Liouville's Theorem (Hamiltonian) - Physical Interpretation

Physical Interpretation

The expected total number of particles is the integral over phase space of the distribution:

A normalizing factor is conventionally included in the phase space measure but has here been omitted. In the simple case of a nonrelativistic particle moving in Euclidean space under a force field with coordinates and momenta, Liouville's theorem can be written

This is similar to the Vlasov equation, or the collisionless Boltzmann equation, in astrophysics. The latter, which has a 6-D phase space, is used to describe the evolution of a large number of collisionless particles moving under the influence of gravity and/or electromagnetic field.

In classical statistical mechanics, the number of particles is very large, (typically of order Avogadro's number, for a laboratory-scale system). Setting gives an equation for the stationary states of the system and can be used to find the density of microstates accessible in a given statistical ensemble. The stationary states equation is satisfied by equal to any function of the Hamiltonian : in particular, it is satisfied by the Maxwell-Boltzmann distribution, where is the temperature and the Boltzmann constant.

See also canonical ensemble and microcanonical ensemble.