Английская Википедия:Helmholtz theorem (classical mechanics)
Шаблон:No footnotes The Helmholtz theorem of classical mechanics reads as follows:
Let <math display="block">H(x,p;V) = K(p) + \varphi(x;V)</math> be the Hamiltonian of a one-dimensional system, where <math display="block">K = \frac{p^2}{2m}</math> is the kinetic energy and <math display="block">\varphi(x;V)</math> is a "U-shaped" potential energy profile which depends on a parameter <math>V</math>. Let <math>\left\langle \cdot \right\rangle _{t}</math> denote the time average. Let
- <math>E = K + \varphi, </math>
- <math>T = 2\left\langle K\right\rangle _{t},</math>
- <math>P = \left\langle -\frac{\partial \varphi }{\partial V}\right\rangle _{t},</math>
- <math>S(E,V)=\log \oint \sqrt{2m\left( E-\varphi \left( x,V\right) \right) }\,dx.</math>
Then <math display="block">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. This in turn allows to define the "thermodynamic state" of a one-dimensional mechanical system. In particular the temperature <math>T</math> is given by time average of the kinetic energy, and the entropy <math>S</math> by the logarithm of the action (i.e., <math display="inline">\oint dx \sqrt{2m\left( E - \varphi \left( x, V\right) \right) }</math>).
The importance of this theorem has been recognized by Ludwig Boltzmann who saw how to apply it to macroscopic systems (i.e. multidimensional systems), in order to provide a mechanical foundation of equilibrium thermodynamics. This research activity was strictly related to his formulation of the ergodic hypothesis.
A multidimensional version of the Helmholtz theorem, based on the ergodic theorem of George David Birkhoff is known as generalized Helmholtz theorem.
Generalized version
The generalized Helmholtz theorem is the multi-dimensional generalization of the Helmholtz theorem, and reads as follows.
Let
- <math>\mathbf{p}=(p_1,p_2,...,p_s),</math>
- <math>\mathbf{q}=(q_1,q_2,...,q_s),</math>
be the canonical coordinates of a s-dimensional Hamiltonian system, and let
- <math> H(\mathbf{p},\mathbf{q};V)=K(\mathbf{p})+\varphi(\mathbf{q};V) </math>
be the Hamiltonian function, where
- <math>K=\sum_{i=1}^{s}\frac{p_i^2}{2m}</math>,
is the kinetic energy and
- <math>\varphi(\mathbf{q};V)</math>
is the potential energy which depends on a parameter <math>V</math>. Let the hyper-surfaces of constant energy in the 2s-dimensional phase space of the system be metrically indecomposable and let <math>\left\langle \cdot \right\rangle_t </math> denote time average. Define the quantities <math>E</math>, <math>P</math>, <math>T</math>, <math>S</math>, as follows:
- <math>E = K + \varphi </math>,
- <math>T = \frac{2}{s}\left\langle K\right\rangle _{t}</math>,
- <math>P = \left\langle -\frac{\partial \varphi }{\partial V}\right\rangle _{t}</math>,
- <math>S(E,V) = \log \int_{H(\mathbf{p},\mathbf{q};V) \leq E} d^s\mathbf{p}d^s \mathbf{q}. </math>
Then:
- <math>dS = \frac{dE+PdV}{T}.</math>
References
- 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).
- 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
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