Helmholtz theorem (classical mechanics) explained

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 is the kinetic energy and \varphi(x;V) is a "U-shaped" potential energy profile which depends on a parameter

V

.Let

\left\langle\right\ranglet

denote the time average. Let

E=K+\varphi,

T=2\left\langleK\right\ranglet,

P=\left\langle-

\partial\varphi
\partialV

\right\ranglet,

S(E,V)=log\oint\sqrt{2m\left(E-\varphi\left(x,V\right)\right)}dx.

Then dS = \frac.

Remarks

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).
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

p=(p1,p2,...,ps),

q=(q1,q2,...,qs),

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

s
K=\sum
i=1
2
p
i
2m
,

is the kinetic energy and

\varphi(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\right\ranglet

denote time average. Define the quantities

E

,

P

,

T

,

S

, as follows:

E=K+\varphi

,

T=

2
s

\left\langleK\right\ranglet

,

P=\left\langle-

\partial\varphi
\partialV

\right\ranglet

,

S(E,V)=log\intH(p,q;V)dspdsq.

Then:

dS=

dE+PdV
T

.

References