Does the Boltzmann principle need a dynamical correction?

TL;DR

This paper investigates the need for dynamical corrections to the Boltzmann principle in small systems, showing that the standard entropy and temperature definitions are sufficient within the studied class of systems.

Contribution

The work clarifies the relationship between dynamical temperature definitions and thermodynamic entropy, demonstrating that no correction is needed for small systems of the type analyzed.

Findings

The temperature from the Fokker-Planck formalism matches an approximate equipartition temperature.

The exact temperature relates to the volume entropy, not the surface entropy.

Numerical results confirm the sufficiency of standard thermodynamic definitions for small systems.

Abstract

In an attempt to derive thermodynamics from classical mechanics, an approximate expression for the equilibrium temperature of a finite system has been derived [M. Bianucci, R. Mannella, B. J. West, and P. Grigolini, Phys. Rev. E 51, 3002 (1995)] which differs from the one that follows from the Boltzmann principle S = k log (Omega(E)) via the thermodynamic relation 1/T= dS/dE by additional terms of "dynamical" character, which are argued to correct and generalize the Boltzmann principle for small systems (here Omega(E) is the area of the constant-energy surface). In the present work, the underlying definition of temperature in the Fokker-Planck formalism of Bianucci et al. is investigated and shown to coincide with an approximate form of the equipartition temperature. Its exact form, however, is strictly related to the "volume" entropy S = k log (Phi(E)) via the thermodynamic relation above for systems of any number of degrees of freedom (Phi(E) is the phase space volume enclosed by the constant-energy surface). This observation explains and clarifies the numerical results of Bianucci et al. and shows that a dynamical correction for either the temperature or the entropy is unnecessary, at least within the class of systems considered by those authors. Explicit analytical and numerical results for a particle coupled to a small chain (N~10) of quartic oscillators are also provided to further illustrate these facts.