1−q=yeylnBy.Thus,(115)y=WLμβ1−re1−r/1−q′D1−q.Consequently, from (113), the Ray energy of the system is(116)R=μD1−q×ln1−q1−q′ξre1−qln1−q′1−rAWLμβ1−re1−r/1−q′D1−q.The specific heat at constant pressure is(117)CP∗=∂R∂T.Let(118)a=1−q1−q′ξRe1−q,c=μ1−re1−r/1−q′D1−q.Then,(119)R=μD1−qlnalnBWLckT,(120)CP∗=μD1−q×−c/kT2e−WLc/kTlnBWLc/kT−1WLc/kTlnBWLc/kTWLc/kT+lnBWLc/kT+1.The specific heat at constant pressure is either positive or negative depending on the values of the deformation parametersq,q′ and r. For some values of the parameters, the specific heat CP∗ is depicted in Figures 5(a) and 5(b).Figure 5
(a) Graph ofCP∗ when q′,r vary and q=0.4, T=2, P=3, D=1, and k=A=1. (b) Graph of CP∗ when q,q′ vary and r=0.8, T=3, P=4, D=1, and k=A=1.
(a)(b)


## 3.1. Microcanonical EnsembleN,V,E

The Hamiltonian of a nonrelativistic classical ideal gas inD dimensions is(39)H=∑iPi22m,Pi=pi,where Pifori=1,2,⋯,N represent the D-dimensional momentum of the gas molecules. This classical nonrelativistic ideal gas is studied in the microcanonical ensemble. In order to compute the entropy of the system, the phase space volume enclosed by the constant energy curve is computed and is given by (see [3])(40)∑N,V,E=VNN!MNΓDN/2+1EDN/2,where(41)M=2πmh2D/2.The three-parameter entropy of the system is(42)Sq,q′,r=klnq,q′,r∑N,V,E=k1−rexp1−r1−q′exp1−q′1−q∑N,V,E1−q−1−1−1.Computing the inner exponential,(43)exp1−q′1−q∑N,V,E1−q−1=exp1−q′1−qVNMNEDN/2N!ΓDN/2+11−q−1=exp1−q′1−qξmc1−q·EDN/21−q−1,where(44)ξmc=VNMNN!ΓDN/2+1.Let(45)u=1−q′1−qξmc1−qEDN/21−q.Then,(46)Sq,q′,r=k1−rexp1−r1−q′euAe−1−r/1−q′−1,where(47)A=e−1−q′1−q.With(48)z=eu,(49)Sq,q′,r=k1−re1−r/1−q′zAe−1−r/1−q′−1.From the definition of temperature,(50)T=∂Sq,q′,r∂E−1.Hence,(51)1T=∂Sq,q′,r∂E=k1−re−1−r/1−q′e1−r/1−q′zA1−r1−q′Adzdu·dudE,where(52)dz=eu=z,dudE=1−q′ξmc1−qDN2EDN/21−q−1.Thus,(53)1T=ke−1−r/1−q′e1−r/1−q′zAzAξmc1−qDN2EDN/21−q−1.Substituting(54)β=1kT,the preceding equation becomes(55)βe1−r/1−q′DN/2A=e1−r/1−q′euAeuξmc1−qEDN/21−q−1.Let(56)y=1−r1−q′Aeu.Then,(57)1−qξmc1−q1−q′lny1−q′1−rA=EDN/21−q.For largeN,(58)DN21−q−1≈DN21−q.Then, (55) is approximated by(59)2βe1−r/1−q′1−r1−qDN=yeylnBy,where(60)B=1−q′1−rA.Let(61)x=2βe1−r/1−q′1−r1−qDN.It follows from (59) that(62)y=WLx.From (57),(63)EDN/21−q=1−qξmc1−q1−q′lnBWL2βe1−r/1−q′1−r1−qDN,which gives(64)E=1−qξmc1−q1−q′lnBWL2βe1−r/1−q′1−r1−qDN2/DN1−q.The specific heat at constant volume is(65)CV=∂E∂T.Let(66)a=1−q1−q′ξmc1−q′,c=21−re1−r/1−q′1−qDN.Then, (64) can be written(67)E=alnBWLckT2/1−qDN.Taking the partial derivative ofE with respect to T,(68)CV=∂E∂T=2a2/DN1−qlnBWLc/kT2/DN1−q−1WLc/kTddtWLckT,where(69)ddTWLckT=−c/kT2e−WLc/kTWLc/kT+1lnBWLc/kT+1.Let us consider the following regions depending on the values of the deformation parametersq,q′, and r. (i) When r>1 and q>1, the argument of WL is positive. If q′>1, then B>0 and WL must be the principal branch WL0. If q′<1, then B<0. With the argument of WL being positive, we shall take WL to be the branch WL,<0. (ii) If r<1 and q<1, then the argument of WL is positive. If q′<1, then B>0 and we take WL to be the principal branch WL0. If q′>1, then B<0 and again we take WL to be the branch WL,<0. (iii) If r<1,q>1, then the argument of WL is negative. If q′<1, thenB is positive and we take WL to be the principal branch WL0. If q′>1, then B is negative. Here, we have two choices for WL, either WL,<1 or WL,<2. Since the heat function must be a continuous function of the deformation parameters, we must in this case restrict WL to WL,<1.The specific heat at constant pressure is either positive or negative depending on the values of the deformation parametersq,q′ and r.The molar heat capacity must be positive in a stable system [22]. In order to find the molar heat capacity of a compound or element, simply multiply the specific heat by the molar mass [23]. For the molar mass to be positive, the specific heat must be positive. Thus, stability of the system requires that specific heat is positive. Looking at (69), stability is guaranteed if BWLc/kT>1 with B and WLc/kT being dependent on the deformation parameters. The analysis above which is based on the deformation parameters will help determine the stability of the system. On the other hand, negative specific heat appears in astrophysical physics [24].For some values of the parameters, the specific heatCV is depicted in Figures 2(a) and 2(b).Figure 2
(a) Graph ofCV when q′,r vary and q=0.4, T=2, V=3, m=2, and h=D=N=k=A=1. (b) Graph of CV when q,q′ vary and r=0.6, T=3, V=4, m=2, h=1, D=1, and N=k=A=1.
(a)(b)In the applications to the three other adiabatic ensembles, the definition ofM, A, z,β, y, and B will be the same as those in (41), (47), (48), (54), (56), and (60), respectively. However, since definition of u differs in every ensemble, z will have different values in every ensemble.


## 3.2. The Isoenthalpic-Isobaric EnsembleN,P,H

A system which exchanges energy and volume with its surroundings in such a way that its enthalpy remains constant is described by the isoenthalpic-isobaric ensemble. In order to calculate the thermodynamic quantities, the phase space volume enclosed by the constant enthalpy curve was computed in [25] and was given by(70)∑N,P,H=MNN!1ΓD+N/2+1∑VVNH−PVDN/2.Since the volume states are very closely spread, the summation is replaced by an integration. But an integral over the volume leads to overcounting of the eigenstates. To overcome this, the shell particle method of counting volume states was employed (see [2, 15]). In this method, only the minimum volume needed to confine in a particular configuration is taken into account. The minimum volume needed to confine a particular configuration is found by imposing a condition that requires at least one particle to lie on the boundary of the system. All the equivalent ways of choosing a minimum volume for a particular configuration are treated as the volume eigenstate and are considered only once. Using this shell particle technique to reject the redundant volume states, the following expression for the phase space volume was obtained (see [15]):(71)∑N,P,H=MN1PNHDN/2+NΓDN/2+N+1.The three-parameter entropy of the system is(72)Sq,q′,r=klnq,q′,r∑N,P,H=k1−re1−r/1−q′zAe−1−r/1−q′−1,where α=DN/2+N,(73)u=1−q′1−qξie1−qHα1−q,ξie=MN1/PNΓDN/2+N+1.From the definition of temperature,(74)1T=∂Sq,q′,r∂H=ke−1−r/1−q′e1−r/1−q′zA·1−q′AdzdududH,where(75)dzdu=eu=z,dudH=α1−q′ξie1−qHα1−q−1.For largeN, approximately(76)Hα1−q−1≈Hα1−q.Then,(77)dudH≈α1−q′ξie1−qHα1−q,(78)1T=ke−1−r/1−q′e1−r/1−q′zAAzαξie1−qHα1−q.Substitutingβ=1/kT, (78) becomes(79)βe1−r/1−q′Aαξie1−q=e1−r/1−q′zAzHα1−q,from which(80)Hα1−q=1−qξie1−q1−q′lny1−q′1−rA,where y is defined in (56). Then, (79) can be written(81)βe1−r/1−q′1−rα1−q=yeylnBy.It follows from (81) that(82)y=WLβe1−r/1−q′1−rα1−q.From (80),(83)H=1−qξie1−q1−q′ln1−q′1−rAWLβe1−r/1−q′1−rα1−q1/α1−q.Let(84)a=1−qξie1−q1−q′,c=e1−r/1−q′1−rα1−q.Then,(85)H=alnBWLckT1/α1−q.The specific heat at constant pressure is(86)CP=∂H∂T.Taking the partial derivative of (85),(87)∂H∂T=1α1−qalnBWLckT1/α1−q−1·ad/dtWLc/kTWLc/kT.Using the derivative ofWLx with respect to x given in Section 2,(88)CP=−c/α1−qkT2a1/α1−qe−WLc/kTlnBWLc/kT1/α1−q−1WLc/kTWLc/kT+1lnBWLc/kT+1.The specific heat at constant pressure is either positive or negative depending on the values of the deformation parametersq,q′ and r. For some values of the parameters, the specific heat CP is depicted in Figures 3(a) and 3(b).Figure 3
(a) Graph ofCP when q′,r vary and q=0.4, T=2, P=3, M=2,D=1, and N=k=A=1. (b) Graph of CP when q,q′ vary and r=0.2, T=3, P=4,M=1, D=1, and N=k=A=1.
(a)(b)


## 3.3. Theμ,V,L Ensemble

The Hill energyL is the heat function corresponding to the μ,V,L ensemble. For large N, an approximate expression of the phase space volume obtained in [3] is(89)∑μ,V,L=expD2LμexpVμD/2eD/2MD/2D/2.This is a first-order approximation of the exact value:(90)∑μ,V,L=∑N=0∞VNN!MNΓDN/2+1L+μNDN/2.The 3-parameter entropy of the classical ideal gas in this adiabatic ensemble is(91)Sq,q′,r=k1−rexp1−r1−q′zAe−1−r/1−q′−1,where z=eu,(92)ξhe=expVμeD/2MD/2D/2,u=1−q′1−qξhe1−qeDL/2μ1−q.From the definition of temperature,(93)1T=∂Sq,q′,r∂H=ke−1−r/1−q′e1−r/1−q′zA·A1−q′AdzdududL,where(94)dudL=1−q′D2μξhe1−qeDL/2μ1−q.Then, (93) becomes(95)2βμexp1−r/1−q′ADξhe1−q=exp1−r1−q′zAzeDL/2μ1−q,and further(96)1−q1−q′ξie1−qlny1−q′1−rA=eDL/2μ1−q,from which(97)L=2μD1−qln1−q1−q′ξie1−qlny1−q′1−rA.To solve fory, rewrite (95) into(98)β1−re1−r/1−q′D/2μ1−q=yeylny1−q′1−rA.Thus,(99)y=WL2βμe1−r/1−q′1−rD1−q.Therefore,(100)L=2μD1−qln1−q1−q′ξhe1−qlnBWL2βμe1−r/1−q′1−rD1−q.The specific heat at constant volume is(101)CV∗=∂L∂T.Let(102)a=1−q1−q′ξie1−q,c=2μ1−re1−r/1−q′D1−q.Then,(103)L=2μD1−qlnalnBWLcβ,∂L∂T=2μD1−q·d/dtlnBWLcβlnBWLcβ,where(104)ddTlnBWLcβ=−c/kT2e−WLc/kTWLc/kTWLckT+1lnBWLckT+1.Thus,(105)CV∗=2μD1−q−c/kT2e−WLc/kTlnBWLc/kT−1WLc/kTWLc/kT+1lnBWLc/kT+1.The specific heat at constant volume is either positive or negative depending on the values of the deformation parametersq,q′ and r. For some values of the parameters, the specific heat CV∗ is depicted in Figures 4(a) and 4(b).Figure 4
(a) Graph ofCV∗ when q′,r vary and q=0.4,T=2, P=3, D=1, and k=A=1. (b) Graph of CV∗ when q,q′ vary and r=0.8, T=3, P=4, D=1, and k=A=1.
(a)(b)


## 3.4. Theμ,P,R Ensemble

The adiabatic ensemble with both the number and volume fluctuations is illustrated here using the classical ideal gas.In the largeN limit, the expression of the phase space volume [3] is(106)∑μ,P,R=expDRμ1−MPμααexpα−1,where α=DN/2+N.The three-parameter entropy is(107)Sq,q′,r=k1−re1−r/1−q′zA·e−1−r/1−q′−1,where z=eu,(108)ξre=1−MPμααeα−1,u=1−q′1−qξre1−qeDR/μ1−q.From the definition of temperature,(109)1T=∂Sq,q′,r∂R=ke−1−r/1−q′e1−r/1−q′zA·A1−q′·dzdu·dudR,where(110)dudR=1−q′Dμξre1−qeDR/μ1−q.Then,(111)1T=ke−1−r/1−q′e1−r/1−q′zAAzDμξre1−qeDR/μ1−q,from which(112)μβe1−r/1−q′ADξre1−q=e1−r/1−q′zAzeDR/μ1−q.Then,(113)R=μD1−qln1−q1−q′ξie1−qlny1−q′1−rA.From (112),(114)μβ1−re1−r/1−q′D1−q=yeylnBy.Thus,(115)y=WLμβ1−re1−r/1−q′D1−q.Consequently, from (113), the Ray energy of the system is(116)R=μD1−q×ln1−q1−q′ξre1−qln1−q′1−rAWLμβ1−re1−r/1−q′D1−q.The specific heat at constant pressure is(117)CP∗=∂R∂T.Let(118)a=1−q1−q′ξRe1−q,c=μ1−re1−r/1−q′D1−q.Then,(119)R=μD1−qlnalnBWLckT,(120)CP∗=μD1−q×−c/kT2e−WLc/kTlnBWLc/kT−1WLc/kTlnBWLc/kTWLc/kT+lnBWLc/kT+1.The specific heat at constant pressure is either positive or negative depending on the values of the deformation parametersq,q′ and r. For some values of the parameters, the specific heat CP∗ is depicted in Figures 5(a) and 5(b).Figure 5
(a) Graph ofCP∗ when q′,r vary and q=0.4, T=2, P=3, D=1, and k=A=1. (b) Graph of CP∗ when q,q′ vary and r=0.8, T=3, P=4, D=1, and k=A=1.
(a)(b)


## 4. Relationship among the Specific Heats and Some Remarks

The relationship among the specific heats in the different ensembles is determined as follows. Writing(121)X=−c/kT2e−WLc/kTWLc/kTlnBWLc/kTWLc/kT+1+1,then the specific heats can be written:(122)CV=2Xa1/α−N1−qlnBWLc/kT1/α−N1−qlnBWLc/kT,(123)CP=Xa1/α1−qα1−qlnBWLc/kT1/α1−qlnBWLc/kT,(124)CV∗=2μXD1−qlnBWLc/kT,(125)CP∗=μXD1−qlnBWLc/kT.From (122) and (123), we have the relation between the specific heats in the microcanonical ensemble and the isoenthalpic-isobaric ensemble:(126)CV=2α1−qX−N/αCPlnBWLckTN/α1/1−N/α.From (124) and (125), we have the relation between the specific heats in the μ,V,L ensemble and the μ,P,R ensemble:(127)CV∗=2CP∗.


### 4.1. Some Remarks

As discussed in [19], if entropy is nonextensive, the physical temperature is not simply the inverse of the Lagrange multiplier associated with the energy constraint but a variable correctly defined through the generalized zeroth law of thermodynamics. The definition of the physical pressure also becomes different from the ordinary one. Taking into account these facts as well as the first law of thermodynamics and Legendre transform structure, it was shown in [19] that Clausius’ definition of the thermodynamic entropy has to be appropriately modified.In the above computation of the heat functions and specific heats, we used the inverse of the Lagrange multiplierβ, in particular T=1/kβ, for the temperature of the system. As this is no longer the physical temperature for nonextensive entropy, we say that we used the “unphysical temperature,” the term being used also to refer to the same expression in [19].The results in [3] corresponding to (64) and (68), (85) and (88), (97) and (105), and (116) and (120) cannot be recovered even when r⟶1. The reason is that in the generalized statistical mechanics, the thermodynamic limit N⟶∞ and the nonextensive limit q⟶1 do not commute with each other [24, 26]. In the context of the translational specific heat, it was observed [24, 26] that the classical q⟶1 limit and the thermodynamic limit do not commute. While equations (4.14) and (4.15) of [25] similarly demonstrate that the thermodynamic limit N⟶∞ of the rotational and the diatomic specific heats does not commute with the corresponding q⟶1 classical limits.


## 4.1. Some Remarks

As discussed in [19], if entropy is nonextensive, the physical temperature is not simply the inverse of the Lagrange multiplier associated with the energy constraint but a variable correctly defined through the generalized zeroth law of thermodynamics. The definition of the physical pressure also becomes different from the ordinary one. Taking into account these facts as well as the first law of thermodynamics and Legendre transform structure, it was shown in [19] that Clausius’ definition of the thermodynamic entropy has to be appropriately modified.In the above computation of the heat functions and specific heats, we used the inverse of the Lagrange multiplierβ, in particular T=1/kβ, for the temperature of the system. As this is no longer the physical temperature for nonextensive entropy, we say that we used the “unphysical temperature,” the term being used also to refer to the same expression in [19].The results in [3] corresponding to (64) and (68), (85) and (88), (97) and (105), and (116) and (120) cannot be recovered even when r⟶1. The reason is that in the generalized statistical mechanics, the thermodynamic limit N⟶∞ and the nonextensive limit q⟶1 do not commute with each other [24, 26]. In the context of the translational specific heat, it was observed [24, 26] that the classical q⟶1 limit and the thermodynamic limit do not commute. While equations (4.14) and (4.15) of [25] similarly demonstrate that the thermodynamic limit N⟶∞ of the rotational and the diatomic specific heats does not commute with the corresponding q⟶1 classical limits.


## 5. Conclusion

We have investigated the adiabatic class of ensembles in the framework of generalized mechanics based on the three-parameter entropy. The derivative and branches of the function were in particular useful in the applications to the thermostatistics of the nonrelativistic ideal gas. In the microcanonical ensemble and isoenthalpic-isobaric ensemble, the formulas for the three-parameter entropy for an arbitrary number of particles were obtained. In the largeN limit, the heat functions were obtained in terms of the temperature and expressed in terms of the logarithmic Lambert function. From the heat functions, the specific heats at constant temperature were computed. In the μ,V,L and the μ,P,R ensembles, an approximate phase volume in the large N limit was used and the three-parameter entropies of the ensembles were computed. From the entropy function, the heat function and the specific heat were found and expressed also in terms of the logarithmic Lambert function.Reexamination of the classical gas model using the appropriate expression for the physical temperature will be explored in a separate paper. The authors, motivated by the applications of the maximum entropy theory (MaxEnt) in ecology [27], also find it interesting to use the q-entropy, q,q′-entropy, and q,q′,r-entropy to formulate a generalized maximum entropy theory in ecology.



---

*Source: 6695559-2021-02-09.xml*