Abstract Disagreements over the meaning of the thermodynamic entropy and howit should be defined in statistical mechanics have endured for well over a century. Inan earlier paper, I showed that there were at least nine essential properties of entropythat are still under dispute among experts. In this paper, I examine the consequencesof differing definitions of the thermodynamic entropy of macroscopic systems.
Two proposed definitions of entropy in classical statistical mechanics are (1) defin-ing entropy on the basis of probability theory (first suggested by Boltzmann in 1877),and (2) the traditional textbook definition in terms of a volume in phase space (alsoattributed to Boltzmann). The present paper demonstrates the consequences of eachof these proposed definitions of entropy and argues in favor of a definition based onprobabilities.
Despite many years of debate, there are still major disagreements among physicistson some of the most fundamental issues in statistical mechanics [1]. Many of thearguments center on the proper definition of entropy and how the thermodynamicentropy should be derived from statistical mechanics. In a recent paper, I listed nine
essential properties of the entropy that are still under dispute among experts [1]. Theextent of the disagreement can be seen from a recent suggestion that entropy is notthe same concept in thermodynamics and statistical mechanics, so that it cannot bederived directly [2]. My own position is that statistical mechanics should provide afoundation for thermodynamics, so that all thermodynamic properties of macroscopicsystems1 can be derived from it.
This paper compares two proposed definitions of the thermodynamic entropy instatistical mechanics. Both definitions have been attributed to Boltzmann, but theirconsequences are quite different. Although the entropy can be defined for both clas-sical and quantum systems, as well as for both distinguishable and indistinguishableparticles, the differences are most clearly seen in the consequences for the classicalideal gas of distinguishable particles.2
The next section gives a brief reminder of the traditional definition of the entropy,which most readers are probably familiar with. The third section gives a detailed dis-cussion of defining the entropy from the probability distribution of extensive variablesin a composite system, with which readers might be less familiar. I believe that thedetailed nature of this description is necessary due to the importance of clearly un-derstanding all assumptions when comparing the consequences of the two definitionsin the fourth section.
2 Traditional Definition of Entropy
Many textbooks follow a tradition of defining the entropy as being proportional tothe logarithm of the volume in phase space, ΩTRAD, with energy less than or equal tothat of a given macrostate. Denoting the traditional entropy by STRAD, we can write
STRAD = kB lnΩTRAD, (1)
where kB is Boltzmann’s constant (first introduced by Max Planck [3] in 1901). Equa-tion (1) has been presented in many textbooks as Boltzmann’s definition. As I haveshown elsewhere [5], (1) is not Boltzmann’s definition of the entropy, but the miscon-ception is well established.
For a classical ideal gas of distinguishable particles, the traditional definition gives
STRAD(U,N,V ) = kBN
[lnV +
(3
2
)ln
(U
N
)+ constant
], (2)
where U is the energy, N is the number of particles, and V is the volume. In Sect. 4,I will present eight consequences of (2) that make it unacceptable as the thermody-namic entropy.
1I am defining macroscopic systems as in Ref. [1]. A macroscopic systems contains a sufficiently largenumber of particles, so that the statistical fluctuations of extensive variables is smaller than the resolutionof experimental measurements.2I use the standard definition of “distinguishability” from quantum mechanics. Two particles are distin-guishable if exchanging them produces a different microscopic state. Since exchanging two classical par-ticles produces a different point in phase space, classical particles are intrinsically distinguishable.
584 Found Phys (2012) 42:582–593
3 Definition of the Entropy from the Probability Distribution of ExtensiveVariables
This section contains a complete derivation of the entropy of an arbitrary thermo-dynamic system based on the probability distribution of the extensive variables in acomposite system. The derivation is given in two forms, each of which highlightsdifferent aspects of the underlying assumptions.
The first subsection contains the derivation of the entropy that I first published in2002 [4]. It differs from the original only in stating the assumptions more explicitlyand in greater detail.
The same basic ideas are seen from a different point of view in the second subsec-tion. This alternative uses the 1877 Boltzmann definition of the entropy of a compos-ite system [5, 6]. It has the advantage of explicitly addressing the question of possibleadditive constants. The derivation given here is not found in Boltzmann’s papers, butit is a direct consequence of his ideas.
Both derivations are based on the principle that thermodynamic properties in gen-eral, and the entropy of macroscopic systems in particular, should be derived from theprobability distribution of the extensive variables in a composite system. I had firstpublished this principle in 2002 [4], without knowing that Boltzmann had beaten meto it by 125 years [6].
3.1 Derivation of Entropy from Probabilities
This subsection derives the entropy of an arbitrary macroscopic system from the prob-ability distribution of a composite system according to my 2002 approach [4]. Thefirst step in the process is to combine the system of interest with another arbitrarymacroscopic system to form a composite system. Let us call the primary system ofinterest system 1, and denote its extensive variables (energy, volume, and number ofparticles) as U1, V1, and N1. The second arbitrary system will be called system 2, andits extensive variables denoted by U2, V2, and N2. The entropies of both systems 1and 2 will be found by the same process. Naturally, more complicated systems can beanalyzed in the same way if additional extensive variables are included. As expected,in all cases the entropy is found to be a state function that does not depend on thehistory of the system or whether it is open or isolated.
After the two systems are combined, we isolate the resulting composite systemfrom any external exchange of energy or particles. Because of the isolation, the sumsof the extensive variables,
U = U1 + U2,
V = V1 + V2,
N = N1 + N2
(3)
are all constant.Initially, the values of the extensive variables in system 1 (U1, V1, and N1) and
system 2 (U2, V2, and N2) are also constrained to be constant. However, these con-straints can be released to allow the composite system to come to a new equilibrium.
Found Phys (2012) 42:582–593 585
For example, a partition separating the two systems might be replaced by a heat con-ducting wall to allow the exchange of energy, which would release the constraint onthe individual values of U1 and U2, while retaining the constraint on their sum, as in-dicated in (3). The partition could also be removed completely to allow the exchangeof both energy and particles. Similarly, if a fixed partition between the systems wereto be replaced by a moveable piston, the constraint on the volumes of the individualsystems would be released.
If an internal constraint is released, the probability distribution for any of the ex-tensive variables can then be expressed as a function,
W12(U1,V1,N1;U2,V2,N2),
with the condition that the sums of the extensive variables in (3) are constant. Theprobability distribution for an extensive variable, U1, V1, or N1, under these condi-tions is well known to be extremely narrow, with the relative width of the thermal fluc-tuations being of the order of 1/
√N1. Because this width is normally much smaller
than the resolution of macroscopic measurements, the predicted values of U1, V1, orN1 are determined by the location of the maximum of W12. This was Boltzmann’sessential insight in 1877 [6]. Since the logarithm is a monotonically increasing func-tion of its argument, the maximum of lnW12 also determines the locations of theequilibrium values of the extensive variables.
Since we are going to use a gas of distinguishable particles as an example inSect. 4, it is appropriate to give an explicit expression for the probability distribu-tion for the extensive variables U1, V1, and N1. Later we will treat the special caseof an ideal gas, but for now we will keep the more general expressions for a systemwith short-range interactions.
W12 = W12(U1,V1,N1;U2,V2,N2)
= N !N1!N2!
∫dq
3N11
∫dp
3N11
∫dq
3N22
∫dp
3N22 δ(U1 − H1) δ(U − H)∫
dq3N∫
dp3Nδ(U − H), (4)
where H = H1 +H2 +H12 is the total Hamiltonian of the composite system. The setof variables {p1, q1} represents the set of momenta and positions describing a pointin phase space for system 1, while {p2, q2} plays the same role for system 2, and{p,q} for the composite system. H1 = H1({p1, q1}) is the Hamiltonian for system 1as a function of the momentum and position variables, while H2 = H2({p2, q2}) isthe Hamiltonian for system 2. The term H12 = H12({q1}, {q2}) represents the sumof direct interactions between a particle in system 1 and a particle in system 2. Thebinomial coefficient, N !/(N1!N2!), in (4) is the number of permutations of the distin-guishable particles between system 1 and system 2.
For an ideal gas, H12 = H12({q1}, {q2}) = 0, because there are no interactions be-tween any particles. For interacting particles, H12 does not vanish exactly, but itscontributions are negligible for short-ranged interactions. Since the range of inter-action is usually extremely small in comparison with the system size, ignoring H12
is an excellent approximation. With this approximation for systems with short-rangeinteractions, (4) can be written as
586 Found Phys (2012) 42:582–593
W12 = W12(U1,V1,N1;U2,V2,N2)
= N !N1!N2!
∫dq
3N11
∫dp
3N11 δ(U1 − H1)
∫dq
3N22
∫dp
3N22 δ(U2 − H2)∫
dq3N∫
dp3Nδ(U − H)(5)
where H = H1 + H2 + H12 ≈ H1 + H2.We can see from (5) that W12 can be expressed in terms of three distinct factors,
W12(U1,V1,N1;U2,V2,N2) = Ω1(U1,V1,N1)Ω2(U2,V2,N2)
Ω(U,V,N), (6)
where Ω1(U1,V1,N1) depends only on the properties of system 1, Ω2(U2,V2,N2)
depends only on the properties of system 2, and Ω(U,V,N) depends only on theconstant values U , V , and N , given in (3). All three factors have the same form.Ω(U,V,N) can be written explicitly as
Ω(U,V,N) = 1
h3N
1
N !∫
dq3N
∫dp3Nδ
(U − H
({p,q})), (7)
with corresponding expressions for Ω1(U1,V1,N1), and Ω2(U2,V2,N2).Note that the value of h in (7) must be a universal constant, but has no special
meaning within purely classical statistical mechanics. Following the usual conven-tion, h is chosen to be Planck’s constant to obtain agreement with the results of quan-tum statistical mechanics in the classical limit. We will see how this works in (15)below.
Taking the logarithm of (6) and multiplying by a positive constant k, we can write
k lnW12(U1,V1,N1;U2,V2,N2) + k lnΩ(U,V,N)
= k lnΩ1(U1,V1,N1) + k lnΩ2(U2,V2,N2). (8)
If we define the first term on the right hand side of (8) as the function
S1(U1,V1,N1) = k lnΩ1(U1,V1,N1), (9)
we see that it depends only on the properties of system 1. Similarly, the function
S2(U2,V2,N2) = k lnΩ2(U2,V2,N2) (10)
depends only on the properties of system 2. This allows us to write (8) in the followingform.
k lnW12(U1,V1,N1;U2,V2,N2) + const
= S1(U1,V1,N1) + S2(U2,V2,N2) (11)
It is important to note that (8)–(11) are valid for any thermodynamic system, includ-ing inhomogeneous systems for which we cannot neglect contributions of surfaces orinteractions with the walls of a container.
After the removal of a constraint on the extensive variables, the left side of (11)will be a maximum at the equilibrium value of the corresponding extensive variable,U1, V1, or N1, for arbitrary systems. At this point, we are able to identify S1 and S2as the entropies of systems 1 and 2, as discussed below.
Thermodynamics postulates the existence of a function, called the entropy, thatdepends on the extensive variables of a system and has certain specific properties
Found Phys (2012) 42:582–593 587
that are listed in Callen’s classic text [7]. When a function with exactly these prop-erties is found in statistical mechanics, it confirms the validity of thermodynamicsand provides a solid microscopic foundation for macroscopic predictions. Callen’sthermodynamic postulates then tell us what to call the function: Entropy.
Equation (11) shows that S1 and S2 satisfy the most important property of entropy.The fact that the sum S12 = S1 + S2 is a maximum at equilibrium allows us to calcu-late equilibrium values and paves the way for identifying partial derivatives of S1 andS2 with the temperature, pressure, and chemical potential. Most importantly, it alsocontains the second law of thermodynamics. Since the entropy goes to a maximumvalue when a constraint is released, (11) shows that
�S12 ≥ 0 (12)
for an arbitrary, isolated composite system.It is easy to confirm that the functions S1 and S2 also satisfy the conditions of
being continuous and differentiable, and having a positive derivative with respectto energy to ensure that the temperature is positive [7]. From the derivative of theentropy with respect to energy, the multiplicative constant k in (8)–(11) is identifiedas Boltzmann’s constant, k = kB , which gives the ratio of units of energy to units oftemperature.
For the classical ideal gas, the Hamiltonian is simply given by the kinetic energy,
HCIG =3N∑j=1
p2j
2m, (13)
where the index j runs over the 3N components of momentum. The entropy of theclassical ideal gas of distinguishable particles can then be evaluated explicitly from(7) as
SPROB(U,N,V ) = kBN
[(3
2
)ln
(U
N
)+ ln
(V
N
)+ X
], (14)
where the constant X is given by
X =(
3
2
)ln
(4πm
3h2
)+
(5
2
). (15)
Equation (14) is the well-known Sakur-Tetrode equation, which can be derived as theclassical limit of the entropy of a quantum gas.
I have denoted the expression for the entropy of an ideal gas of distinguishableparticles in (14) with the subscript “PROB” to indicate that it was derived from theprobability distributions of the extensive thermodynamic variables. It has been shownelsewhere that (14) is also valid for a classical ideal gas of indistinguishable parti-cles [4].
3.2 Derivation Using Boltzmann’s 1877 Definition of Entropy
This subsection presents an alternative derivation of the entropy of an arbitrarymacroscopic system using the 1877 Boltzmann definition [6], which is also basedon the probability distribution in a composite system.
588 Found Phys (2012) 42:582–593
We use the same notation for systems 1 and 2 as in the previous subsection, andagain consider an isolated composite system formed from them. We again base thederivation on the probability distribution W12.
According to the 1877 Boltzmann definition, the total entropy of the compositesystem is given by
S12(U1,V1,N1;U2,V2,N2) = k lnW(U1,V1,N1;U2,V2,N2) + X, (16)
where k and X are arbitrary constants that Boltzmann did not specify [6]. We will,of course, identify k = kB as the Boltzmann constant because it is determined simplyby the ratio of the units of energy to the units of temperature. The additive constantcan depend on the constant values of U , V , and N , and the value of X = X(U,V,N)
will be determined below by a consistency condition. It plays no direct role in thedetermination of the entropies of systems 1 and 2.
Using (11) in the previous subsection, we can rewrite (16) for arbitrary systemswith short-range interactions as
S12(U1,V1,N1;U2,V2,N2) = S1(U1,V1,N1) + S2(E2,V2,N2) + Y, (17)
where the functions S1 and S2 are defined as in (9) and (10). As discussed above inconnection with (11), S1 depends only on the properties of system 1, and S2 dependsonly on the properties of system 2. Y is an additive constant that we will determinebelow. Without loss of generality, we can remove any additive constants from S1 andS2 and include them in the value of Y .
Equation (17) is sufficient to identify S1 and S2 as the entropies of systems 1 and2 to within additive constants, since the maximum of the sum of these two functionsis located at the equilibrium values of the extensive variables for arbitrary systems.
Next we evaluate the constant Y in (17) from a consistency condition. Since theentropy of any thermodynamic system can be calculated by the procedure describedabove, we can also use it to find the entropy of a composite system. Take the com-posite system formed from systems 1 and 2, and form a new composite system bycombining it with system 3, which may also be chosen arbitrarily. Repeating the pro-cedure outlined above, a straightforward calculation shows that the entropy of theoriginal composite system is given by
where the functions S1 and S2 are again the same as given above in (9) and (10). Con-sistency between (17) and (18) requires Y = 0. This completes the general derivationof the entropy for an arbitrary thermodynamic system, including the additivity con-dition in (18).
If the containers of systems 1 and 2 have the same shape, their entropies, S1 andS2, can be represented by a common function S, where
S1(U1,V1,N1) = S(U1,V1,N1) (19)
and
S2(U2,V2,N2) = S(U2,V2,N2). (20)
Found Phys (2012) 42:582–593 589
This would be true even the systems were inhomogeneous, so that their entropieswere not extensive.
We can go further for homogeneous systems, for which the contributions offree surfaces and interactions with container walls can be ignored. The functionS(U,V,N) is always extensive for a homogeneous system due to the presence ofthe factor of 1/N !. In this case, the value of X(U,V,N) is
X(U,V,N) = S(U,V,N), (21)
where U , V , and N are the total energy, volume, and number of particles definedin (3).
4 Consequences of the Differences in the Two Definitions
The traditional definition of the entropy and the definition based on probability distri-butions in composite systems both purport to describe the same macroscopic system,but their expressions for the entropy, as shown in (2) and (14), are different. To putthese differences in perspective, it is useful to examine their consequences for a clas-sical ideal gas of distinguishable particles. Eight consequences are discussed in thefollowing subsections.
4.1 Extensivity
Although the entropy of an inhomogeneous system—such as a system for whichinteractions with the walls of a container cannot be neglected—is not extensive, theentropy of a homogeneous system is expected to be extensive. An example is given bycolloids, for which the particles are obviously distinguishable, but the experimentallyfound entropy is well known to be extensive.
Equation (2) shows that STRAD is not extensive, but the definition based on prob-abilities produces an extensive entropy for a classical ideal gas of distinguishableparticles, as seen in (14) for SPROB.
4.2 Additivity
The calculation of the entropy of a composite system using the traditional definitioncan give a result that differs from the sum of their expressions for the entropies ofthe subsystems, so that additivity is violated. This has been noted by several authors,most recently by Versteegh and Dieks [2].
The definition based on probabilities gives an entropy that is additive for idealgases, and for any system with short-range forces [4].
4.3 Definition of the Chemical Potential
The thermodynamic definition of the chemical potential μ is given in terms of apartial derivative of the entropy.
590 Found Phys (2012) 42:582–593
−μ
T=
(∂S
∂N
)U,V
. (22)
Thermodynamic equilibrium between two subsystems that exchange particles meansthat their chemical potentials are equal.
Unfortunately, due to the lack of additivity for STRAD, the partial derivatives in(22) for two subsystems are not equal in equilibrium, so that (22) does not give aconsistent definition of the chemical potential from STRAD.
Using SPROB, (22) does give a consistent definition of the chemical potential.
4.4 Euler Equation
The validity of the Euler equation,
U = T S − PV + μN, (23)
relies on the assumption of extensivity, so it is not valid for STRAD.Since SPROB is extensive, the Euler equation is valid.
4.5 Gibbs-Duhem Relation
The Gibbs-Duhem relation,
dμ =(
S
N
)dT −
(V
N
)dP, (24)
relies on the assumption of extensivity. That it is not valid for STRAD is clear from(24) since the ratio STRAD/N is not independent of size.
Using SPROB, the Gibbs-Duhem relation is valid.
4.6 Second Law of Thermodynamics: �S ≥ 0
The inequality �S ≥ 0 is an expression of the second law of thermodynamics. Itshould be true for any thermodynamic process, with equality being achieved only forquasi-static processes.
The traditional definition gives rise to �STRAD < 0 when a partition is insertedinto an ideal gas of distinguishable particles.
The definition based on probabilities always gives �SPROB ≥ 0, which I find com-forting.
4.7 Stability Criteria
There are thermodynamic stability criteria that restrict the second derivatives of ther-modynamic potentials. Several of these criteria are violated by STRAD and its Leg-endre transforms. As an example, consider the Helmholtz free energy, F(T ,V,N) =U − T S. One stability condition is
(∂2F
∂N2
)T ,V
> 0. (25)
Found Phys (2012) 42:582–593 591
A simple calculation using (2) shows that the traditional definition violates it.(∂2FTRAD
∂N2
)T ,V
= 0. (26)
If the ideal-gas model is modified to include weak repulsive interactions, the inequal-ity is saved. However, if there are weak attractive interactions, the second derivativecan even be negative.
To understand why (25) can be violated by STRAD, we must look at the derivationsof stability criteria. These derivations invariably invoke the thermodynamic conceptof a “wall” or “partition,” which has important consequences that depend on the na-ture of the wall chosen.
The use of the term “wall” might, of course, refer to a solid, insulating barrierthat lets nothing pass between subsystems. However, it could also behave like a cop-per wall and transmit energy between subsystems while preventing the exchange ofparticles. It could be a moveable wall (or piston) that allows the volumes of the sub-systems to change. It could even have a hole in it, so that particles and energy mightpass between subsystems. In fact, the wall could even be a non-restrictive boundarythat defines the volumes of the subsystems without restricting flow of particles andenergy at all.
In the thermodynamic derivation of the inequality in (25), a non-restrictive wall isintroduced to divide the volume into two parts, and the free energy of the resultingcomposite system is written as the sum of the free energies of the two subsystems.Since the total free energy of the composite system must be a minimum at equilibriumby the second law, (25) follows immediately. However, since this procedure clearlyrequires the free energy to be additive, the derivation fails for STRAD.
As with the other properties, SPROB gives expressions for the free energy thatsatisfy the stability criterion in (25).
4.8 Fundamental Relation
The equation
S = S(U,V,N), (27)
where U is the energy, V is the volume, and N is the number of particles, is knownin thermodynamics as a fundamental relation. Callen has expressed its importance bywriting that, “if the fundamental relation of a particular system is known all conceiv-able thermodynamic information about the system is ascertainable from it.” [7].
Unfortunately, the lack of additivity means that STRAD = STRAD(U,V,N) is nota fundamental relation. As Versteegh and Dieks have shown, knowing STRAD foreach individual subsystem is not sufficient to predict the properties of two ideal gasesthat can exchange particles, and a separate calculation for the composite system isrequired [2].
The equation
SPROB = SPROB(U,V,N), (28)
does constitute a fundamental relation from which all thermodynamic informationcan be correctly obtained.
592 Found Phys (2012) 42:582–593
Table 1 Comparison ofproperties of the entropy of aclassical ideal gas ofdistinguishable particles. Thecolumn labeled “Traditionaldefinition” refers to definingentropy in terms of a volume inphase space, while that labeled“Probabilities” refers to thedefinition of entropy based onprobability distributions incomposite systems
Property Traditional definitionSTRAD
ProbabilitiesSPROB
Extensivity NO YES
Additivity NO YES−μT
= (∂S∂N
)U,V
NO YES
Euler equation NO YES
Gibbs-Duhem relation NO YES
�S ≥ 0 NO YES(∂2F
∂N2
)T ,V
> 0 NO YES
Fundamental relation NO YES
4.9 Summary of Comparisons
Table 1 presents a summary of comparisons between the properties of STRAD andthose of SPROB.
5 Conclusions
The differences in the eight properties of STRAD and SPROB given in the previoussection demonstrate clearly that the traditional definition of the entropy in statisticalmechanics is not viable as a foundation of thermodynamics.
Some of the failures of the traditional definition, like the lack of extensivity, havebeen known for some time. There have been many interpretations of their signifi-cance, along with attempts to patch up the traditional definition by including an adhoc term of the form −kB lnN ! [2, 8]. But why should we need an extra ad hoc term?For that matter, why should the N -dependence be treated in a fundamentally differentmanner than the U - or V -dependence?
Since I—and Boltzmann long before me—have shown that it is possible to definethe entropy in statistical mechanics in such a way as to be completely consistentwith thermodynamics, I believe that there is no advantage gained from a definitionin statistical mechanics that is inconsistent with the properties of the thermodynamicentropy.
In conclusion, I believe that Table 1 tells the story. The eight properties of theentropy that (1) gives up are a high price to pay for tradition—especially when tradi-tion’s attribution of (1) to Boltzmann is incorrect [5, 6].
Acknowledgements I would like to thank Jan Tobochnik for very useful comments and suggestions.I would also like to thank Dennis Dieks for an interesting discussion. Finally, I would like to thank ErwinFrey and the members of the Arnold Sommerfeld Center for Theoretical Physics in Munich for theirgracious hospitality during this work.
References
1. Swendsen, R.H.: How physicists disagree on the meaning of entropy. Am. J. Phys. 79(4) (2011)2. Versteegh, M.A.M., Dieks, D.: The Gibbs paradox and the distinguishability of identical particles. Am.
J. Phys. 79, 741–746 (2011)
Found Phys (2012) 42:582–593 593
3. Planck, M.: Über das Gesetz der Energieverteilung im Normalspektrum. Drudes Ann. 553–562 (1901).Reprinted: Die Ableitung der Strahlungsgesteze. In: Ostwalds Klassiker der exakten Wissenschaften,Bd. 206, pp. 65–74
4. Swendsen, R.H.: Statistical mechanics of classical systems with distinguishable particles. J. Stat. Phys.107, 1143–1165 (2002)
5. Swendsen, R.H.: Statistical mechanics of colloids and Boltzmann’s definition of the entropy. Am. J.Phys. 74, 187–190 (2006)
6. Boltzmann, L.: Über die Beziehung zwischen dem zweiten Hauptsatze der mechanischen Wärmetheo-rie und der Wahrscheinlichkeitsrechnung respektive den Sätzen über das Wärmegleichgewicht. Wien.Ber. 76, 373–435 (1877). Reprinted: Boltzmann defines the entropy on second and third pages of the ar-ticle. In: Wissenschaftliche Abhandlungen von Ludwig Boltzmann (Chelsea, New York, 1968), vol. II,pp. 164–223
7. Callen, H.B.: Thermodynamics and an Introduction to Thermostatistics, 2nd edn. Wiley, New York(1985)
8. Cheng, C.-H.: Thermodynamics of the system of distinguishable particles. Entropy 1, 326–333 (2009)
