Two different definitions of entropy, S = k ln W, in the microcanonical ensemble have been competing for over 100 years. The Boltzmann/Planck definition is that W is the number of states accessible to the system at its energy E (also called the surface entropy). The Gibbs/Hertz definition is that W is the number of states of the system up to the energy E (also called the volume entropy). These two definitions agree for large systems but differ by terms of order N−1 for small systems, where N is the number of particles in the system. For three analytical examples (a generalized classical Hamiltonian, identical quantum harmonic oscillators, and the spinless quantum ideal gas), neither the Boltzmann/Planck entropy nor heat capacity is extensive because it is always proportional to N − 1 rather than N, but the Gibbs/Hertz entropy is extensive and, in addition, gives thermodynamic quantities, which are in remarkable agreement with canonical ensemble calculations for systems of even a few particles. In a fourth example, a collection of two-level atoms, the Boltzmann/Planck entropy is in somewhat better agreement with canonical ensemble results. Similar model systems show that temperature changes when two subsystems come to thermal equilibrium are in better agreement with expectations for the Gibbs/Hertz temperature than for the Boltzmann/Planck temperature, except when the density of states is decreasing. I conclude that the Gibbs/Hertz entropy is more useful than the Boltzmann/Planck entropy for comparing microcanonical simulations with canonical molecular dynamics simulations of small systems.

Engraved on Ludwig Boltzmann’s tomb in Vienna is an equation, S = k log W, defining entropy for an isolated system (the microcanonical ensemble). For over a century, two competing interpretations have been used for W: the Boltzmann/Planck (surface) definition that includes states accessible to the system at its energy, E, and the Gibbs/Hertz (volume) definition that includes all states up to E.1 These two entropy definitions, defined formally in Sec. II B, agree for large systems, with a difference of order N−1, where N is the number of particles.2,3

Gibbs4 was the first, in 1902, to compare the utility of these two definitions. Among many who have contributed to the argument, Hertz5 favored the volume entropy because it is an adiabatic invariant, as did Rugh.6 Pearson et al.7 favored the volume entropy because it obeys equipartition. Gross and Kenney8 favored the surface energy in describing phase transitions and because it includes only the states accessible to the system, in agreement with ideas of entropy using information theory, whereas the volume entropy includes inaccessible states. On the other side, Campisi1 showed that the volume entropy, but not the surface entropy, satisfies the generalized Helmholtz theorem dS = (dE + PdV)/T, where T is the Gibbs/Hertz temperature, TG (defined below), on condition that the system is ergodic. Campisi1 also showed that the equipartition theorem is satisfied if and only if the temperature is TG. In another manuscript, Campisi9 starts from the volume entropy and derives the δ-function in energy as the phase space density for the microcanonical ensemble by maximizing the entropy subject to normalization and energy constraints and an assumption that the dynamics is ergodic. In a later paper, Campisi10 showed that, among the integrating factors that makes the heat an exact differential, the Gibbs/Hertz temperature is the only one that agrees with the ideal gas law, leading directly to the Gibbs/Hertz entropy. The Boltzmann/Planck entropy does not satisfy any of these requirements.

Recently, Hilbert et al.11 exhaustively compared the agreement of both definitions with the fundamental laws of thermodynamics and strongly favored the volume entropy. Frenkel and Warren12 rebutted Hilbert et al. and argued that the surface entropy conveniently allows for negative temperatures (population inversion) in systems with a decreasing density of states. They and Swendsen and Wang13 argued that the differences between predictions of volume and surface entropies are not measurable for large systems where statistical mechanics and thermodynamics apply. More recently, Lustig14 compared classical Monte Carlo results for three- and four-atom Lennard-Jones clusters and confirmed that the two entropy definitions gave measurably different results but, although calculations using the surface entropy were numerically more difficult, their relative suitability remained debatable. Lavis15 confirmed the volume entropy as being more accurate for systems with increasing density of states but useless for systems with a decreasing density of states and thus described by negative temperature. Cerino et al.16 also argued for the surface entropy in cases with a bounded phase space.

This paper leans toward the position taken by Hilbert et al.11 and demonstrates that the Gibbs/Hertz definition of entropy is in remarkable agreement with thermodynamic quantities calculated in the canonical ensemble, in perhaps a surprising agreement for systems with only a few particles. The more common Boltzmann/Planck definition of entropy is not extensive and is in worse agreement with canonical values for systems with increasing density of states.

The most obvious criticism of the Gibbs/Hertz entropy is that it includes inaccessible states in the “partition function,” making it inconsistent with standard statistical mechanics,17 especially since the Gibbs/Hertz entropy is not an integral of ln(ρ) over the microcanonical density. However, measurable thermodynamic quantities involve only derivatives of the partition function rather than the partition function itself, and the volume entropy has derivatives that give values in agreement with the canonical ensemble and that are determined at the surface of the energy shell and depend on the correct states included in the microcanonical ensemble. When used this way, the microcanonical “partition function” must be understood on a different footing than in other ensembles.

Uline et al.17 also criticized the use of the Gibbs/Hertz entropy by demonstrating that using the Heaviside step-function density gives incorrect probabilities by sampling all states up to the energy of the system. The rebuttal to this claim is that the Heaviside step-function density is the incorrect density function for microcanonical ensemble expectation values, the Dirac delta function being the correct microcanonical density.

Some commenters have rejected the Gibbs/Hertz entropy with the comment that thermodynamics only deals with macroscopic variables, and the Boltzmann/Planck definition is perfectly satisfactory for macroscopic systems.13 However, as I developed classical simulations of hard sphere systems for pedagogical purposes, I was forced to adopt the Gibbs/Hertz definition to maintain agreement between classical quantities, such as average kinetic energy and temperature, and also for distributions of velocities and kinetic energies in simulations.18 The Gibbs/Hertz entropy is a pragmatic choice for systems of a few to a few thousand open degrees of freedom, the kind many numerical simulations study and now obtainable in many nanoscale experiments like clusters or droplets and also in unimolecular reactions.

In most common energetically open systems, those where the density of states increases with energy, the Gibbs/Hertz definition of entropy gives thermodynamic quantities in agreement with the canonical ensemble and with fundamental principles, in particular additivity and extensivity (which are equivalent terms for examples examined here). Here, I take the canonical ensemble as giving “correct” values and judge microcanonical definitions on their ability to produce values for quantities in agreement with those of the canonical ensemble. The canonical ensemble is useful as a reference because its results for entropy per particle and heat capacity per particle are independent of the size of the system (precisely extensive). I use a generalized classical Hamiltonian (Sec. II) and two quantum examples of non-interacting particles, identical harmonic oscillators (Sec. III A) and the spinless quantum ideal gas (Sec. III B), for which I present new analytic results and which demonstrate this agreement. The preference shifts to the Boltzmann/Planck definition for systems with a finite state space, as shown in an example of a collection of two-level atoms (Sec. III C). Whereas much of the previous debate has revolved around the admissibility of negative temperatures under the Boltzmann/Planck definition,12,13,15,16 I here focus on the agreement between canonical and microcanonical thermodynamic quantities, particularly in the important area of molecular dynamics where negative temperatures represent nonequilibrium states, and thus outside the realm of equilibrium statistical mechanics.

It can be argued that microcanonical systems and canonical systems describe physically distinct scenarios and may be expected to disagree for small systems. However, I claim that the uncanny agreement for calculated fundamental physical quantities between canonical results and microcanonical results using the Gibbs/Hertz entropy is a strong argument for using the Gibbs/Hertz entropy, especially when comparing simulations using the different ensembles.

When two macroscopic systems are brought into thermal contact, heat flows from the system at higher temperature to the system at lower temperature, resulting in a final, intermediate temperature for the combined equilibrium system. For similar examples of small systems, Sec. IV demonstrates that temperatures defined by the Gibbs/Hertz entropy satisfy this requirement better than temperatures defined by the Boltzmann/Planck entropy. Again, however, the Boltzmann/Planck temperature works better for systems with a decreasing density of states.

For a wide variety of classical systems, Sec. 1 of the supplementary material shows that the number of states at or below an energy E (phase volume) is given by

(1)

where A and ln(B) are constants proportional to the size of the size of the system [Eq. (S11)] and where Γ(x) is the gamma function.19 Thus, the density of states is

(2)

The generalized Hamiltonian given in the supplementary material by Eq. (S1) includes only power-law potential energy terms. However, many classical systems not given by Eq. (S1) also have Φ(E) proportional to Ep, where p is proportional to the number of particles, and the following results still apply.

This section summarizes canonical quantities for classical systems to which microcanonical quantities will be compared. For systems described by Eq. (1), the energy moments of the canonical partition function are given by20 

(3)

where kT = β−1. The energy moments in Eq. (3) allow calculations of standard canonical thermodynamic quantities,

(4)
(5)
(6)
(7)

The most probable energy for the system is the maximum of the phase space density (the integrand in Q0), which is given by

(8)

The width of the phase space distribution around the maximum of the phase space density is given by [using Eqs. (4) and (5)]

(9)

Thus, the average energy, E, of the system is extensive, according to Eq. (4), because A is proportional to the size of the system for a system of identical subsystems, and the most probable (or mode) energy for the system is exactly one increment of kT lower than E but still within the expected energy fluctuations of the system, σE, even for systems with just a few particles. Note also that the entropy [Eq. (7)] is extensive because B is a constant with units of inverse energy raised to the Nth power for a system made up of identical subsystems (see Sec. 1 of the supplementary material).

The Boltzmann/Planck entropy definition is11 

(10)

where Δ is an arbitrary but small energy resolution required for dimensional reasons. Because G(E) is the (2N − 1)-th dimensional surface area of the energy shell in 2N-dimensional phase space, SG has been called the surface entropy.1 The Gibbs/Hertz definition of entropy is

(11)

Because Φ is the 2N-dimensional phase space volume enclosed by the energy shell, SG has been called the volume entropy.1 

For systems satisfying Eq. (1), I now evaluate in the microcanonical ensemble the same thermodynamic quantities calculated above for the canonical ensemble to verify the level of agreement with canonical quantities,

(12)
(13)

where the prime in Eq. (13) indicates a derivative with respect to E. For identical particles, the microcanonical energy, E, is not proportional to the number of particles times TB, whereas E is proportional to the number of particles times TG, an indication that SG preserves equipartition.7 Note that TB for a given microcanonical energy has a higher value than TG.

Comparing Eqs. (12) and (13) to Eqs. (4) and (8) clarifies the relationship between these two microcanonical temperatures. The microcanonical temperature TG is the same as the canonical heat bath temperature when the microcanonical energy equals the average energy of the canonical ensemble. However, TB is the canonical temperature where the microcanonical energy equals the most probable canonical energy.

I now obtain expressions for the entropy and heat capacity in terms of the temperature according to both definitions. Substituting Eq. (12) into Eq. (11), the result is

(14)

Substituting Eq. (13) into Eq. (10),

(15)

The nonextensive terms in Eq. (14) are corrections to Stirling’s approximation for the factorials (gamma functions).19 Up to these terms, which are negligible compared to N for large systems, SG has the same dependence on kTG as the canonical S has on kT [Eq. (7)]. However, SB does not have the same dependence on kTB as canonical S; it is not extensive, agreeing more with a system with one fewer degrees of freedom.

For the same reasons [cf. Eq. (6)],

(16)

where primes denote energy derivatives. Likewise,

(17)

Thus, the microcanonical heat capacity calculated with G(E) is not extensive again and does not give the same value as the canonical ensemble, but the heat capacity calculated with Φ(E) is extensive and agrees with the canonical ensemble.

Examining Eqs. (1) and (2), it is physically easy to see why SG is proportional to N and SB is proportional to N − 1. The phase space volume, Φ, is proportional to a power of E raised to the Nth power, and its derivative, G(E), is the “surface area” of that volume in which E is raised to a power smaller by 1. Everything in this paper for energetically open systems [those satisfying Eq. (1)] follows from this geometrical fact.

In the usual statistical ensembles studied in textbooks, the classical partition function is an integral of the appropriate phase space density, and the expectation value of a chosen phase space function is given by the corresponding phase space integral over that density. In addition, all thermodynamic quantities can be calculated from the logarithm of the partition function. The canonical ensemble phase space density is eβH, and the microcanonical ensemble phase space density is δ(EH)Δ. The above discussion highlights the continuing confusion regarding the microcanonical ensemble: Using δ(EH)Δ as the microcanonical density correctly gives expectation values by phase space averages over this density; however, one arrives at the partition function G(E)Δ, which gives nonextensive results in disagreement with canonical ensemble results for small systems (here, small means N is experimentally distinguishable from N − 1). To get results in agreement with canonical values, one must use θEH, the Heaviside step function, as the phase space density to arrive at the partition function Φ(E). This “partition function” includes states inaccessible at the energy E. However, the derivatives of the step function, which involve δ-functions at the microcanonical energy, E, have the correct behavior for proper microcanonical averages and thermodynamic functions. The use of Φ(E) as the “partition function” has been criticized because expectation values of the corresponding density include states not accessible at the microcanonical energy, E, and thus does not give correct results.17 However, if one uses δ(EH)Δ as the phase space density for expectation values and Φ(E) to calculate thermodynamic functions, one obtains consistent results in agreement with canonical ensemble results for systems with all N because measurable thermodynamic functions depend on the derivative of Φ(E), and the derivative of the density θEH with respect to energy is δ(EH). On the contrary, using the Boltzmann/Planck definition of entropy means calculated measurable quantities depend on the derivative of the δ-function and disagree with those calculated in the canonical ensemble.

I have demonstrated the utility of the Gibbs/Hertz entropy for a wide variety of classical systems of distinguishable particles given by Eq. (1). In addition, similar results have been observed in other classical systems like hard spheres and particles interacting through more realistic interaction potentials.7,21 In addition, the arguments of Pearson et al.7 and Hilbert et al.11 extend these arguments to classical systems not included in Eq. (1).

It is well known2 that different ensembles give results that can disagree to order N−1, so it is not surprising that SB gives microcanonical results that are in disagreement in comparison to the canonical ensemble by a factor of (N − 1)/N. However, this section has demonstrated that nearly all of this disagreement for many classical systems can be avoided by using SG instead.

In this section, I demonstrate the utility of the Gibbs/Hertz entropy in simple quantum systems. Because of the discrete nature of quantum states, the quantum canonical heat capacity as a function of temperature departs from its classical value at high temperature and approaches zero at low temperature. For the same reason, entropy limits to zero (Third Law) when temperature approaches T = 0 when a nondegenerate quantum ground state is the only accessible state. Here, for simplicity, I retain Boltzmann statistics and neglect spin so as to avoid complications of Fermi–Dirac or Bose–Einstein statistics. In this case, quantum canonical ensemble calculations make use of the canonical partition function, a sum rather than a continuous integral, but Eqs. (4)(7) still apply for thermodynamic quantities,

(18)

To evaluate microcanonical thermodynamic quantities, I need to evaluate energy derivatives of the quantum expressions for G(E) (for the Boltzmann/Planck entropy) or Φ(E) (for the Gibbs/Hertz entropy) for use in Eqs. (12)(17). I have examined several alternative finite difference schemes discussed in Sec. 2 of the supplementary material. These differentiation schemes are unsatisfactory compared to converting the functions describing G(E) or Φ(E) into continuous functions and then differentiating them directly (analytic continuation). I will compare the resulting formulas to canonical quantities, in particular heat capacity and entropy.

For N harmonic oscillators with identical circular frequency, ν, and quantum numbers, ni (i = 1, 2, …, N), the energy is

(19)

where h is Planck’s constant and M is the total quantum number. It is convenient to choose the zero of energy to be the ground state energy, E0. For convenience, I take Δ = , the harmonic oscillator unit of energy, in Eq. (10).

1. Canonical ensemble

In this case, the canonical partition function simplifies

(20)

In addition,

(21)

where the arrow represents limiting behavior at high temperature. In addition,

(22)

Note again, in agreement with classical mechanics in which ai = 1/2, (i = 1, 2, …, 2N) in Eq. (S1), that the entropy and heat capacity are both extensive properties proportional to N (for all N), even at low temperatures where they depart from their classical values.

Since the quantum heat capacity and entropy differ from their classical values primarily at low temperature, it is useful to express Eqs. (21) and (22) as high-temperature expansions. The expansions were evaluated using Mathematica,22 

(23)
(24)

The asymptotic expansion coefficients are sj = B2j/[(2j − 2)!(2j)2], where B2j are Bernoulli numbers,23 tj = −2sj, and where the initial values are s1 = +1/24 so t1 = −1/12. Values up to j = 8 are tabulated in Sec. 3 of the supplementary material.

2. Microcanonical ensemble

To calculate thermodynamic quantities in the microcanonical ensemble, the allowed energies are still given by Eq. (19), and the density of states is given by the number of ways M quanta can be distributed among the N oscillators,

(25)

and similarly,

(26)

In the right-most expressions of Eqs. (25) and (26), the factorials have been converted to the continuous gamma function, whose derivatives can be expressed in terms of the digamma function ψ(z) = d ln Γ(z)/dz.19 

We can now obtain thermodynamic variables in the quantum microcanonical ensemble using the analytic derivatives of the Γ-function.19 The results are

(27)

where I have used dΦ/dE = (/ε)dΦ/dM [middle equality in Eq. (16)] and the corresponding second derivative in place of G and dG/dE, which are no longer exactly, respectively, equal. [Note that ψ′(x) is sometimes called the trigamma function.] Similarly,

(28)

Equations (27) and (28) are new results, although Lavis15 exhibited the first part of Eq. (28) without applying it and Miranda and Bertoldi24 treated the system using a backward difference approach. Miranda25 also treated this system numerically, and the analytic results shown here are in agreement and extend those results. In applying these equations, note that differences in ψ values are sums of reciprocals of consecutive integers and differences in ψ′ values are sums of reciprocals of squares of consecutive integers.

3. Heat capacity

The Gibbs/Hertz heat capacity for N = 6 is plotted in Fig. 1. It agrees with the canonical heat capacity extremely well, while the Boltzmann/Planck heat capacity is precisely the same formula for a system with N − 1 oscillators. In Fig. 1, the solid line is the canonical heat capacity [Eq. (22)], which can be evaluated at arbitrary temperature. For the two microcanonical functions, the temperature is evaluated for each value of M using Eq. (27) or Eq. (28) and then the corresponding heat capacity is evaluated. One point is obtained for each quantum level, M, this way, and the plot consists of a series of points rather than a continuous line. The Gibbs/Hertz heat capacity limits to Nk for high temperature, in agreement with the classical result, but it also agrees surprisingly well with the canonical calculation at low temperature. Again, the Boltzmann/Planck heat capacity limits to (N − 1)k (the classical result) and is thus not extensive.

FIG. 1.

Heat capacity, CV/k, vs kT/ for six harmonic oscillators. Solid line: canonical Cv [Eq. (22)]. Blue diamonds: Gibbs/Hertz definition (CVG vs TG) for first 21 quantum states. Green triangles: Boltzmann/Planck definition (CVB vs TB) for the first 18 quantum states.

FIG. 1.

Heat capacity, CV/k, vs kT/ for six harmonic oscillators. Solid line: canonical Cv [Eq. (22)]. Blue diamonds: Gibbs/Hertz definition (CVG vs TG) for first 21 quantum states. Green triangles: Boltzmann/Planck definition (CVB vs TB) for the first 18 quantum states.

Close modal

To examine the agreement at low temperature between the canonical heat capacity and the two definitions of microcanonical entropy, it is instructive to treat the excitation value M as a continuous variable again. In that case, M = 0 properly gives Φ = 1 and SG = 0, but this does not correspond to TG = 0. To obtain the equivalent of absolute zero temperature, let M approach −1 for which Γ(M + 1) → −∞ and βG → ∞ (as simple poles), corresponding to TG → 0. In order to evaluate ψ and ψ′ functions for arbitrary real values instead of integers, I employed a FORTRAN subroutine (see supplementary material, Sec. 14) that uses asymptotic expansions for large argument and recurrence relations to step the argument value down to the desired value.26 Then, Eq. (27) can be used to evaluate TG and CVG for any real value of M greater than −1 or Eq. (28) can likewise be used to evaluate TB and CVB.

The Gibbs/Hertz heat capacity, CVG, approaches 1 at low temperature, so the heat capacity per oscillator CVG/N, as well as all its derivatives, vanish as N−1. This behavior is demonstrated in Fig. 2, showing CVG/Nk as a continuous function of TG for five different values of N (3, 6, 12, 24, and 48) along with the canonical heat capacity. One can clearly see that the Gibbs/Hertz definition for low temperature heat capacity clearly differs from the canonical heat capacity by an amount inversely proportional to N. However, the difference is perhaps surprisingly small. The reason the Gibbs/Hertz definition of entropy gives a heat capacity larger than the canonical heat capacity is clarified in Sec. 5 of the supplementary material.

FIG. 2.

Low temperature heat capacity per oscillator, CVG/Nk vs TG, for five values of N. Blue line: N = 3, red line: N = 6, green line: N = 12, violet line: N = 24, and orange line: N = 48. The black line is the canonical expression [Eq. (22), CV vs T].

FIG. 2.

Low temperature heat capacity per oscillator, CVG/Nk vs TG, for five values of N. Blue line: N = 3, red line: N = 6, green line: N = 12, violet line: N = 24, and orange line: N = 48. The black line is the canonical expression [Eq. (22), CV vs T].

Close modal

The Boltzmann/Planck heat capacity is precisely the same formula for a system with N − 1 oscillators, so even if rescaled by N − 1 rather than N, agreement is slightly worse because corrections of order (N − 1)−1 are greater than those of order N−1.

4. Entropy

I now compare the Gibbs/Hertz entropy and the Boltzmann/Planck entropy to the canonical entropy in Fig. 3. All three of these obey the third law of thermodynamics in that the entropy goes to zero at the ground state. However, as stated above, the microcanonical temperature for M = 0 as determined by Eq. (27) or Eq. (28) is not T = 0. This fact shifts the Gibbs/Hertz entropy to the right on the temperature scale. This feature shows up as a nearly constant vertical difference between the canonical entropy and the Gibbs/Hertz entropy (see Fig. 4). The asymptotic value of this difference for large M is γN = ln(Γ(N + 1))/N − ln N + 1 ≈ ln(2πN)/2N + 1/(12N2) + O(N−4) and is related to the error in Stirling’s approximation for factorials. For large systems, γN is negligible compared to N. The Gibbs/Hertz entropy is proportional to N[ln(kTG/) + 1 − γN] for large M, in agreement with the leading terms of the canonical expression [Eq. (23)]. A constant difference in entropy exhibited by the canonical vs Gibbs/Hertz definition is not experimentally measurable, resulting in the same changes in entropy for thermodynamic processes. Figure 4 plots the slight deviation from γN as a function of kTG/ for N = 6, for which γ6 = 0.304 78…. The deviation from γ6 is in the third decimal place except for the first few values of M.

FIG. 3.

Entropy, S/k, vs kT/ for six harmonic oscillators. Solid line: canonical S [Eq. (21) multiplied by 6]. Blue diamonds: Gibbs/Hertz definition [using Eq. (26) vs TG] for first 33 quantum states. Green triangles: Boltzmann/Planck definition [using Eq. (25) vs TB] for the first 28 quantum states.

FIG. 3.

Entropy, S/k, vs kT/ for six harmonic oscillators. Solid line: canonical S [Eq. (21) multiplied by 6]. Blue diamonds: Gibbs/Hertz definition [using Eq. (26) vs TG] for first 33 quantum states. Green triangles: Boltzmann/Planck definition [using Eq. (25) vs TB] for the first 28 quantum states.

Close modal
FIG. 4.

Difference between canonical entropy per oscillator and the Gibbs/Hertz entropy per oscillator for the first 31 quantum states for N = 6 plotted vs TG. The asymptotic difference given by γ6 is due to the nonzero value of TG for the ground state.

FIG. 4.

Difference between canonical entropy per oscillator and the Gibbs/Hertz entropy per oscillator for the first 31 quantum states for N = 6 plotted vs TG. The asymptotic difference given by γ6 is due to the nonzero value of TG for the ground state.

Close modal

On the other hand, the difference between the Boltzmann/Planck entropy and the canonical entropy is not constant because the Boltzmann/Planck entropy is precisely that of a system with N − 1 oscillators. The Boltzmann/Planck entropy is proportional to (N − 1)[ln(kTB/) + 1 − γN−1] for large M. Again, this is the same correspondence seen in classical systems—extensivity and agreement for the Gibbs/Hertz entropy, but nonextensivity and disagreement for the Boltzmann/Planck entropy.

5. Microcanonical high temperature expansions

The Gibbs/Hertz high temperature expansions for both the heat capacity and entropy were generated analytically using Mathematica22 and Eqs. (25)(28). These are tabulated in Secs. 4 and 5 of the supplementary material. To leading order, all coefficients agree with those for the canonical heat capacity and entropy described by Eqs. (23) and (24) through order β,16 the highest order attempted. The expansion coefficients differ only in terms proportional to N−2 and higher. The N-dependence of the coefficients describes the deviation from the constant offset of the microcanonical entropy from the canonical entropy, an example of which is shown in Fig. 4. The full dependence of the deviation is given by the N-dependent terms in the high temperature expansion, the leading terms of which are γN + (βhν)2/(24N2). The Boltzmann/Planck entropy divided by N − 1 instead of N also differs from the canonical entropy per particle by a constant but by γN−1, a slightly larger number. The high temperature coefficients are identical to those of the Boltzmann/Planck entropy with N replaced by N − 1.

6. Summary

For a set of harmonic oscillators, not only does the Gibbs/Hertz definition of entropy demonstrate nearly perfect extensivity for entropy and heat capacity, it even gives low temperature behavior in agreement with canonical quantities, with differences of order N−1. Moreover, it also reproduces the canonical expansion coefficients in the high temperature expansions with corrections of order N−2. The Boltzmann/Planck entropy and heat capacity does neither, although it does have the expected expansion coefficients, but for a system with N − 1 oscillators.

For N identical distinguishable particles of mass m in a cubic box of edge length L in d dimensions, the allowed quantum energy levels are (where h is Planck’s constant)

(29)

for which the values of E/ε are all integers as the quantum numbers, ni, are integers. I take Δ = ε in Eq. (10).

1. Canonical ensemble

The energy moments of the canonical partition function for d = 1 have the form

(30)

These sums are straightforward to evaluate numerically for low temperatures when only a few states are occupied. For higher temperatures when many states are occupied, these quantities can be evaluated by converting the sum to an integral using the Euler–Maclaurin summation formula with corrections as shown in the right-most expression in Eq. (30).27 The natural high temperature expansion variable can be seen to be x = (βε/π)1/2. Quantities of interest are (for high T)

(31)
(32)
(33)
(34)
(35)

For a system of N particles in d dimensions, Eqs. (32)(35) or the results of direct summation are multiplied by dN. Now, I will examine microcanonical ensemble expressions for agreement with these expressions.

2. Microcanonical ensemble

The phase volume state count, ΦdN(j), where j = E/ε = Ẽ, is given approximately by the following function (which is a new result):28 

(36)

where

(37)

is the m-dimensional volume of a sphere of radius R. The first term in Eq. (36) can be understood geometrically by assuming that each quantum state occupies a cube of unit dN-dimensional volume centered around the point with quantum numbers (n1, n2, n3, …, ndN). To a first approximation, the states up to E are those whose dN-dimensional radius from the origin is less than (E/ε)1/2. Additional terms are motivated in Sec. 7 of the supplementary material. In particular, for larger values of dN, it is convenient to focus on the dominant terms for comparison with classical systems. Then, using f = dN/2,

(38)

where

(39)
(40)

(four additional terms are written out in Sec. 7 of the supplementary material), where the number of terms in the expansion is limited to dN + 1. The leading term is the same as the classical phase volume given by Eqs. (S2)–(S9), and additional terms represent quantum corrections. The exponent of energy in Φ, f, is loosely the number of “degrees of freedom” because only kinetic energy terms contribute to thermodynamic quantities [in classical terms, a1 = 1/2, but a2 = 0 for each dimension in Eq. (S1)]. I now use the continuous phase space volume given by Eq. (38) to calculate microcanonical thermodynamic quantities. I will not consider finite difference differentiation in this case because discreteness of the density of states make differentiation unstable, although smoothed results are similar to those given below.

To obtain microcanonical expressions for the heat capacity and entropy, take the analytic derivative of the phase space volume [Eq. (36) or Eq. (38)] with respect to E to obtain G(E) = Φ′(E), giving an expansion for βG = Φ′(E)/Φ(E) in terms of the expansion variable (/(πE))1/2. Reverting the series using Mathematica22 gives an expansion for (/(πE))1/2 in terms of x = (βGε/π)1/2, which can be substituted into expressions SG/k = ln Φ and CVG/k = 1/(1 − ΦΦ″/Φ′2). The resulting high temperature expansion coefficients for the Gibbs/Hertz entropy and the Gibbs/Hertz heat capacity are exactly the same as the high temperature expansions for the canonical entropy [Eq. (35)] and canonical heat capacity [Eq. (34)] for all orders attempted except for corrections proportional to f−1 and higher (see Secs. 7 and 8 of the supplementary material). Again, there is a nearly constant entropy difference, this time γf/2 because dN = 2f, between the canonical entropy and the Gibbs/Hertz microcanonical entropy. As stated before, this constant entropy difference has no measurable consequence and becomes negligible for large systems and again results from the nonzero microcanonical temperature at the zero-point energy. The leading terms of the entropy difference at high temperature is γf/2 + (β/π)1/2/(16f), so the variation in this difference is slowly decaying with both increasing temperature and system size than in the harmonic oscillator case [which behaved like γN + β2/(24N2)].

The Boltzmann/Planck expressions were also evaluated: SB/k = ln() and CVB/k = 1/(1 − GG″/G2). Similar to previous examples, the Boltzmann/Planck heat capacity limits to d(N − 2)k/2 = (f − 1)k rather than dNk/2 = fk and has a high temperature expansion of a system more appropriate to f − 1 degrees of freedom. Similarly, the Boltzmann/Planck entropy is more like that of a system of f − 1 degrees of freedom. However, in this case, even when rescaled, the dN-dependent corrections are larger for CVB and SB, in this case nine times larger (see Secs. 7 and 8 of the supplementary material). The Boltzmann/Planck entropy also contains an additional additive term of ln(π/4) because of the factor raised to the fth power in Eq. (38). Thus, the asymptotic difference between the canonical S/k and SB/[2k(f − 1)] is γf−1/2 − ln(π/4)/[2(f − 1)].

The low temperature Gibbs/Hertz heat capacity can be examined by allowing E to take on continuous values below the zero-point energy. Figure 5 shows the Gibbs/Hertz heat capacity, CVG, divided by dN for dN particles in a d-dimensional box for dN = 4, 8, 12, and 24 along with the canonical ensemble heat capacity. Like in the case of harmonic oscillators, the deviations are inversely proportional to N.

FIG. 5.

Heat capacity vs temperature for ideal gas. Black line: canonical CV/k. Other curves are microcanonical CVG/(kdN) vs TG. Red line: dN = 4, green line: dN = 8, purple line: dN = 12, and blue line: dN = 24. Symbols on top of curves correspond to quantum energies.

FIG. 5.

Heat capacity vs temperature for ideal gas. Black line: canonical CV/k. Other curves are microcanonical CVG/(kdN) vs TG. Red line: dN = 4, green line: dN = 8, purple line: dN = 12, and blue line: dN = 24. Symbols on top of curves correspond to quantum energies.

Close modal

Not shown in Fig. 5 are plots for the Boltzmann/Planck heat capacity, CVB, which must be divided by dN − 2 to approach the same high temperature limit of one half. Both CVG/[2kf] and CVB/[2k(f − 1)] approach the canonical heat capacity as dN = 2f increases, with corrections proportional to powers of N−1. Both CVG/[2kf] and CVB/[2k(f − 1)] also approach the canonical heat capacity in the neighborhood of the maximum. In this region, the order N−1 corrections for CVB/[2k(f − 1)] are three times as big as those for CVG/[2kf] even when rescaled for extensivity. The low temperature behavior of CVB and CVG when E is considered a continuous variable and extended below the quantum zero-point energy is discussed in Sec. 8 of the supplementary material.

Consider a system with N identical noninteracting atoms, each with two energy levels, E0 = 0 and E1 = ε. This system is isomorphic to a system of identical noninteracting spin-one-half spins in a magnetic field.29 This example exhibits a decreasing quantum density of states.

1. Canonical ensemble

The energy moments of the canonical partition function for the system are given by

(41)

The entropy and heat capacity (and their high temperature expansions) are thus

(42)
(43)

where u1 = −1/8 and v1 = 1/4, and additional terms up though j = 8 are tabulated in Sec. 9 of the supplementary material. Note that the maximum entropy for N spins is Nk ln 2 because there are 2N total states. A broad peak in the heat capacity described by Eq. (43) is commonly known as a Schottky anomaly.29 

2. Microcanonical ensemble

The density of states for N two-level atoms with M excited atoms is given by the number of ways M quanta can be distributed among the N atoms,

(44)

and using the derivatives of the continuous gamma function19 as in the harmonic oscillator case,

(45)

Note that, as a function of excitation, M, G(M) has a maximum when M = N/2, so βB changes sign, corresponding to negative temperatures and population inversion for M > N/2. Equation (45) is a new result but similar to that of Miranda and Bertoldi.24 

A continuous analytic function that expresses the phase volume,

(46)

is not known, so I attempted polynomial fits discussed in Sec. 10 of the supplementary material.

Figure 6 exhibits computed heat capacity data, CVG and CVB, as a function of temperature for N = 6 compared with computed canonical ensemble data. The heavier, black line is the canonical heat capacity curve, and the integral under the canonical CV curve from T = 0 to T → ∞ is N/2 = 3. Infinite temperature corresponds to all six one atom excited states occupied with probability one half (M = N/2). To have an average excitation greater than three quanta would correspond to a population inversion and require a negative temperature, i.e., a nonequilibrium condition.

FIG. 6.

Heat capacity vs temperature for N = 6 two-level atoms. The black line is the canonical ensemble function [Eq. (43) multiplied by 6]. The blue line is CVG [Eqs. (S52) and (S53) and derivatives]. The red line is CVB [Eq. (45)]. Red and blue circles indicate quantum values. Red and blue lines connect the circles for non-integer values.

FIG. 6.

Heat capacity vs temperature for N = 6 two-level atoms. The black line is the canonical ensemble function [Eq. (43) multiplied by 6]. The blue line is CVG [Eqs. (S52) and (S53) and derivatives]. The red line is CVB [Eq. (45)]. Red and blue circles indicate quantum values. Red and blue lines connect the circles for non-integer values.

Close modal

In Fig. 6, red (CVB) and blue (CVG) circles correspond to quantum states (integer values of M = 0, 1, 2, and 3). The red line connects the red circles by using the FORTRAN program supplied in Sec. 14 of the supplementary material using Eq. (45) for TB and CVB at non-integer values of M. The blue line does similarly using the parameterization of Eqs. (S52) and (S53) and TG = Φ/Φ′ and CVG/k = 1/[1 − ΦΦ″/(Φ′)2].

The area under the blue CVG curve is 7 = N + 1 because M is parameterized from M = −1, corresponding to TG = 0, to M = 6 (all states excited), corresponding to TG → ∞. CVG limits to 2 at low temperature because I have chosen to lead Eq. (S52) with quadratic terms (k = 2). The area under the red CVB curve is 4 = N/2 + 1 because M is parameterized in Eq. (45) from M = −1, corresponding to TB = 0, to M = 3 [maximum in G(M) in Eq. (44)], corresponding to TB → ∞. Even though both CVG and CVB both have reasonable form for such a small system, as N increases, CVB looks more and more like the canonical heat capacity with difference proportional to N−1. However, for large N, CVG will continue to be much too large, having an area a bit more than double the canonical curve. In addition, whereas CVG is not defined for further excitation (M = N being the limit), CVB is repeated in mirror image as M is increased from N/2 to N as temperature goes through a simple pole and becomes negative, approaching zero from below as M approaches N.

Figure 7 exhibits computed entropy per particle data for SG and SB as a function of temperature for N = 6 compared with computed canonical ensemble data. The heavier, black line is the canonical entropy curve, which limits to N ln 2 = 4.1588… as T → ∞. The blue SG curve also correctly limits to N ln 2 but is offset to the right because S(0) = 0(Φ(0) = 1) corresponds to a nonzero value of TG, just as in the harmonic oscillator example. However, SG no longer has a nearly constant offset from the canonical curve as in the harmonic oscillator case, a weakness of the fitting method used [Eqs. (S52) and (S53)]. In comparison, SB limits a value less than N ln 2 because fewer than the total number of states are degenerate at M = N/2, where TB → ∞ and G(M) reaches a maximum. The additional states do not show up to make the SB curve approach the value of N ln 2, however, as M approaches N because G(M) decreases back to zero. This limit, however, makes statistical sense because for M = N, all atoms are excited, and this state is nondegenerate just like the ground state. The high temperature expansion coefficients for SB and CVB are exhibited in Secs. 11 and 12 of the supplementary material, and they agree with the canonical values [Eqs. (42) and (43)] up to corrections terms of order N−1. Thus, SB for the two-level atom example has the same property as SG has in the harmonic oscillator case in that it approaches a constant difference from the canonical entropy in a way analogous to Fig. 4, except that the asymptotic value for the entropy difference per oscillator is ln(πN/2)/(2N) (a more complete expansion is in Sec. 11 of the supplementary material). One surprise, however, is that the high temperature expansions for SB and CVB can be recast more compactly in terms of N + 1, in contrast to the harmonic oscillator case where these expansions were in terms of N − 1.

FIG. 7.

Entropy vs temperature for N = 6 two-level atoms. The black line is the canonical ensemble function [Eq. (43) multiplied by 6]. The blue line is SG [logarithm of Eqs. (S52) and (S53) using Φ(M)/Φ′(M) for temperature]. The red line is SB [logarithm of Eq. (44) using temperature defined by Eq. (45)]. Red and blue circles are for quantum values. Red and blue lines connect the circles at non-integer values of M.

FIG. 7.

Entropy vs temperature for N = 6 two-level atoms. The black line is the canonical ensemble function [Eq. (43) multiplied by 6]. The blue line is SG [logarithm of Eqs. (S52) and (S53) using Φ(M)/Φ′(M) for temperature]. The red line is SB [logarithm of Eq. (44) using temperature defined by Eq. (45)]. Red and blue circles are for quantum values. Red and blue lines connect the circles at non-integer values of M.

Close modal

In summary, for this example of N two-level atoms, the Gibbs/Hertz definition of entropy gives somewhat better agreement with the canonical entropy than the Boltzmann/Planck definition, although SG, as parameterized here, does not approach a constant difference as in the harmonic oscillator case. For the heat capacity, CVB is in better agreement with canonical heat capacity than CVG, even though CVB scales more like a system with N + 1 atoms. CVG describes all N units of excitation before TG diverges, whereas CVB is more comparable with CV in describing only approximately N/2 units of excitation before TB diverges. For this directly measurable quantity, the Boltzmann/Planck heat capacity is both more parallel in interpretation and quantitatively more accurate than the Gibbs/Hertz heat capacity for the system of N two-level atoms.

In this section, I consider whether, under the Gibbs/Hertz and Boltzmann/Planck definitions of temperature, energy flows from a system of high temperature to a system of low temperature in accordance the expected behavior of macroscopic systems, an issue of concern to previous authors.8,11–13,16

Consider two classical subsystems satisfying Eq. (1) with energies E1 and E2 with A1 and A2 degrees of freedom brought into thermal contact to form a combined system with energy E1 + E2 and A1 + A2 degrees of freedom. Neglecting any interaction terms between the subsystems and using Eq. (12), the Gibbs/Hertz temperature change undergone by each of the subsystems in the process is described by

(47)

Using Eq. (47), the ratio of the temperature changes is

(48)

showing that the temperature changes are always in the opposite direction and in inverse proportion to the heat capacity of the two subsystems. Equation (48) is more easily recognized by rewriting it as

(49)

where opposite sides of the equation now indicate the energy gained or lost by each subsystem.

The Gibbs/Hertz entropy change of the total system when the two classical subsystems combine is [using Eq. (11)]

(50)

where γA is defined in Sec. III and may again be neglected for large arguments. Equation (50) may be rewritten as

(51)

which, when the correction terms involving γA are neglected, is just the entropy change for macroscopic systems with constant heat capacities, where TGT is the final Gibbs/Hertz temperature of the combined system. Note that including the small correction terms in Eq. (51) always gives an incremental increase in entropy when initial and final temperatures are all equal (because γA is a decreasing function of A) in agreement with the second law of thermodynamics.

Turning now to examine the behavior of the Boltzmann/Planck temperature for classical systems, let Ãj=Aj − 1, which is proportional to the Boltzmann/Planck heat capacity of a system [Eq. (17)]. Now, consider bringing two subsystems with energies E1 and E2 and Boltzmann/Planck temperatures kTB1=E1/Ã1 and kTB2=E2/Ã2 into thermal contact to form a combined system with energy E1 + E2 and Boltzmann/Planck temperature kTBT=E1+E2/A1+A21=ET/ÃT. Consider the ratio

(52)

This characteristic ratio was identically −1 for the Gibbs/Hertz temperature changes [Eq. (48)]—but for Boltzmann/Planck temperature changes only approaches −1 when E1E2 or E2E1. In particular, this ratio vanishes and changes sign when the numerator in Eq. (52) vanishes,

(53)

The characteristic ratio also changes sign when the denominator of Eq. (52) vanishes,

(54)

Thus, there is a narrow range of temperature inversely proportional to the sizes of the subsystems where the Boltzmann/Planck temperature changes of small classical systems contradict the expectation of macroscopic thermodynamics. This occurs when the initial temperatures are nearly the same but where both subsystems change their Boltzmann/Planck temperature in the same direction when placed in thermal contact. The most convenient way to examine this contradiction is to consider two identical subsystems at the same Boltzmann/Planck temperature being brought into thermal contact. In the case of the Gibbs/Hertz temperature, no heat is exchanged, the Gibbs/Hertz temperatures remain unchanged, and the entropy of the total system increases but by a negligible amount. The change in Boltzmann/Planck temperature when two identical systems with energy E and initial temperature TB = E/kà are brought in to thermal contact is, using Eq. (13),

(55)

Thus, the Boltzmann/Planck temperature always decreases when two identical subsystems of equal temperature are placed in thermal contact. This decrease is inversely proportional to the size of the system, but the change increases with the initial temperature. This behavior contradicts the expectation of macroscopic thermodynamics. The explanation, according to critics of the Boltzmann/Planck definition, is that the Boltzmann/Planck temperature is slightly too high compared to the Gibbs/Hertz temperature, but the error is reduced by a factor of two when the system size is doubled [see Eq. (13)]. This conclusion casts in doubt the conclusion of Cerino et al.,16 who found no unexpected temperature changes using TB in systems where d2SB/dE2 ≥ 0, as it is for the classical systems examined above.

Now consider the Boltzmann/Planck entropy change when two identical subsystems of equal Boltzmann/Planck temperature are brought into thermal contact using Eq. (13) in Eq. (10),

(56)

where I have now included the largest two correction terms in γA from the gamma functions. The corresponding entropy change for the Gibbs/Hertz entropy is ΔSG/k=12lnπA+18A+OA3. The ln(kTB/Δ) term in the change in Boltzmann/Planck entropy change [Eq. (56)] is another contradiction to the expectations of macroscopic thermodynamics. If Δ is taken to be sufficiently large, the entropy change becomes negative, in violation of the second law of thermodynamics, reinforcing the claim after Eq. (10) that Δ must be small. However, if Δ is too small, the entropy change in Eq. (56) is arbitrarily large, again in contradiction of macroscopic thermodynamics for identical subsystems placed in thermal equilibrium. Even with an optimal choice for Δ, the contradiction increases with temperature. An equivalent term is present in the Boltzmann/Planck entropy changes, for example, quantum systems (see Table II).

The anomalies in both the temperature change and the entropy change for the Boltzmann/Planck definition occur because the density of states has the energy raised to the power A − 1 rather than A, so its logarithm is not proportional to the size of the system. The Boltzmann/Planck entropy is nonextensive and nonadditive in nanoscale systems, and this fact is again the source of these problems as well as in the following examples for quantum systems.

For each example quantum system used earlier, I now examine the temperature and entropy change when two identical subsystems at the same temperature are brought into thermal contact using both the Gibbs/Hertz and Boltzmann/Planck definitions of entropy. For the first two systems, I focus on the system at high energy, looking for quantum effects in the first three terms in the high temperature expansion of the phase volume and density of states. The density of states for each is then G(E) = /dE. From these, one calculates kTG = Φ/G and kTB = G/G′. Table I exhibits the results for temperature changes, and Table II exhibits results for entropy changes. These results are expressed in inverse powers of the system size (A, N, or f = dN/2) and dimensionless temperature, T̃ = kT/ε (ε having a slightly different meaning in each example). In each case, the Boltzmann/Planck change has a term similar to that seen in the classical system and which is an order larger than the corresponding Gibbs/Hertz change, and that term derives from the same nonextensivity discussed above.

TABLE I.

Temperature changes when two identical subsystems at the same temperature are brought into thermal contact. Higher order terms inversely proportional to the square or cube of the size of the system are neglected. Dimensionless temperatures (T̃) are T for the classical example, kT/() for harmonic oscillators, and kT/ε for ideal gas, with ε = h2/(2mL2).

SystemΔT̃GΔT̃B
Classical example T̃B2A1 
N identical harmonic oscillators 14N1+14NT̃G T̃B2N18NT̃B 
Ideal gas 116fT̃Gπ121+332f116πf T̃B2f516fT̃Bπ1/2 
SystemΔT̃GΔT̃B
Classical example T̃B2A1 
N identical harmonic oscillators 14N1+14NT̃G T̃B2N18NT̃B 
Ideal gas 116fT̃Gπ121+332f116πf T̃B2f516fT̃Bπ1/2 
TABLE II.

Leading terms in the high temperature expansion for entropy changes when two identical subsystems at the same temperature are brought into thermal contact. Higher order terms inversely proportional to the size of the system and inversely proportional to the temperature are neglected. Note that f̃=f1, and Ã=A1. Dimensionless temperatures defined in Table I.

SystemΔSG/kΔSB/k
Classical example lnπA+18A lnkTBΔ+lnπÃ38à
Identical harmonic oscillators 12T̃G+12lnπN lnkTBΔ+12lnπN 
+18N18NT̃G2 78N124T̃B21+3N 
Ideal gas 12lnπf+14πT̃G121+332f lnkTBΔ+12lnπf̃38f̃ 
+18f116πT̃G1+78f +7πT̃B121+213224f̃ 
10116πT̃B1+148101f̃ 
SystemΔSG/kΔSB/k
Classical example lnπA+18A lnkTBΔ+lnπÃ38à
Identical harmonic oscillators 12T̃G+12lnπN lnkTBΔ+12lnπN 
+18N18NT̃G2 78N124T̃B21+3N 
Ideal gas 12lnπf+14πT̃G121+332f lnkTBΔ+12lnπf̃38f̃ 
+18f116πT̃G1+78f +7πT̃B121+213224f̃ 
10116πT̃B1+148101f̃ 

In both quantum systems, nonzero terms appear in the results that are not present for the classical example. For the Gibbs/Hertz definition, these terms become negligible for high excitation and for large system size. However, corresponding nonzero terms for the Boltzmann/Planck definition are much larger than for the Gibbs/Hertz definition, and the leading term in each case grows with temperature, in contradiction to expected behavior.

1. Quantum example No. 3: N two-level atoms

Neither the Boltzmann/Planck nor the Gibbs/Hertz procedure agrees with our macroscopic expectations for changes in temperature for the two-level system as well as they do for the previous examples, but the Gibbs/Hertz procedure fails spectacularly for excitations with M > N/2.25 For M < N/2, both procedures behave about as in the previous examples. However, before examining the temperature changes when two identical subsystems are brought into thermal contact, it is useful to note that, for both definitions, entropy changes are always positive (provided Δ is chosen equal to ε), and the entropy change is smallest when the subsystems are about the same temperature. Although no analytic formula is known, this property has been verified numerically for hundreds of values of N and M.

As an illustrative example, consider two subsystems with equal temperature with N = 14 placed in thermal contact to form a system with N = 28. If each subsystem has a moderate excitation of M1 = M2 = 5, initially kTB/ε = 1.83 and kTG/ε = 1.36 (values are rounded) and the total system with M = 10 has kTB/ε = 1.77 and kTG/ε = 1.46. So TB has decreased about 4%, about as expected based on the behavior in the previous examples; however, TG has increased about 7%, not as good as in the previous examples but not too bad for such a small system. However, for subsystems with M1 = M2 = 9, initially kTB/ε = −1.83 and kTG/ε = 10.2 and the total system with M = 18 has kTB/ε = −1.77 and kTG/ε = 26.7. In this case, in the top half of the energy range, TB is negative and has moved in the opposite direction by symmetry, again by 4%, but now indicating higher excitation; TG has also increased to higher excitation; however, TG has more than doubled. Worse, if we combine M1 = 9 (kTG1/ε = 10.2) with M2 = 11 (kTG2/ε = 91), both subsystems rise to kTG/ε = 140 for M = 20. This behavior gets worse by orders of magnitude as M approaches N as has been noted by Miranda.25 Because the phase volume cannot decrease and, indeed, flattens out as M approaches N, TG is unbounded but can never indicate a population inversion.

2. Example No. 4

One final example combines two previous examples and consists of one subsystem with N1 two-level atoms with excitation energy and a second subsystem with N2 harmonic oscillators with equal excitation energy. The first three terms in the high temperature expansion of the phase volume of the combined system are given by

(57)

and the leading terms of the resulting Gibbs/Hertz temperature has the form

(58)

When combining one subsystem of N1 two-level atoms with M1 quanta with TG1 given by a suitably parameterized phase volume, Φ1, and a second subsystem of N2 harmonic oscillators with M2 quanta with TG2 given by Eq. (27) to form a system with MT = M1 + M2 quanta with TGT given by Eq. (58), the resultant change in temperature for sufficiently high values of M2 is

(59)

Thus, for low excitation, M1 < N1/2, where TG1 < TG2 for sufficiently high M2, and ΔTG2 will be negative, as expected in a macroscopic system. However, for higher excitation, M1 > N1/2, and where still TG1 < TG2, both ΔTG1 and ΔTG2 will be positive, contradicting the expectations for macroscopic systems. Even if a better parameterization of Φ1 might give a higher value of TG1, TG2 will eventually exceed the new value of TG1 for some sufficiently high value of M2, and the contradiction re-emerges. In contrast, the Boltzmann/Planck definition of temperature for this example displays no such contradiction. The expression for TBT for the combined system and ΔTB2 has the same form as Eqs. (58) and (59), respectively, with N2 replaced by N2 − 1, but ΔTB1 is always negative for M1 > N1/2 because G(M1) is decreasing, and a negative temperature must always be considered to be higher than any positive TBT calculated for the total system.

I have evaluated the performance of two competing definitions for microcanonical entropy for temperature and entropy changes as two subsystems are brought into thermal contact. For quantum systems with a density of states decreasing with energy, the Gibbs/Hertz definition of entropy and temperature cannot usefully describe the phenomenon of temperature inversion, and the results show temperature changes in contradiction to expected behavior for macroscopic systems. Even though example No. 4 has a total density of states that is increasing, the decreasing density of states of subsystem No. 1 is sufficient to produce the same problem for the Gibbs/Hertz temperature.

However, for systems with density of states increasing with energy, the Gibbs/Hertz definition of entropy is superior to the Boltzmann/Planck definition because both entropy and temperature changes when two subsystems are brought into thermal contact are qualitatively in agreement with the expected behavior for macroscopic systems. In classical systems, changes in Gibbs/Hertz temperature are exactly as expected for macroscopic systems, even for systems with a few particles. In quantum systems, the changes in Gibbs/Hertz temperature for identical subsystems are as expected for large systems and high levels of excitation. Similar quantum results for the Boltzmann/Planck temperature are worse, even increasing with excitation level where the Bohr correspondence principle predicts agreement with classical physics.30 

Entropy changes when two identical, classical subsystems at the same energy are brought into thermal contact in the Gibbs/Hertz definition are minimal, with deviations in quantum systems diminishing with high excitation. Corresponding entropy changes in the Boltzmann/Planck definition are larger and increase with excitation level, again in contradiction to the results expected for macroscopic systems. The contradiction associated with the unphysical entropy change term ln(kTB/Δ) might be ignored for quantum systems by arguing that G is a dimensionless number specifying the degeneracy of states at excitation level M, obviating the need for the Δ parameter in the harmonic oscillator and two-level atom examples. However, that argument is less useful in more complicated systems with incommensurate energies and only accidental degeneracies. Furthermore, the arbitrariness of the Δ parameter in the Boltzmann/Planck definition of entropy is itself a weakness suggesting a more fundamental problem.11 

I have shown that the Gibbs/Hertz (or volume) entropy gives results for thermodynamic quantities (namely, heat capacity, entropy, and both temperature and entropy changes when subsystems are placed in thermal equilibrium) that favorably agree with the results for the canonical ensemble for a wide class of classical systems and for two ideal quantum systems with increasing density of states and for which I presented new analytical results. This agreement holds even for systems of only a few particles and goes beyond just the proportionality to the number of particles in the system (extensivity) but to at least the first several terms in the high temperature expansions of both entropy and heat capacity—and even to the low temperature behavior of the heat capacity. For these examples, the Boltzmann/Planck (or surface) entropy gives results that are not extensive but instead proportional to a system with one fewer degrees of freedom. The agreement shown by the Gibbs/Hertz definition becomes important in nanoscale systems where N is a few hundred or less and where theoretical results are to be compared with numerical simulations or experiments.

For N two-level atoms, the Boltzmann/Planck definition of entropy gives better agreement, in some ways, with canonical heat capacity and entropy. Moreover, it allows the inclusion of negative temperature, which is not a feature of canonical equilibrium statistical mechanics, including an approach to zero entropy when all atoms are excited. For the Gibbs/Hertz entropy, the agreement was not as good as in the other examples examined. In any case, Vilar and Rubi31 argued that SG cannot usefully describe such a system past the maximum in the density of states. On the other hand, Campisi10 has shown that the Gibbs/Hertz entropy more successfully describes the magnetization of N spin-1/2 particles in a magnetic field (isomorphic to N two-level atoms) better than the Boltzmann/Planck entropy.

Most of the recent debate regarding entropy definitions has concerned the theoretical and philosophical foundations of statistical mechanics, particularly the admissibility of negative temperature.12,13,15 Setting aside negative temperature as irrelevant to equilibrium classical dynamics, this paper concerns practical aspects of molecular dynamics (MD) simulation, where the density of states increases with energy. Most modern MD calculations are done in the canonical ensemble (constant NVT) simulating classical many-body systems. For comparing MD simulations with fewer than several hundred particles with microcanonical (constant NVE) simulations, I claim that TG is the correct temperature for comparison.

Several authors have analyzed which definition of entropy is either more consistent with the laws of thermodynamics or more accurate or more useful for systems of a few particles.4–16 In weighing whether one should use TG or TB, it is important to remember that temperature is a derived property in the microcanonical ensemble, not a control variable like E, V, and N. In fact, Hilbert, et al.11 showed that neither TG(E) nor TB(E) is always invertible and thus uniquely determined. On the other hand and more importantly, changes in entropy are more fundamental, and entropy under both definitions always increases in approaching equilibrium—in agreement with the second law.

Boltachev and Schmeltzer32 showed than when a small system is in contact with a heat bath (canonical ensemble) at temperature T0, the temperature of the small system fluctuates with an average higher than T0 by an amount inversely proportional to the size of the system: T = T0(1 + k/CV) [Eq. (40) in Ref. 32] as result of increasingly large fluctuations for small systems. This relationship is almost the same as the relationship between TG and TB: TB = TG(1 + k/CVG) + O(N−2) equating E in Eqs. (12) and (13) and using Eq. (16), which I argued above was due to the difference between and average and most probable energies in the canonical distribution, that is, due to fluctuations in a small system. This difference in temperature between the bath and small system has been investigated by others previously.33,34 This corresponds well to the claim by Hilbert et al.11 that TB is the effective canonical temperature of a subsystem within a larger system. It is also provocative to note that many of the averages of physical quantities in a microcanonical system differ from those for an infinite system by amounts inversely proportional to the size of the system because the probability distribution differs from the Boltzmann distribution,18 which is only exact in the limit of large systems. Further research is needed in exploring similar questions concerning the thermodynamics of nanoscale systems.

Measurable thermodynamic quantities always depend on derivatives of the partition function. The derivative of the Heaviside step function, which describes the Gibbs/Hertz entropy, is the Dirac delta function, which concentrates all action on the energy shell, precisely as needed for the microcanonical ensemble. Thus, the volume or Gibbs/Hertz entropy provides thermodynamic quantities in the microcanonical ensemble that agree with corresponding quantities in the canonical ensemble for intensely studied systems with an increasing density of states like molecular dynamics.

Additional derivations, formulas, explanations, tables, and computer code are supplied as the supplementary material as described in the text.

A preliminary version of these results for the first two quantum systems was presented as a poster at the American Conference on Theoretical Chemistry at Telluride, Colorado, held on 20–24 July 2011. Preliminary results were also presented as posters at the Spring 2007 American Chemical Society National Meeting in Washington DC and the Fall 2009 American Chemical Society National Meeting in Chicago, IL.

The author has no conflicts of interest to declare.

The data that support the findings of this study are available within the article and its supplementary material.

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