Helmholtz theorem (classical mechanics)

The Helmholtz theorem of classical mechanics reads as follows:

Let [math]\displaystyle{ H(x,p;V) = K(p) + \varphi(x;V) }[/math] be the Hamiltonian of a one-dimensional system, where [math]\displaystyle{ K = \frac{p^2}{2m} }[/math] is the kinetic energy and [math]\displaystyle{ \varphi(x;V) }[/math] is a "U-shaped" potential energy profile which depends on a parameter [math]\displaystyle{ V }[/math]. Let [math]\displaystyle{ \left\langle \cdot \right\rangle _{t} }[/math] denote the time average. Let

• [math]\displaystyle{ E = K + \varphi, }[/math]
• [math]\displaystyle{ T = 2\left\langle K\right\rangle _{t}, }[/math]
• [math]\displaystyle{ P = \left\langle -\frac{\partial \varphi }{\partial V}\right\rangle _{t}, }[/math]
• [math]\displaystyle{ S(E,V)=\log \oint \sqrt{2m\left( E-\varphi \left( x,V\right) \right) }\,dx. }[/math]

Then [math]\displaystyle{ dS = \frac{dE+PdV}{T}. }[/math]

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 [math]\displaystyle{ T }[/math] is given by time average of the kinetic energy, and the entropy [math]\displaystyle{ S }[/math] by the logarithm of the action (i.e., [math]\displaystyle{ \oint dx \sqrt{2m\left( E - \varphi \left( x, V\right) \right) } }[/math]).
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

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