Helmholtz theorem (classical mechanics)

The Helmholtz theorem of classical mechanics reads as follows:

Let

${\displaystyle H(x,p;V)=K(p)+\varphi (x;V)}$

be the Hamiltonian of a one-dimensional system, where

${\displaystyle K={\frac {p^{2}}{2m}}}$

is the kinetic energy and

${\displaystyle \varphi (x;V)}$

is a "U-shaped" potential energy profile which depends on a parameter ${\displaystyle V}$. Let ${\displaystyle \left\langle \cdot \right\rangle _{t}}$ denote the time average. Let

${\displaystyle E=K+\varphi ,}$

${\displaystyle T=2\left\langle K\right\rangle _{t},}$

${\displaystyle P=\left\langle -{\frac {\partial \varphi }{\partial V}}\right\rangle _{t},}$

${\displaystyle S(E,V)=\log \oint {\sqrt {2m\left(E-\varphi \left(x,V\right)\right)}}\,dx.}$

Then

${\displaystyle dS={\frac {dE+PdV}{T}}.}$

Remarks

The thesis of this theorem of classical mechanics reads exactly as the heat theorem of thermodynamics. This fact shows that thermodynamic-like relations exist between certain mechanical quantities. This in turn allows to define the "thermodynamic state" of a one-dimensional mechanical system. In particular the temperature ${\displaystyle T}$ is given by time average of the kinetic energy, and the entropy ${\displaystyle S}$ by the logarithm of the action (i.e.${\displaystyle \oint dx{\sqrt {2m\left(E-\varphi \left(x,V\right)\right)}}}$).
The importance of this theorem has been recognized by Ludwig Boltzmann who saw how to apply it to macroscopic systems (i.e. multidimensional systems), in order to provide a mechanical foundation of equilibrium thermodynamics. This research activity was strictly related to his formulation of the ergodic hypothesis. A multidimensional version of the Helmholtz theorem, based on the ergodic theorem of George David Birkhoff is known as generalized Helmholtz theorem.

References

• Helmholtz, H., von (1884a). Principien der Statik monocyklischer Systeme. Borchardt-Crelle’s Journal für die reine und angewandte Mathematik, 97, 111–140 (also in Wiedemann G. (Ed.) (1895) Wissenschafltliche Abhandlungen. Vol. 3 (pp. 142–162, 179–202). Leipzig: Johann Ambrosious Barth).
• Helmholtz, H., von (1884b). Studien zur Statik monocyklischer Systeme. Sitzungsberichte der Kö niglich Preussischen Akademie der Wissenschaften zu Berlin, I, 159–177 (also in Wiedemann G. (Ed.) (1895) Wissenschafltliche Abhandlungen. Vol. 3 (pp. 163–178). Leipzig: Johann Ambrosious Barth).
• Boltzmann, L. (1884). Über die Eigenschaften monocyklischer und anderer damit verwandter Systeme.Crelles Journal, 98: 68–94 (also in Boltzmann, L. (1909). Wissenschaftliche Abhandlungen (Vol. 3,pp. 122–152), F. Hasenöhrl (Ed.). Leipzig. Reissued New York: Chelsea, 1969).
• Gallavotti, G. (1999). Statistical mechanics: A short treatise. Berlin: Springer.
• Campisi, M. (2005) On the mechanical foundations of thermodynamics: The generalized Helmholtz theorem Studies in History and Philosophy of Modern Physics 36: 275–290