Helmholtz theorem (classical mechanics) explained
The Helmholtz theorem of classical mechanics reads as follows:
Let be the Hamiltonian of a one-dimensional system, where is the kinetic energy and is a "U-shaped" potential energy profile which depends on a parameter
.Let
\left\langle ⋅ \right\ranglet
denote the time average. Let
T=2\left\langleK\right\ranglet,
P=\left\langle-
\right\ranglet,
S(E,V)=log\oint\sqrt{2m\left(E-\varphi\left(x,V\right)\right)}dx.
Then
Remarks
is given by time average of the kinetic energy, and the
entropy
by the logarithm of the
action (i.e.,
).
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 the generalized Helmholtz theorem.
Generalized version
The generalized Helmholtz theorem is the multi-dimensional generalization of the Helmholtz theorem, and reads as follows.
Let
be the
canonical coordinates of a
s-dimensional
Hamiltonian system, and let
H(p,q;V)=K(p)+\varphi(q;V)
be the Hamiltonian function, where
,
is the kinetic energy and
is the potential energy which depends on a parameter
.Let the hyper-surfaces of constant energy in the 2
s-dimensional phase space of the system be
metrically indecomposable and let
\left\langle ⋅ \right\ranglet
denote time average. Define the quantities
,
,
,
, as follows:
,
T=
\left\langleK\right\ranglet
,
P=\left\langle-
\right\ranglet
,
S(E,V)=log\intH(p,q;V)dspdsq.
Then:
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