On the Helmholtz-Boltzmann thermodynamics of mechanical systems

In an 1884 paper, Boltzmann showed that for a one-dimensional mechanical system with a convex potential energy phi that depends on a parameter V, it is possible to define a temperature T, pressure p, and entropy S that satisfy the Gibbs relation TdS = de + p dV, where e = kin.en+phi. In the paper we review the extension of the Boltzmann construction to general natural mechanical systems endowed with a fibration over the (possibly multidimensional) space of macroscopic parameters. Moreover, for certain discrete mechanical systems with non-convex potential energies, which are used as models for phase transitions in solids, we compare the thermodynamic pressure p = p(e,V) introduced above with the quasi-static, macroscopic, stress-strain relation.