It has recurrently been proposed that the Boltzmann textbook definition of entropy S(E) = k ln Ω(E) in terms of the number of microstates Ω(E) with energy E should be replaced by the expression |$S_G (E) = k\ln \sum\nolimits_{E^\prime < E} {\Omega (E^\prime)} $| examined by Gibbs. Here, we show that SG either is equivalent to S in the macroscopic limit or becomes independent of the energy exponentially fast as the system size increases. The resulting exponential scaling makes the realistic use of SG unfeasible and leads in general to temperatures that are inconsistent with the notions of hot and cold.
I. INTRODUCTION
The well-established textbook definition of entropy S(E) = k ln Ω(E) in terms of the number of microstates Ω(E) with energy E was introduced by Boltzmann, reformulated by Planck in its present form, and subsequently generalized by Gibbs through the ensemble approach.1 Since then, this formula has been the cornerstone of statistical physics. In his seminal monograph,2 Gibbs also explored the properties of the continuous phase space counterpart of the alternative definition |$S_G (E) = k\ln \sum\nolimits_{E^\prime < E} {\Omega (E^\prime)} $|, mainly as a calculation device because of its equivalence to S in the macroscopic limit of the systems of interest at the time. This macroscopic equivalence has been exploited backwards recurrently over the years to propose that the standard definition of entropy S should be replaced by SG.3–6 Specially prominent has been the use of SG to negate the existence of negative temperatures,5,6 which might seem unappealing and counterintuitive but which are inevitable in systems with bounded energy spectra.7 In these types of systems, which range from nuclear spins8,9 to trapped ultracold atoms,10,11 the number of microstates Ω(E) has a maximum for finite energies and S and SG do not coincide with each other in the thermodynamic limit.
Here, we show that SG, in contrast to S, ceases to be a function of the energy for decreasing Ω(E) in the macroscopic limit and that it does so exponentially fast. Such exponential dependence makes meaningful use of SG unfeasible not only for macroscopic systems but also for small systems with over tens of elements and leads to temperatures that are inconsistent with the notions of hot and cold.
II. RESULTS
The fact that SG ceases to be a function of the energy for decreasing Ω(E) in the macroscopic limit follows straightforwardly from the maximum-term approach,12 which shows that the logarithm of a sum can be approximated by the logarithm of the maximum term. It leads to SG(E) = k ln Ω(E*) in the macroscopic limit, where E*(⩽E) is the energy that maximizes the number of microstates. This result can be worked out explicitly by considering the energy levels indexed by u from u = 0 to u = U so that Eu < Eu+1 and EU = E. The value of the sum |$\sigma (E) = \sum\nolimits_{u = 0}^U {\Omega (E_u)}$| is greater than the value of the largest term, Ω(E*), and smaller than the number of terms, 1 + U, times the value of the largest term. In mathematical terms, these conditions are expressed as Ω(E*) ⩽ σ(E) ⩽ (1 + U)Ω(E*). Taking logarithms and multiplying by k/N gives S(E*)/N ⩽ SG(E)/N ⩽ S(E*)/N + [k ln (1 + U)]/N. Since the number of energy levels grows subexponentially with the system size, these bounds imply that SG(E) = S(E*) for large N. Therefore, both definitions of entropy are the same in the macroscopic limit if the number of microstates increases continuously with the energy since E* = E for all E. However, when the number of microstates decreases with the energy, SG becomes constant for all E > E*.
This general result is illustrated explicitly by the prototypical ensemble of N two-level units with energies 0 and ɛ and total energy E = ɛU, where U is the number of units in the higher energy level.13 In this case, the number of microstates is given by |$\Omega (E) = \frac{{N!}}{{(N - E/\varepsilon)!(E/\varepsilon)!}}$|, which leads to |$S_G = k\ln \sum_{u = 0}^{E/\varepsilon } {\frac{{N!}}{{(N - u)!u!}}} $|. As the system size increases, SG looses its dependence on E for E > ɛN/2 (Fig. 1).
System-size scaling of SG. The normalized entropy SG of the ensemble of two-level units is shown as a function of the normalized energy for system sizes N = 3, 10, 30, and 100 (from lighter to darker colors).
System-size scaling of SG. The normalized entropy SG of the ensemble of two-level units is shown as a function of the normalized energy for system sizes N = 3, 10, 30, and 100 (from lighter to darker colors).
A fundamental question for the validity of SG as feasible thermodynamic quantity is how fast SG ceases to be a function of the energy for E > E*. Explicitly, the key question is whether SG can physically provide information on the thermal properties of the system for large but finite N.
The loss of thermal information can be quantified explicitly through the difference ΔSG = SG(EU) − SG(EU−1), which indicates how SG changes between two contiguous energy levels. This quantity is related to the associated temperature through TG = (EU − EU−1)/ΔSG, the discrete counterpart of the macroscopic expression TG = (∂SG/∂E)−1. Using Ω(E) = eS(E)/k in SG leads to
An upper bound that indicates explicitly how fast ΔSG goes to zero as N increases for E > E* can be obtained by making use of two inequalities. The first one, ln (1 + x) < x, leads to
which together with the second one, |$\sum\nolimits_{u = 0}^{U - 1} {e^{S(E_u)/k} }\break > e^{S(E^*)/k}$|, valid for E = EU > E* or equivalently EU−1 ⩾ E*, results in
Since the entropy S is an extensive quantity and S(E) < S(E*), this result explicitly shows that SG ceases to be a function of the energy exponentially fast for E > E* as the system size increases. As a result, SG cannot physically provide feasible information on the thermal properties of the system for E > E*.
The resulting temperature TG = (EU − EU−1)/ΔSG, in turn, is not consistent with an intensive quantity, as required by thermodynamics, but instead it grows exponentially with the system size indefinitely for all E = EU > E*. This exponential behavior makes impossible a meaningful association of TG with a physical quantity since doubling the system size, for instance, increases TG several orders of magnitude even for relatively small systems far below the macroscopic or mesoscopic limit.
The prototypical ensemble of N two-level units discussed previously clearly illustrates the implications of this pathological behavior. In this case, making use of |$\Omega (E)\break = \frac{{N!}}{{(N - E/\varepsilon)!(E/\varepsilon)!}}$| and |$\Omega (E^*) = \frac{{N!}}{{[(N/2)!]^2 }}$| in the formula of the entropy S, we obtain
It is possible to use Stirling's approximation in the previous expression to obtain an approximate bound but using n! ⩽ e nn + 1/2e−n in the numerator and |$\sqrt {2\pi } \;n^{n + 1/2} e^{ - n} \le n!$| in the denominator allows us to obtain the precise bound
where x = U/N. Consequently, the resulting temperature TG = ɛ/ΔSG grows exponentially with the system size as TG > υeαN with α = ln 2 + (1 − x)ln (1 − x) + x ln x and |$\upsilon = \varepsilon 2\pi \sqrt {(1 - x)x} /ke^2$| as illustrated in Fig. 2. In this case, doubling the system size from N = 100 to N = 200 already increases TG over 5 orders of magnitude for x = 0.75.
Exponential growth of TG with the system size N. The dimensionless temperature kTG/ɛ of the ensemble of two-level units is shown as a function of the system size N for x = 0.75 (continuous black line). The lower bound kυeαN/ɛ is shown as a dashed gray line.
Exponential growth of TG with the system size N. The dimensionless temperature kTG/ɛ of the ensemble of two-level units is shown as a function of the system size N for x = 0.75 (continuous black line). The lower bound kυeαN/ɛ is shown as a dashed gray line.
What happens then when systems of different sizes exchange energy with each other? Consider for instance a system A with N = 100 and U = 55 and a system B with N = 10 000 and U = 5000. When the two systems are allowed to exchange energy with each other, the energy will be redistributed so that the average energy of each element is the same and heat will flow from system A to system B (Fig. 3). The entropy SG, however, assigns a temperature TG = 17.33ɛ/k to system A, which is lower than the temperature TG = 62.67ɛ/k that it assigns to system B. Therefore, TG is not consistent with the notions of hot and cold and it would imply heat spontaneously flowing from low to high temperatures.
Inconsistency of TG with the notions of hot and cold. The averages over 300 realizations of the time evolution of U for the ensemble of two-level units are shown for systems A (N = 100) and B (N = 104) upon coupling. In each time step, the state of the system is updated by swapping the states of two randomly picked elements in systems A and B. In this case, heat spontaneously flows from system A to system B even though TG for system A is lower than TG for system B.
Inconsistency of TG with the notions of hot and cold. The averages over 300 realizations of the time evolution of U for the ensemble of two-level units are shown for systems A (N = 100) and B (N = 104) upon coupling. In each time step, the state of the system is updated by swapping the states of two randomly picked elements in systems A and B. In this case, heat spontaneously flows from system A to system B even though TG for system A is lower than TG for system B.
III. DISCUSSION
The definition of entropy is the main cornerstone of statistical physics. Our results have shown that the recurrent proposal |$S_G (E) = k\ln \sum\nolimits_{E^\prime < E} {\Omega (E^\prime)} $| as fundamental entropy cannot faithfully describe physical systems at any scale. Macroscopically, SG is either identical to S or unrealistically independent of the energy. For finite systems, from small to large, the resulting temperature TG is not consistent with the notions of hot and cold, implying that heat can spontaneously flow from low to high TG. Thus, our results strongly support the concept of absolute negative temperature as measured experimentally in nuclear spins8,9 and trapped ultracold atoms,10,11 which has so prominently been contested recently.6