Generalized Helmholtz theorem

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The generalized Helmholtz theorem on skates 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 2s-dimensional phase space of the system be metrically indecomposable and let \left\langle \cdot \right\rangle_t denote time average. Define the quantities E, P, T, S, as follows:

E = K + \varphi ,
T = \frac{2}{s}\left\langle K\right\rangle _{t},
P = \left\langle -\frac{\partial \varphi }{\partial V}\right\rangle _{t},
S(E,V) = \log \int_{H(\mathbf{p},\mathbf{q};V) \leq E} d^s\mathbf{p}d^s \mathbf{q}.

Then:

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

Remarks[edit]

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[edit]

Further reading[edit]

  • 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