On the mechanical foundations of thermodynamics: The generalized Helmholtz theorem

Citations of this article
Mendeley users who have this article in their library.
Get full text


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 continuum phase 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 logarithm of the phase-space volume enclosed by the constant energy hyper-surface (volume entropy). © 2005 Elsevier Ltd. All rights reserved.




Campisi, M. (2005). On the mechanical foundations of thermodynamics: The generalized Helmholtz theorem. Studies in History and Philosophy of Science Part B - Studies in History and Philosophy of Modern Physics, 36(2), 275–290. https://doi.org/10.1016/j.shpsb.2005.01.001

Register to see more suggestions

Mendeley helps you to discover research relevant for your work.

Already have an account?

Save time finding and organizing research with Mendeley

Sign up for free