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

$\mathbf{𝐩}=(p_1,p_2,...,p_s),$

$\mathbf{𝐪}=(q_1,q_2,...,q_s),$

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

$H(\mathbf{𝐩},\mathbf{𝐪};V)=K(\mathbf{𝐩})+\phi (\mathbf{𝐪};V)$

be the Hamiltonian function, where

$K=\sum _{i=1}^s\frac{p_i^2}{2m}$,

is the kinetic energy and

$\phi (\mathbf{𝐪};V)$

is the potential energy which depends on a parameter $V$. Let the hyper-surfaces of constant energy in the 2s-dimensional phase space of the system be metrically indecomposable and let ${⟨\cdot ⟩}_t$ denote time average. Define the quantities $E$, $P$, $T$, $S$, as follows:

$E=K+\phi$,

$T=\frac{2}{s}{⟨K⟩}_t$,

$P={⟨-\frac{\partial \phi }{\partial V}⟩}_t$,

$S(E,V)=\log \underset{H(\mathbf{𝐩},\mathbf{𝐪};V)\le E}{\int }{d}^s\mathbf{𝐩}{d}^s\mathbf{𝐪}.$

Then:

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

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 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 $T$ is given by twice the time average of the kinetic energy per degree of freedom, and the entropy $S$ 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