Helmholtz theorem (classical mechanics)

For other uses, see Helmholtz theorem (disambiguation).

The Helmholtz theorem of classical mechanics reads as follows:

Let

H(x,p;V)=K(p)+\varphi(x;V)

be the Hamiltonian of a one-dimensional system, where

K=\frac{p^2}{2m}

is the kinetic energy and

\varphi(x;V)

is a "U-shaped" potential energy profile which depends on a parameter V. Let \left\langle \cdot \right\rangle _{t} denote the time average. Let

E = K + \varphi,
T = 2\left\langle K\right\rangle _{t},
P = \left\langle -\frac{\partial \varphi }{\partial V}\right\rangle _{t},
S(E,V)=\log \oint \sqrt{2m\left( E-\varphi \left( x,V\right) \right) }\,dx.

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. This in turn allows to define the "thermodynamic state" of a one-dimensional mechanical system. In particular the temperature T is given by time average of the kinetic energy, and the entropy S by the logarithm of the action (i.e.\oint dx\sqrt{2m\left( E-\varphi \left( x,V\right) \right) }).
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.

References