Liouville's Theorem (Hamiltonian)

Liouville Equations

These Liouville equations describe the time evolution of the phase space distribution function. Although the equation is usually referred to as the "Liouville equation", this equation was in fact first published by Gibbs in 1902. It is referred to as the Liouville equation because its derivation for non-canonical systems utilises an identity first derived by Liouville in 1838. Consider an Hamiltonian dynamical system with canonical coordinates and conjugate momenta, where . Then the phase space distribution determines the probability that the system will be found in the infinitesimal phase space volume . The Liouville equation governs the evolution of in time :

$\frac{d\rho}{dt}= \frac{\partial\rho}{\partial t} +\sum_{i=1}^n\left(\frac{\partial\rho}{\partial q^i}\dot{q}^i +\frac{\partial\rho}{\partial p_i}\dot{p}_i\right)=0.$

Time derivatives are denoted by dots, and are evaluated according to Hamilton's equations for the system. This equation demonstrates the conservation of density in phase space (which was Gibbs's name for the theorem). Liouville's theorem states that

The distribution function is constant along any trajectory in phase space.

A simple proof of the theorem is to observe that the evolution of is defined by the continuity equation:

That is, the tuplet is a conserved current. Notice that the difference between this and Liouville's equation are the terms

$\rho\sum_{i=1}^n\left( \frac{\partial\dot{q}^i}{\partial q^i} +\frac{\partial\dot{p}_i}{\partial p_i}\right) =\rho\sum_{i=1}^n\left( \frac{\partial^2 H}{\partial q^i\,\partial p_i} -\frac{\partial^2 H}{\partial p_i \partial q^i}\right)=0,$

where is the Hamiltonian, and Hamilton's equations have been used. That is, viewing the motion through phase space as a 'fluid flow' of system points, the theorem that the convective derivative of the density, is zero follows from the equation of continuity by noting that the 'velocity field' in phase space has zero divergence (which follows from Hamilton's relations).

Another illustration is to consider the trajectory of a cloud of points through phase space. It is straightforward to show that as the cloud stretches in one coordinate – say – it shrinks in the corresponding direction so that the product remains constant.

Equivalently, the existence of a conserved current implies, via Noether's theorem, the existence of a symmetry. The symmetry is invariant under time translations, and the generator (or Noether charge) of the symmetry is the Hamiltonian.

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Liouville's Theorem (Hamiltonian) - Liouville Equations