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

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

be the Hamiltonian function, where

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

is the kinetic energy and

:$varphi(mathbf\{q\};V)$

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

:$E\; =\; K\; +\; varphi$,

:$T\; =\; frac\{2\}\{s\}leftlangle\; K\; ight\; angle\; \_\{t\}$,

:$P\; =\; leftlangle\; -frac\{partial\; varphi\; \}\{partial\; V\}\; ight\; angle\; \_\{t\}$,

:$S(E,V)\; =\; log\; int\_\{H(mathbf\{p\},mathbf\{q\};V)\; leq\; E\}\; d^smathbf\{p\}d^s\; mathbf\{q\}.$

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.

References

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