Liouville's theorem (Hamiltonian) Guide, Meaning , Facts, Information and Description

In Hamiltonian mechanics, Liouville's theorem, named after the French mathematician Joseph Liouville, predicts how probability distributions evolve over time. Typically, ρ is the probability that a physical system will be found in an infinitesimal volume of phase space, τ standing for both position and momentum coordinates. In a system of N particles, τ is a convenient shorthand for the set of variables

In a system with Hamiltonian H and distribution function ρ, the theorem states that

where the curly braces denote a Poisson bracket.

An interesting corollary of this theorem is that time evolution preserves volumes in phase space. If a system is known to begin within a particular volume of phase space, then after an interval of time passes, the system will reside in a subspace of equal volume.

This theorem is of fundamental importance in statistical mechanics of classical systems, where it is also known (after J. Willard Gibbs) as the conservation of density in phase space. It can be proved (to the satisfaction of a physicist) by considering motion of a 'cloud' of points through phase space. The local density of points D is given by N/V where N is the number of points in the cloud, of volume V.

Constancy of N - in a deterministic system, phase-space trajectories can never cross. Were two trajectories to intersect, it would imply that some configuration of the system would have two possible futures. Assuming that the system is deterministic (given perfect knowledge of its condition), then such intersections are impossible. Thus systems neither enter nor leave V.

Constancy of V - this follows because any expansion of the volume along a co-ordinate q_i is exactly balanced by the shrinking of the volume in the direction of the conjugate momentum p_i. This balance follows from Hamilton's relations between p_i, q_i and their rates of change.

In more detail:-

Consider the time rate-of-change (taken with the flow, ie convective derivative) of a small phase-space volume

Now the rate of separation of a line element say is given by the difference in 'velocity' between its two ends ie

and similarly for other q & p.

Thus

and on substituting the Hamilton's relations for , this last bracket is seen to be zero.

The motion of phase-space points therefore resembles the flow of an incompressible fluid. Consequently, one can apply the analogue of the fluid flow mass-continuity equation to the "flow" of representative points through phase space, together with Hamilton's equations for the co-ordinates q_i and momenta p_i.

Canonical quantization yields a quantum-mechanical version of this theorem. This procedure, often used to devise quantum analogues of classical systems, involves describing a classical system using Hamiltonian mechanics. Classical variables are then re-interpreted as quantum operators, while Poisson brackets are replaced by commutators. In this case, the resulting equation is

where ρ is the density matrix.

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