Comment on “Which is greater: $eπ$ or π^e? An unorthodox physical solution to a classic puzzle” [Am. J. Phys. 92(5), 397–398 (2024)]

In a recent Note,¹ Vallejo and Bove provide a physical argument based nominally on the second law of thermodynamics as a way of resolving the mathematical question appearing in the title. A remarkable aspect of their argument is that it does not depend on the numerical value of π, because $ex ⩾ xe$ for all positive x, with equality occurring only when x = e. Moreover, their argument does not depend on the validity of the second law but is rather a limited proof of it for this particular case.

Their argument is based on a scenario in which an incompressible solid body A with constant heat capacity C at initial temperature $T1=π$ in the units of some absolute temperature scale is placed in contact with an ideal reservoir B at initial temperature $T2=e$ in the same units. The system evolves irreversibly to equilibrium at the temperature of the reservoir. In these units $ΔSA=C(1−ln π)$ and $ΔSB=C(π/e−1)$⁠, leading to an overall entropy change $ΔS=C(π/e−ln π)$⁠. Invoking the second law $ΔS>0$ for an irreversible process, the authors obtain $π>ln πe$ and thus $eπ ⩾ πe$⁠.

The argument appears to depend on the value of e through $ln e=1$ as one of the steps in obtaining $ΔSA$ (the more general case is discussed below) but does not make use of the numerical value of π, for example by determining the direction of heat flow through its relation with e. Thus, the argument and consequently the result must be independent of the value of π provided it is real and positive. This comment investigates this situation more fully.

Following on from the arguments of Vallejo and Bove, Eq. (2), when coupled with the second law, might be taken as a “proof” that $1−1/x$ is a lower bound for $ln (x)$⁠, as is well known. However, the inequality follows straightforwardly from the analytical properties of the function, and the second law need never be invoked. Rather, the inequality acts as a demonstration that the second law is valid for this model system.

Interest in thermodynamic “proofs” of mathematical inequalities appears to have begun with Landsberg's short, citation-free article applying the first and second laws to n identical heat reservoirs initially at different temperatures to affirm the inequality between the arithmetic and geometric means.⁵ As noted in a brief historical article by Deakin,⁶ however, the argument dates back to P. G. Tait in 1868,⁷ and was used as an exercise in Sommerfeld's book on thermodynamics and statistical mechanics;⁸ by 1980, Landsberg had become aware of Sommerfeld's work.⁹ A collection by Tykodi of similar inequalities supported by model systems was published in this journal in 1996,¹⁰ and a demonstration by Plastino et al. of thermodynamic support for Jensen's inequality, of which the inequality of the arithmetic and geometric means is a consequence, was published the following year.¹¹ Over time, the framing of these examples has shifted, noting that they are not strictly “proofs”⁶ and are more correctly characterized as demonstrating mathematical inequalities.¹⁰ 

A recent article in this journal by Johal¹² returns to the source from which Tait built his original observation, namely a paper by William Thomson (Lord Kelvin) on the extraction of all available work from an unequally heated space by means of a heat engine.¹³ Tait updated and discretized Thomson's result to determine that for a set of identical masses, the final temperature after such a process is the geometric mean of their initial temperatures, while the temperature achieved by thermal equilibration is the higher arithmetic mean.⁷ Limiting consideration to two masses for simplicity, Johal notes that a more edifying interpretation of the thermalization process can be obtained by dividing it into two steps: a reversible one in which all available work is extracted until the bodies are at the same temperature (the geometric mean of their original temperatures), and a second one in which the same quantity of energy is returned as heat and the bodies are warmed to the arithmetic mean of their original temperatures. The input of heat in this second step is a useful pedagogical illustration that the final entropy of the system must be higher in accordance with the second law. Similar arguments were made previously by Pyun¹⁴ and Leff.¹⁵ All such arguments depend on the positivity of the heat capacity of material bodies, which may not be universally valid.¹⁶ An additional point made by Leff and worth reiterating here is that although the second law in the form of entropy increase is demonstrated rather than assumed by Vallejo and Bove, it is a central requirement of their example that temperature equilibration—one of the observed macroscopic phenomena leading to the invention of the entropy concept—takes place.

The author states that there is no conflict of interest to be disclosed.