Liouville's theorem (Hamiltonian)

Liouville's theorem (also sometimes known as the Liouville equation) is a key theorem in statistical mechanics of classical systems, studied in the physical sciences. It is also important in the mathematical study of Hamiltonian mechanics and symplectic topology. The two disclipines make rather different expressions of the key result, and so both are given here. The quantum mechanical analogue of the key result is also given.

History

Liouville's theorem is named after the French mathematician Joseph Liouville (1809-1882). It is one of two such-named theorems, the other being in the field of complex analysis

Physical sciences expression

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 demonstrates that the density D of systems in the neighbourhood of some given system in phase space remains constant with time (ie the 'convective derivative ' is zero). That is

${\frac {dD}{dt}}={\frac {\partial D}{\partial t}}+\sum _{i=1}^{N}\left[{\frac {\partial D}{\partial q_{i}}}{\dot {q}}_{i}+{\frac {\partial D}{\partial p_{i}}}{\dot {p}}_{i}\right]=0.$

(This expression can be rendered more compact by using Hamillton's relations and noting that the resulting 2nd and 3rd terms are then a Poisson bracket - see below)

If the ensemble is statistically stationary (partial time derivative of D zero) then this equation is satisfied by D=D(H) where H is the Hamiltonian function of the system.

Informal demonstration

The result can be demonstrated (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, made up of a 'cloud' of points:-

$\Delta V=\Delta q_{1}\Delta p_{1}\dots \Delta q_{i}\Delta p_{i}\dots$

Now the rate of separation of a line element (made up of system points) $\Delta q_{i}$ say is given by the difference in 'velocity' between its two ends ie

${\frac {d}{dt}}{\Delta q_{i}}=({\frac {\partial }{\partial q_{i}}}{{\dot {q}}_{i}}).{\Delta q_{i}}$

and similarly for other q & p.

Thus

${\frac {d}{dt}}({\Delta p_{i}}{\Delta q_{i}})=\Delta p_{i}\Delta q_{i}.({\frac {\partial }{\partial p_{i}}}{{\dot {p}}_{i}}+{\frac {\partial }{\partial q_{i}}}{{\dot {q}}_{i}})$

and on substituting the Hamilton's relations for ${{\dot {p}}_{i}},{{\dot {q}}_{i}}$, this last bracket is seen to be zero.

The result can be derived with more rigour by noting that the motion of a system point in time is given by a 'contact transformation' of the canonical co-ordinates in phase space.

The motion of phase-space points therefore resembles the flow of an incompressible fluid.

Mathematical expression

Typically, in an appropriately normalised system, ρ is the probability that a physical system will be found in an infinitesimal volume $d\tau$ 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

$\{\,{\vec {x}}_{1},{\vec {x}}_{2},\ldots ,{\vec {x}}_{N},;{\vec {p}}_{1},{\vec {p}}_{2},\ldots ,{\vec {p}}_{N}\,\}.$

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

${\frac {\partial }{\partial t}}\rho (\tau ,t)=-\{\,\rho (\tau ,t),H\,\}$

where the curly braces denote a Poisson bracket.

A closely similar expression can be written by using the Liouvillian operator

${\hat {L}}=\sum _{i=1}^{N}\left[{\frac {\partial H}{\partial p_{i}}}{\frac {\partial }{\partial q_{i}}}-{\frac {\partial H}{\partial q_{i}}}{\frac {\partial }{\partial p_{i}}}\right].$

so that

${\frac {\partial \rho }{\partial t}}+{\hat {L}}\rho =0$

Quantum Mechanical expression

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

${\frac {\partial }{\partial t}}\rho =-{\frac {i}{\hbar }}[\rho ,H]$

where ρ is the density matrix.