Physics:Generalized Helmholtz theorem

The generalized Helmholtz theorem is the multi-dimensional generalization of the Helmholtz theorem which is valid only in one dimension. The generalized Helmholtz theorem reads as follows. Let

[math]\displaystyle{ \mathbf{p}=(p_1,p_2,...,p_s), }[/math]

[math]\displaystyle{ \mathbf{q}=(q_1,q_2,...,q_s), }[/math]

be the canonical coordinates of a s-dimensional Hamiltonian system, and let

[math]\displaystyle{ H(\mathbf{p},\mathbf{q};V)=K(\mathbf{p})+\varphi(\mathbf{q};V) }[/math]

be the Hamiltonian function, where

[math]\displaystyle{ K=\sum_{i=1}^{s}\frac{p_i^2}{2m} }[/math],

is the kinetic energy and

[math]\displaystyle{ \varphi(\mathbf{q};V) }[/math]

is the potential energy which depends on a parameter [math]\displaystyle{ V }[/math]. Let the hyper-surfaces of constant energy in the 2s-dimensional phase space of the system be metrically indecomposable and let [math]\displaystyle{ \left\langle \cdot \right\rangle_t }[/math] denote time average. Define the quantities [math]\displaystyle{ E }[/math], [math]\displaystyle{ P }[/math], [math]\displaystyle{ T }[/math], [math]\displaystyle{ S }[/math], as follows:

[math]\displaystyle{ E = K + \varphi }[/math],

[math]\displaystyle{ T = \frac{2}{s}\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 \int_{H(\mathbf{p},\mathbf{q};V) \leq E} d^s\mathbf{p}d^s \mathbf{q}. }[/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 in multidimensional ergodic systems. This in turn allows to define the "thermodynamic state" of a multi-dimensional ergodic mechanical system, without the requirement that the system be composed of a large number of degrees of freedom. In particular the temperature [math]\displaystyle{ T }[/math] is given by twice the time average of the kinetic energy per degree of freedom, and the entropy [math]\displaystyle{ S }[/math] by the logarithm of the phase space volume enclosed by the constant energy surface (i.e. the so-called volume entropy).

Further reading

• 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).
• Khinchin, A. I. (1949). Mathematical foundations of statistical mechanics. New York: Dover.
• 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