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