Liouville's theorem (Hamiltonian)

It asserts that the phase-space distribution function is constant along the trajectories of the system—that is that the density of system points in the vicinity of a given system point traveling through phase-space is constant with time.

This time-independent density is in statistical mechanics known as the classical a priori probability.

Liouville's theorem ignores the possibility of chemical reactions, where the total number of particles may change over time, or where energy may be transferred to internal degrees of freedom.

There are extensions of Liouville's theorem to cover these various generalized settings, including stochastic systems.

[2] The Liouville equation describes the time evolution of the phase space distribution function.

: 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).

That is, viewing the motion through phase space as a 'fluid flow' of system points, the theorem that the convective derivative of the density,

[7] The theorem above is often restated in terms of the Poisson bracket as or, in terms of the linear Liouville operator or Liouvillian, as In ergodic theory and dynamical systems, motivated by the physical considerations given so far, there is a corresponding result also referred to as Liouville's theorem.

More generally, one can describe the necessary and sufficient condition under which a smooth measure is invariant under a flow[8].

On our phase space symplectic manifold we can define a Hamiltonian vector field generated by a function

, we get where we utilized Hamilton's equations of motion and the definition of the chain rule.

[9] In this formalism, Liouville's Theorem states that the Lie derivative of the volume form is zero along the flow generated by

Canonical quantization yields a quantum-mechanical version of this theorem, the von Neumann equation.

Classical variables are then re-interpreted as quantum operators, while Poisson brackets are replaced by commutators.

When applied to the expectation value of an observable, the corresponding equation is given by Ehrenfest's theorem, and takes the form where

Note the sign difference, which follows from the assumption that the operator is stationary and the state is time-dependent.

In the phase-space formulation of quantum mechanics, substituting the Moyal brackets for Poisson brackets in the phase-space analog of the von Neumann equation results in compressibility of the probability fluid, and thus violations of Liouville's theorem incompressibility.

This, then, leads to concomitant difficulties in defining meaningful quantum trajectories.

, the side lengths change as To find the new infinitesimal phase-space volume

The Hamiltonian for this system is given by By using Hamilton's equations with the above Hamiltonian we find that the term in parentheses above is identically zero, thus yielding From this we can find the infinitesimal volume of phase space: Thus we have ultimately found that the infinitesimal phase-space volume is unchanged, yielding demonstrating that Liouville's theorem holds for this system.

As a result, a region of phase space will simply rotate about the point

To see an example where Liouville's theorem does not apply, we can modify the equations of motion for the simple harmonic oscillator to account for the effects of friction or damping.

This time, we add the condition that each particle experiences a frictional force

Instead, as depicted in the animation in this section, a generic phase space volume will shrink as it evolves under these equations of motion.

To see this violation of Liouville's theorem explicitly, we can follow a very similar procedure to the undamped harmonic oscillator case, and we arrive again at Plugging in our modified Hamilton's equations, we find Calculating our new infinitesimal phase space volume, and keeping only first order in

As can be seen from the equation as time increases, we expect our phase-space volume to decrease to zero as friction affects the system.

As for how the phase-space volume evolves in time, we will still have the constant rotation as in the undamped case.

Again we can solve for the trajectories explicitly using Hamilton's equations, taking care to use the modified ones above.

As the system evolves the total phase-space volume will spiral in to the origin.