1 Introduction

Entropy is a familiar concept often understood as representing uncertainty, the mathematical expression of which is recognized popularly as the logarithm of the number of accessible states (for example, \(\log 6\) for a dice). The celebrated formula due to Boltzmann, \(S_{B}=k_{B} \ln W\) with \(k_{B}\) being the Boltzmann constant, represents the entropy of a physical system. The definition of W is involved in statistical mechanics and appears in the microcanonical ensemble (ME) theory. The ME is a collection of isolated N-particle systems in equilibrium, each having a specified energy E.

In the Boltzmann formula, W denotes the total number of accessible microstates of the systems in the ME. The conventional way to obtain W is to introduce an energy window of width \(\Delta \) around energy E and to express W as [1, 2]

$$\begin{aligned} W(E,\lambda ) = \Omega (E,\lambda )-\Omega (E{-}\Delta ,\lambda ) \approx \omega (E,\lambda ) \Delta , \end{aligned}$$
(1)

where \(\lambda \) is an externally controllable parameter, \(\Omega (E,\lambda )\) is the number of states with energy less than E and \(\omega (E,\lambda )\) is the density of states. Accordingly, the Boltzmann formula obtains the form

$$\begin{aligned} S_{\Gamma }= k_{B} \ln [\omega (E,\lambda ) \Delta ], \end{aligned}$$
(2)

which is referred to as surface entropy. The window width \(\Delta \) is regarded conventionally as a small constant with no physical consequences. The approximation of Eq. (1) is the leading-order term for small \(\Delta \), in which the derivative of \(\Omega (E,\lambda )\) with respect to the energy E provides the density of states:

$$\begin{aligned} \omega (E,\lambda )=\frac{\partial \Omega (E,\lambda )}{\partial E}~. \end{aligned}$$
(3)

In a quantum system, the cumulative density of states is given by \(\Omega (E,\lambda ) = \sum _n \Theta (E-E_n)\) with the Heaviside step function \(\Theta (E-E_n)\), where the dependence on \(\lambda \) comes through the energy level \(E_n\). On the other hand, for a classical system having continuous energy spectra [1], \(\Omega (E,\lambda )\) corresponds to the volume of the phase space enclosed by the surface \(H(p,q;\lambda )=E\) with Hamiltonian \(H(p,q;\lambda )\) of an N-particle system in a ME, where p and q denote the collective coordinates and momenta, respectively, of the particles moving in the d-dimensional space. More explicitly, one can express the phase-space volume as

$$\begin{aligned} \Omega (E,\lambda )= & {} \int \frac{d^N\!p \,d^N\!q}{h^{Nd}}\Theta [E-H(p,q;\lambda )], \end{aligned}$$
(4)

where h is the Planck constant. Then, Eq. (1) corresponds to taking W as the volume of a thin shell bounded by the two surfaces \(H=E\) and \(H=E-\Delta \) in the phase space; the energy window \(\Delta \) measures the thickness of the shell. Henceforth, for simplicity of notation, we occasionally omit function arguments E and \(\lambda \) without loss of clarity. The number N of particles is kept fixed and does not play as a variable.

A closely related quantity is the volume entropy, which is given by

$$\begin{aligned} S_{\Sigma }=k_{B}\ln \Omega (E,\lambda )~. \end{aligned}$$
(5)

Usually, \(S_{\Gamma }\) and \(S_{\Sigma }\) are used without distinction for systems with many degrees of freedom, which are mostly the subject of thermodynamics and statistical mechanics. Specifically, the equivalence between \(S_{\Gamma }\) and \(S_{\Sigma }\) is emphasized in the statement [1] “if these definitions [Eqs. (2) and (5)] were not equivalent, the validity of statistical mechanics would be in doubt.”

On the other hand, there has been controversy as to the validity of the two expressions. For instance, it was disclosed that Eq. (2) does not comply with the fundamental relation of thermodynamics [3, 4]:

$$\begin{aligned} T{\text {d}}S = {\text {d}}E - F {\text {d}}\lambda , \end{aligned}$$
(6)

where T and E are the temperature and internal energy of the system, respectively. Here, F is a generalized force conjugate to a system parameter \(\lambda \), which is given by the microcanonical ensemble average of \(\partial H/\partial \lambda \):

$$\begin{aligned} F= & {} \left\langle \frac{\partial H}{\partial \lambda } \right\rangle \nonumber \\= & {} \frac{1}{\omega }\int \frac{d^N\!p\, d^N\!q}{h^{Nd}}\frac{\partial H}{\partial \lambda }\delta (E-H)~. \end{aligned}$$
(7)

Consequently, only Eq. (5) was taken to be consistent with thermodynamic relations [3, 4].

Taking seriously this argument that \(S_{\Gamma }\) disagrees with the thermodynamic relation, we seem forced to accept only \(S_{\Sigma }\) as the thermodynamic entropy in the ME. However, the problem is that at the cost of accepting \(S_{\Sigma }\), we should also give up a few critical notions which, based on \(S_{\Gamma }\), describe thermodynamic and statistical mechanical properties of large systems effectively well. Additivity of entropy and the concept of negative temperature, which \(S_{\Sigma }\) does not support, are such examples. Note that while the non-additive property of \(S_{\Sigma }\) was analyzed [5,6,7], justifications of negative temperature were provided along with its usefulness [8,9,10,11,12,13,14,15,16,17]. Then the question is which one to accept. Specifically, one may ask whether we must abandon \(S_{\Gamma }\) and reformulate thermodynamics and statistical mechanics based only on \(S_{\Sigma }\).

In this study, we probe this issue to provide a correct expression of \(S_{\Gamma }\) as the consistent thermodynamic entropy. Our study focuses on the ME of classical systems, except that we use some quantum mechanical content to give an idea of the energy width \(\Delta \) related to the level spacing. We should mention a delicate issue in applying the thermodynamic relation to quantum mechanical systems. For a quantum mechanical ME, one can naturally identify W with the degree of the degeneracy of the energy level occupied by the systems in the ME. As will be discussed later in this paper, the logarithm of the degree of degeneracy serves as an eligible entropy of a quantum mechanical ME, thanks to its adiabatic invariance property. The problem is that energy levels are, in the strict sense, discrete for a quantum mechanical system: Energy eigenvalues form a countable set, and there are no accessible states at energies other than the eigenvalues. Namely, W vanishes unless energy \(E=E_{n}\), which makes it problematic to define \(\ln W\) as a continuous function of E. Therefore, mathematical proof that the logarithm of the degeneracy satisfies the thermodynamic relations between the differentials in Eq. (6) is inherently infeasible. Only some particular systems allow analytic relations between E and W, for which we can explicitly prove that the logarithm of the degree of degeneracy satisfies the thermodynamic relations (see Sect. 4 and Appendix). For general systems, we should rely upon an interpolating scheme presenting the degree of degeneracy as a smooth function of E and \(\lambda \), for instance, allowing an energy window of width \(\Delta \) around energy E and taking the approximation in Eq. (1) [1, 2]. Alternatively, we may need to extend the relation given by Eq. (6) to cope with the discrete nature of the quantum mechanical ME. This may raise a potentially crucial problem in dealing with the thermodynamics of quantum systems; however, it is beyond the scope of this study, which examines the nature of the width \(\Delta \) appearing in the entropy definition given by (2) for classical systems.

This paper is organized as follows: For a clear manifestation, in Sect. 2 we take a simple example to illustrate a problem arising from the non-compliance of \(S_{\Gamma }\) with the thermodynamic relation. In Sect. 3, we show that for \(S_{\Gamma }\) to be consistent with the thermodynamic relations, \(\Delta \) should be not a constant but a non-extensive function satisfying a specific differential equation. Subsequently, the differential equation is shown to bear non-unique solutions; in other words, the entropy form consistent with thermodynamic relations is not unique. It is discussed that \(S_{\Sigma }\) is just an outcome of one solution to the differential equation and brings up the problem of the extensive temperature scale for systems having the density of states as a non-monotonic function of energy. Another possible solution is thus suggested in Sects. 4 and 5, based on the adiabatic invariance of thermodynamic entropy and the Weyl correspondence: the energy width \(\Delta \), taken to be an energy level spacing in the quantum counterpart of the classical ensemble considered. With a few model systems, we demonstrate that the expression \(S_{\Gamma }\) with \(\Delta \) taken in this way can serve as correct thermodynamic entropy. Finally, Sect. 6 summarizes our study.

2 Surface entropy and volume entropy

Recall that the thermodynamic relation in Eq. (6) holds only for quasi-static processes. One relevant quasi-static process for the ME is realized by slowly varying a system parameter \(\lambda \) while thermally isolated from external surroundings. Such a thermally isolated quasi-static process is an adiabatic process in which no heat transfers into the system and also an isentropic process according to the Clausius relation: \({\text {d}}Q = T{\text {d}}S\) with \({\text {d}}Q\) being an infinitesimal amount of heat transfer. For the thermally isolated ME, the work expended in the system by varying the parameter (say, from \(\lambda _{i}\) to \(\lambda _{f}\)) equals the energy change \(\Delta E\). Hence we have a link between work and energy change occurring in the thermally isolated quasi-static (or isentropic) process. Integrating Eq. (6), we obtain

$$\begin{aligned} \int _{\lambda _{i}}^{\lambda _{f}}{\text {d}}\lambda \,F|_{S} = \Delta E |_{S}~, \end{aligned}$$
(8)

where the notation \(|_{S}\) indicates the constant-S condition.

Let us see how Eq. (8) applies to a concrete example: the ME of N identical harmonic oscillators in one dimension. The system Hamiltonian reads \(H(p,q, \lambda ) = \sum _{i=1}^{N}[p_{i}^{2}/(2\,m)+m\lambda q_{i}^{2}/2]\equiv K+V\) with \(p \equiv \{p_{1}, p_2, \ldots , p_N \}\) and \(q \equiv \{q_{1}, q_2 , \ldots , q_N \}\). The curvature of the harmonic potential plays a role of an externally controllable parameter \(\lambda \). The volume enclosed by the surface \(H(p,q,\lambda )=E\) in the 2N-dimensional phase space is given by

$$\begin{aligned} \Omega (E,\lambda )=\frac{1}{N!h^{N}}\left( \frac{2\pi E}{\sqrt{\lambda }}\right) ^{N}~, \end{aligned}$$
(9)

which leads to the density of states \(\omega (E,\lambda )=\partial \Omega (E,\lambda )/\partial E = (N/E)\Omega (E,\lambda )\). Omitting constants irrelevant to the differential relation, Eq. (6), setting the Boltzmann constant equal to unity (\(k_{B}\equiv 1\)), henceforth, we find that entropies given by Eqs. (2) and (5) take the following forms:

$$\begin{aligned} S_{\Gamma }(E,\lambda )= & {} (N-1)\ln E - N\ln \sqrt{\lambda }, \nonumber \\ S_{\Sigma }(E,\lambda )= & {} N\ln E - N\ln \sqrt{\lambda }~. \end{aligned}$$
(10)

Meanwhile, using \(\partial H(p,q,\lambda )/\partial \lambda = V/\lambda \) in this example, we find the generalized force given by

$$\begin{aligned} F= \frac{\langle V\rangle }{\lambda } = \frac{\langle H\rangle }{2\lambda }= \frac{E}{2\lambda }~. \end{aligned}$$
(11)

One can compute each side of Eq. (8) with S taken to be either \(S_{\Gamma }\) or \(S_{\Sigma }\) and find that the equality holds only for \(S_{\Sigma }\), as we see below.

First, consider the \(S_{\Sigma }\)-invariant process. During the process, the energy variations follow the relation \(E(\lambda )=\sqrt{\lambda }e^{S_{\Sigma }/N}\) obtained from Eq. (10). In this case, work performed along the \(S_{\Sigma }\)-invariant path validates the relation in Eq. (8) as follows:

$$\begin{aligned} \int _{\lambda _{i}}^{\lambda _{f}}{\text {d}}\lambda \, F_{\lambda }|_{S_{\Sigma }}= & {} \int _{\lambda _{i}}^{\lambda _{f}}{\text {d}}\lambda \, \frac{E(\lambda )}{2\lambda } \nonumber \\= & {} e^{S_{\Sigma }/N}\left( \sqrt{\lambda _{f}}-\sqrt{\lambda _{i}}\right) \nonumber \\= & {} E(\lambda _{f})-E(\lambda _{i})~. \end{aligned}$$
(12)

On the other hand, in the \(S_{\Gamma }\)-invariant process, Eq. (10) allows one to express the energy as a function of \(\lambda \):

$$\begin{aligned} E(\lambda )=\lambda ^{\alpha /2}e^{S_{\Gamma }/(N-1)}, \end{aligned}$$
(13)

with \(\alpha \equiv N/(N-1)\). Evaluating the integral in Eq. (8),

$$\begin{aligned} \int _{\lambda _{i}}^{\lambda _{f}}{\text {d}}\lambda \, F_{\lambda }|_{S_{\Gamma }}= & {} \int _{\lambda _{i}}^{\lambda _{f}}{\text {d}}\lambda \, \frac{E(\lambda )}{2\lambda } \nonumber \\= & {} \alpha ^{-1} e^{S_{\Gamma }/(N-1)} \left( \lambda _{f}^{\alpha /2}-\lambda _{i}^{\alpha /2}\right) \nonumber \\= & {} \alpha ^{-1}[E(\lambda _{f})-E(\lambda _{i})], \end{aligned}$$
(14)

we find that the result is different from the energy change by the factor \(\alpha \). Only when N is very large, \(\alpha \) approaches unity and Eq. (8) becomes approximately valid.

This example indicates that what is invariant in an isentropic process is not \(S_{\Gamma }\) but \(S_{\Sigma }\). Accordingly, one can conclude that \(S_{\Gamma }\) is not precisely consistent with the thermodynamic entropy appearing in the fundamental relation given by Eq. (6). Attempting to formulate the thermodynamics of a small system in terms of \(S_{\Gamma }\), one will inevitably run into the problem of incorrectly predicting the energy change made by an isentropic process. This fact was accentuated by the statement [18], “if we make \(S_{\Gamma }\) the analog of entropy, we are embarrassed by the necessity of making numerous exceptions for systems of one or two degrees of freedom.” Note, however, that the degree of inconsistency of \(S_{\Gamma }\) depends on the system size and becomes supposedly minute for many-particle systems, which is addressed in detail in the next section.

3 Equation for energy window

The thermodynamic relation in Eq. (6), as establishing the mathematical equality that eligible entropy should equip with, provides the following partial derivatives:

$$\begin{aligned} \frac{\partial S}{\partial E}= & {} \beta , \nonumber \\ \frac{\partial S}{\partial \lambda }= & {} -\beta F~. \end{aligned}$$
(15)

The first equation defines the inverse temperature \(\beta \), and the second one relates the partial derivative of entropy to the generalized force F with the proportion given by \(\beta \).

It is straightforward to show that \(S_{\Sigma }=\ln \Omega \) obeys Eq. (15). From Eq. (4), we obtain

$$\begin{aligned} \frac{\partial S_{\Sigma }}{\partial \lambda }= & {} -\frac{1}{\Omega } \int \frac{d^N\!q\, d^N\!p}{h^{Nd}} \frac{\partial H}{\partial \lambda } \delta (E-H) \nonumber \\= & {} -\frac{\omega }{\Omega } F \nonumber \\= & {} -\beta _{\Sigma } F~, \end{aligned}$$
(16)

where we have used Eq. (7) together with the relation \(d\Theta (x)/{\text {d}}x = \delta (x)\). The inverse temperature \(\beta _{\Sigma }\) in Eq. (16) is defined to be

$$\begin{aligned} \beta _{\Sigma }=\frac{\partial \ln \Omega }{\partial E}=\frac{\omega }{\Omega }~, \end{aligned}$$
(17)

which is always non-negative. Equation (16) thus takes the same form as Eq. (15).

In contrast, the partial derivative of \(S_{\Gamma }=\ln [\omega \Delta ]\) with the energy width \(\Delta \) regarded as a constant deviates from Eq. (15) as follows:

$$\begin{aligned} \frac{\partial S_{\Gamma }}{\partial \lambda }= & {} -\frac{1}{\omega }\frac{\partial }{\partial E}\int d^N\!q\, d^N\!p \,\delta (E-H)\frac{\partial H}{\partial \lambda } \nonumber \\= & {} -\frac{1}{\omega }\frac{\partial }{\partial E}(\omega F ) \nonumber \\= & {} -\beta _{\Gamma }F-\frac{\partial F}{\partial E} \end{aligned}$$
(18)

with \(\beta _{\Gamma }= \omega ^{-1} \partial \omega /\partial E\), which manifests the term \(\partial F/\partial E\) as the source of the thermodynamic inconsistency.

Let us remark on two noteworthy points regarding this observation. First, note that the term \(\partial F/\partial E\) is negligible (in comparison with \(\beta _{\Gamma }F\)) for the ME consisting of a large number of particles (\(N \gg 1\)) because the ratio \(|\beta _{\Gamma }F|^{-1} |\partial F/\partial E| = {{\mathcal {O}}}(E)^{-1}\) and energy E is extensively large: \(E = {{\mathcal {O}}}(N)\). We thus conclude: For the ME having \(N \gg 1\), \(S_{\Gamma }\) serves as entropy asymptotically consistent with Eq. (15).

Second, for the entropy expression \(\ln [\omega \Delta ]\) to be exactly consistent with the thermodynamic relation for any finite-sized systems, \(\Delta \) should be not a constant but a function of E and \(\lambda \), satisfying

$$\begin{aligned} \frac{\partial }{\partial \lambda }\left( \frac{1}{\Delta }\right) =-\frac{\partial }{\partial E} \left( \frac{F}{\Delta }\right) . \end{aligned}$$
(19)

To discriminate from \(S_{\Gamma }\) with \(\Delta \) being a constant, we drop the subscript and write the expression with a function \(\Delta (E,\lambda )\) as

$$\begin{aligned} S=\ln [\omega \Delta (E,\lambda )], \end{aligned}$$
(20)

which, with the function \(\Delta (E,\lambda )\) satisfying Eq. (19), indeed fulfills Eq. (15). Therefore, we come to the second conclusion: For any N, Eq. (20) is the thermodynamic entropy, provided that \(\Delta (E,\lambda )\) is a solution of Eq. (19).

It is also required that S should be indiscernible from \(S_{\Gamma }\) in the limit \(N\rightarrow \infty \); we thus supplement this statement by imposing the condition that \(\Delta \) should not be extensive. In short, the problem now reduces to finding the non-extensive quantity \(\Delta (E,\lambda )\), which is the solution of the partial differential equation (19).

In principle, once \(F(E,\lambda )\) is given for a specific system, one can solve Eq. (19) and obtain solutions. Finding a unique solution of the partial differential equation of the type in Eq. (19) with appropriate boundary conditions is the Cauchy problem, which may be handled analytically or numerically. Unfortunately, however, in the present one searching the functional form of the width \(\Delta \) itself, the boundary conditions or the specific form of \(\Delta \) at given energy or parameter values are unavailable. This leads Eq. (19) to carry multiple solutions and accordingly makes inevitable non-unique entropy expressions. For a more clear understanding, let us examine the case of harmonic oscillators considered in Sec. II. The generalized force is given by Eq. (11): \(F=E/(2\lambda )\). We see that apart from a proportional constant, both \(\Delta (E,\lambda )=E/N\) (where N put here makes \(\Delta \) intensive) and \(\Delta (E,\lambda )=\sqrt{\lambda }\) provide the solutions to Eq.(19). Their entropy expressions, \(S=\ln [\omega \Delta (E,\lambda )]\) with corresponding \(\Delta \), conform to the thermodynamic relations and yet differ from each other.

Note that \(\Delta = \Omega /\omega \) gives a solution to Eq. (19). This solution converts \(S_{\Gamma }\) into \(S_{\Sigma }\) and itself provides the temperature derived from \(S_{\Sigma }\):

$$\begin{aligned} S_{\Gamma }= & {} \ln (\omega \Delta )=\ln \Omega =S_{\Sigma }, \end{aligned}$$
(21)
$$\begin{aligned} \Delta= & {} \frac{1}{\beta _{\Sigma }}~. \end{aligned}$$
(22)

Nevertheless, we cannot settle on this solution: A severe problem that the one given by Eq. (22) poses is that the temperature scale \(\beta _{\Sigma }^{-1}\) and, therefore, \(\Delta \) increases extensively with the system size when the density \(\omega \) of states is a non-monotonic function of energy. Let us, for simplicity, assume that there exists only one energy value \(E^{*}\) at which \(\omega \) becomes maximum. For the energy range \(E>E^{*}\), the phase volume \(\Omega (E)\) at given \(\lambda \) satisfies

$$\begin{aligned} \Omega (E^{*})+(E-E^{*})\omega (E) \le \Omega (E)~, \end{aligned}$$
(23)

which leads to the inequality

$$\begin{aligned} \frac{\Omega (E^{*})}{\omega (E^{*})} +(E-E^{*}) \le \beta _{\Sigma }^{-1}(E) \end{aligned}$$
(24)

for \(\beta ^{-1}_{\Sigma }(E)=\Omega (E)/\omega (E)\). This relation indicates that as E grows with the system size, \(\beta ^{-1}_{\Sigma }(E)\) also grows. Such size-dependent behavior of \(\beta _{\Sigma }^{-1}\) (equivalently \(\Delta \)) contradicts the condition that \(\Delta \) should be non-extensive and goes against our traditional notion that temperature is an intensive variable. This necessitates seeking an alternative solution other than \(\Delta =\Omega /\omega \).

4 Thermodynamic entropy as an adiabatic invariant

In this section, we discuss adiabatic invariance as an essential property of thermodynamic entropy. In mechanics, when a system is subject to a quasi-static change of its parameter, quantities that remain unchanged are called adiabatic invariants. No thermal reservoir is involved in this context, and naturally, such slow processes referred to in mechanics are thermally isolated processes. In view of mechanics, entropy, which keeps constant during a thermally isolated quasi-static process in thermodynamics, should thus be an adiabatic invariant. Note that the phase-space volume \(\Omega \) is an adiabatic invariant for classical systems [19,20,21], which underlies the early acknowledgment of \(S_{\Gamma }=\ln \Omega \) as the thermodynamic entropy [22].

More explicitly, consider a thermally isolated quasi-static process that begins with energy and parameter value (\(E,\lambda \)) and ends with (\(E^{\prime },\lambda ^{\prime }\)). In this process, the adiabatic invariance of the phase volume implies

$$\begin{aligned} \Omega (E,\lambda )=\Omega (E^{\prime },\lambda ^{\prime })~. \end{aligned}$$
(25)

The entropy \(S_{\Sigma }\), given by \(\Omega \) at the beginning, stays the same at the end and is thus eligible to account for the isentropic process considered. This contrasts with the density of states, which is not an adiabatic invariant. Differentiating Eq. (25) with respect to E yields

$$\begin{aligned} \omega (E,\lambda )=\omega (E^{\prime },\lambda ^{\prime }) \left[ \frac{\partial E^{\prime }(E,\lambda ;\lambda ^{\prime })}{\partial E}\right] , \end{aligned}$$
(26)

which manifests that the density of states at the beginning differs from that at the end by the Jacobian factor in the square brackets. In consequence, the corresponding entropy expression, \(\ln (\omega \Delta )\) with \(\Delta \) constant, is inadequate for describing an isentropic process.

Regarding the adiabatic invariance of a mechanical system, although our interest is in classical systems, it is illuminating to notice that the degeneracy of a quantum system is also an adiabatic invariant [23, 24]. Therefore, the Boltzmann expression of entropy \(\ln W\), viewed as the logarithm of the degeneracy at specified energy in the ME of quantum systems, is qualified as appropriate entropy.

As an instructive example, we consider the ME consisting of N two-level systems. Each system takes the energy value of either \(\lambda \) or \(-\lambda \), and the energy of the whole ME system reads \(E=(2N_{+}-N)\lambda \) with \(N_{+}\) being the number of systems occupying the energy level \(\lambda \). The generalized force is given by \(F(E,\lambda )=E/\lambda \); in the isentropic process changing \(\lambda \), the energy varies following the curve

$$\begin{aligned} \left. \frac{{\text {d}}E}{{\text {d}}\lambda } \right| _{S}=F|_{S}~, \end{aligned}$$
(27)

which is the differential form of Eq. (8) and bears a solution:

$$\begin{aligned} \frac{E}{\lambda } = \text{ constant }. \end{aligned}$$
(28)

The degeneracy accommodated in this ensemble is described by the number

$$\begin{aligned} W(N_{+})=\left[ \frac{N!}{N_{+}! (N-N_{+})!}\right] _{N_{+}(E,\lambda )=\frac{1}{2}\left( \frac{E}{\lambda }+N\right) }~, \end{aligned}$$
(29)

which manifests that along the curve depicted by Eq. (28), \(N_{+}\) remains fixed and accordingly, preserves the degeneracy number W. Considering that W remains constant in the isentropic process, we can regard \(\ln W\) as entropy and also explicitly show that \(\ln W\) satisfies the thermodynamic relations in Eq. (15) (see the Appendix).

This quantum mechanical example offers insight into what \(\Delta \) can be in the case of classical systems. Note here that the density of states multiplied by the level spacing \(\lambda \),

$$\begin{aligned} \lambda \omega (E,\lambda ) = \sum _{N_{+}=0}^{N}\delta [E/\lambda {-}(2N_{+}{-}N)] W(N_{+})~, \end{aligned}$$
(30)

remains constant as E varies according to Eq. (28). This manifests that \(\lambda \omega (E,\lambda )\) is an adiabatic invariant; \(\omega \) by itself cannot thus be an adiabatic invariant. It is of interest that \(\lambda \), which is the level spacing in this discrete model system, plays the role of the window width \(\Delta \) and that \(\Delta (E,\lambda )=\lambda \) is indeed the solution of Eq. (19). With this hint that the level spacing serves as \(\Delta \), we move on to the issue concerning an alternative solution of \(\Delta \).

5 Alternative solution: level spacing

To find \(\Delta \) making \(\ln (\omega \Delta )\) consistent with the thermodynamic relation is equivalent to identifying the classical counterpart of the degeneracy number of a quantum system. As addressed in the previous section, estimating the degeneracy number with constant \(\Delta \) is inadequate. The degeneracy number \(W_{{\text {qm}}}(E,\lambda )\) for a quantum system with energy E can be expressed, in terms of the cumulative number \(\Omega _{{\text {qm}}}\) of energy eigenstates lying below energy E, as

$$\begin{aligned} W_{{\text {qm}}}(E,\lambda )= & {} \Omega _{{\text {qm}}}(E,\lambda ) -\Omega _{{\text {qm}}}(E{-}\delta (E,\lambda ),\lambda )~, \end{aligned}$$
(31)

where \(\delta (E,\lambda )\) denotes the energy spacing between the nearest energy levels below E. Here, \(\Omega _{{\text {qm}}}\) is defined as

$$\begin{aligned} \Omega _{{\text {qm}}}(E,\lambda ) \equiv \text{ Tr }\,\Theta [E-H({{\hat{p}}},{{\hat{q}}};\lambda )]~, \end{aligned}$$
(32)

where symbol \({{\hat{O}}}\) denotes the quantum observable corresponding to the classical degree of freedom \(O\,(= p, q)\). In search of the classical correspondence of this expression and, thus, that of Eq. (31), one needs to replace the trace operations with the integrals over the phase variables, which can be achieved with the help of the Weyl correspondence.

The Weyl correspondence represents a trace operation of a function of quantum mechanical observables in terms of the phase space integral of the corresponding classical function [25,26,27]:

$$\begin{aligned} \Omega _{{\text {qm}}} = \int \frac{d^{N}\!p\,d^{N}\!q}{h^{Nd}}\, G(p,q; E,\lambda )~, \end{aligned}$$
(33)

where \(G(p,q;E,\lambda )\) is a function of classical degrees of freedom:

$$\begin{aligned} G = \int {\text {d}}x\, e^{-ipx/\hbar } \left\langle q-\frac{x}{2}\Big |\Theta [E-H({{\hat{p}}},{{\hat{q}}};\lambda )] \Big |q+\frac{x}{2}\right\rangle ~. \end{aligned}$$
(34)

Expanding \(\Omega _{{\text {qm}}}\) in powers of h: \(\Omega _{{\text {qm}}} = \Omega ^{(0)}+h\Omega ^{(1)}+\cdots \) with all \(\Omega ^{(i)}\)’s having the same power of h, one finds that the leading-order term for small h is nothing but the phase-space volume in Eq. (4) [27]:

$$\begin{aligned} \ \Omega ^{(0)} = \int \frac{d^N\!p\,d^N\!q}{h^{Nd}} \Theta [E-H(p,q;\lambda )] = \Omega (E,\lambda ). \end{aligned}$$
(35)

This discloses the correspondence between the phase-space volume of a classical system and the cumulative number of states in the corresponding quantum system:

$$\begin{aligned} \Omega _{{\text {qm}}}(E,\lambda ) \,\leftrightarrow \, \Omega (E,\lambda )~. \end{aligned}$$
(36)

These quantities compose the entropy expression in Eq. (5). Note that \(\ln \Omega _{{\text {qm}}}\) for the quantum ME and its classical counterpart \(\ln \Omega \) conforms with the thermodynamic relations [28].

We apply the procedure stated above to find the classical counterpart of Eq. (31): Writing first the relation (31) as

$$\begin{aligned} W_{{\text {qm}}}(E,\lambda )= & {} \sum _{n=1}^{N}\frac{\delta ^{n}(E,\lambda )}{n!}\frac{\partial ^{n}\Omega _{{\text {qm}}}(E,\lambda )}{\partial E^{n}}~, \end{aligned}$$
(37)

and expanding the above as

$$\begin{aligned} W_{{\text {qm}}}(E,\lambda )=W^{(0)}(E,\lambda )+hW^{(1)}(E,\lambda )+\cdots , \end{aligned}$$
(38)

we extract the leading-order term \(W^{(0)}\). Here we should also consider that the level spacing \(\delta \) contains the Planck constant on a par with the expansion in Eq. (38):

$$\begin{aligned} \delta = \delta ^{(0)} + h\delta ^{(1)} + \cdots ~. \end{aligned}$$
(39)

From Eq. (37), taking only the leading-order term for small h, we obtain

$$\begin{aligned} W^{(0)}(E,\lambda )= \delta ^{(0)}(E,\lambda )\, \omega (E,\lambda )~, \end{aligned}$$
(40)

where we have used \(\partial \Omega ^{(0)}/\partial E = \omega \). This provides the classical correspondence to the degeneracy number:

$$\begin{aligned} W_{{\text {qm}}}(E,\lambda )\, \leftrightarrow \, W^{(0)} = \delta ^{(0)}(E,\lambda )\,\omega (E,\lambda )~. \end{aligned}$$
(41)

Now, we come to one of our main assertions: If a quantum system has its classical counterpart, \(W^{(0)}\) is an adiabatic invariant because \(W_{{\text {qm}}}\) should be an adiabatic invariant at any order of h. We can therefore take \(\ln [\omega \delta ^{(0)}]\) as the entropy for a classical system, equivalent to taking \(\Delta \) in Eq. (2) to be \(\delta ^{(0)}\). We recall here that for \(\ln [\omega \Delta ]\) to comply with the thermodynamic relations, the width \(\Delta \) should be the solution of Eq. (19). Accordingly, the thermodynamic consistency of \(\ln [\omega \delta ^{(0)}]\) requires that \(\delta ^{(0)}\) should satisfy Eq. (19). We have analytically tractable examples demonstrating this point, which include the two-level system discussed in the previous section. Let us take two more.

For non-interacting systems, the energy level spacing reduces to the spacing between single-particle energy eigenvalues. In the case of a one-dimensional (1D) simple harmonic oscillator (SHO), we have

$$\begin{aligned} \delta ^{(0)} (E,\lambda ) = \hbar \sqrt{\lambda }, \end{aligned}$$
(42)

where the level spacing is uniform over the energy spectrum and per se corresponds to the leading-order term, namely, \(\delta = \delta ^{(0)}\). Another example is N monoatomic molecules of mass m in a 1D box of size \(\lambda \). The single-particle energy levels are quantized as \(\epsilon _{n} = \pi ^{2}n^{2}\hbar ^{2}/(2\,m\lambda ^{2})\) with n being positive integers. For the energy \(E = \epsilon _{n}\), the level spacing \(\delta (E,\lambda )=[ \epsilon _{n+1}-\epsilon _{n}]_{E=\epsilon _n}\) amounts, in the lowest order in \(\hbar \), to

$$\begin{aligned} \delta ^{(0)} (E,\lambda ) \approx \frac{h\sqrt{E}}{\sqrt{2m\lambda ^{2}}}. \end{aligned}$$
(43)

It is straightforward to check that Eqs. (42) and (43) indeed conform to Eq. (19), with the generalized forces \(F=E/(2\lambda )\) and \(F=-2E/\lambda \) in the case of 1D SHO and 1D box, respectively. We also see that \(\delta ^{(0)}\)’s given by Eqs. (42) and (43) fulfill the requirement that they should be intensive or sub-extensive. Generally, however, direct confirmation of whether or not \(\delta ^{(0)}\) is the solution of Eq (19) may resist analytical investigation and have one resort to numerical calculation. Our reasoning that \(\ln [\omega \delta ^{(0)}]\) is acceptable as thermodynamic entropy is based on adiabatic invariance, and why the leading-order term of the level spacing should satisfy Eq. (19) remains to be an intriguing question.

6 Summary

We have revisited the thermodynamic consistency of the entropy expression in Eq. (2) given in terms of the density states with window width \(\Delta \) or the phase-space volume of an energy shell with thickness \(\Delta \). It has been shown that Eq. (2) can be the thermodynamically consistent entropy of a classical microcanonical ensemble when \(\Delta \) is a non-extensive function satisfying Eq. (19). We have discussed the non-uniqueness of the solution of the differential equation and pointed out that the entropy formula given by Eq. (5) is one particular form of the possible solution. When the density of states is a non-monotonic function of energy, this solution becomes problematic in that the associated temperature becomes extensive, opposing the traditional notion of temperature. In search of an alternative, we have paid attention to the property that thermodynamic entropy, as well as the degeneracy number of a quantum system, should be an adiabatic invariant. Based on the Weyl correspondence, we have proposed \(\Delta \) to be energy level spacing in the quantum counterpart of a classical ensemble under consideration and also presented several paradigmatic model systems that exemplify our argument.

In conclusion, the Boltzmann formula \(\ln W\) is valid for serving as thermodynamic entropy. One reservation is that approximating W of a classical ME as the product of the density of states and a small constant \(\Delta \) brings about some thermodynamic inconsistency. For the ME having many degrees of freedom, such inconsistency hardly arises. Should one wish to describe a system with a few degrees of freedom (let aside whether it is worthwhile), paying some effort in finding the energy width \(\Delta \) adequate for the thermodynamic relations is necessary.