On the mechanical foundations of thermodynamics: The generalized Helmholtz theorem

An elegant but seldom appreciated effort to provide a mechanical model of equilibrium thermodynamics dates back to the Helmholtz theorem (HT). According to this theorem, the thermodynamic relations hold mechanically (without probabilistic assumptions) in the case of one-dimensional monocyclic systems. Thanks to a discrete picture of the phase space, Boltzmann was able to apply the HT to multi-dimensional ergodic systems, suggesting that the thermodynamic relations we observe in macroscopic systems at equilibrium are a direct consequence of the microscopic laws of dynamics alone. Here I review Boltzmann’s argument and show that, using the language of the modern ergodic theory, it can be safely re-expressed on a continuumphase space as a generalized Helmholtz theorem(GHT), which can be readily proved. Along the way the agreement between the Helmholtz–Boltzmann theory and that of P. Hertz (based on adiabatic invariance) is revealed. Both theories, in fact, lead to define the entropy as the logarithmof the phase-space volum e enclosed by the constant energy hypersurface (volume entropy). r 2005 Elsevier Ltd. All rights reserved.