In physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics. 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.
There are related mathematical results in symplectic topology and ergodic theory; systems obeying Liouville's theorem are examples of incompressible dynamical systems.
There are extensions of Liouville's theorem to stochastic systems.
Liouville equation
The Liouville equation describes the time evolution of the phase space distribution function. Although the equation is usually referred to as the "Liouville equation", Josiah Willard Gibbs was the first to recognize the importance of this equation as the fundamental equation of statistical mechanics. 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 a 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 :
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 proof of Liouville's theorem uses the n-dimensional divergence theorem. This proof is based on the fact that the evolution of obeys an 2n-dimensional version of the continuity equation:
That is, the 3-tuple is a conserved current. Notice that the difference between this and Liouville's equation are the terms
where is the Hamiltonian, and Hamilton's equations as well as conservation of the Hamiltonian along the flow 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, for instance, it shrinks in the corresponding direction so that the product remains constant.
Other formulations
Poisson bracket
The theorem above is often restated in terms of the Poisson bracket as
or, in terms of the linear Liouville operator or Liouvillian,
as
Ergodic theory
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. In Hamiltonian mechanics, the phase space is a smooth manifold that comes naturally equipped with a smooth measure (locally, this measure is the 6n-dimensional Lebesgue measure). The theorem says this smooth measure is invariant under the Hamiltonian flow. More generally, one can describe the necessary and sufficient condition under which a smooth measure is invariant under a flow. The Hamiltonian case then becomes a corollary.
Symplectic geometry
We can also formulate Liouville's Theorem in terms of symplectic geometry. For a given system, we can consider the phase space of a particular Hamiltonian as a manifold endowed with a symplectic 2-form
The volume form of our manifold is the top exterior power of the symplectic 2-form, and is just another representation of the measure on the phase space described above.
On our phase space symplectic manifold we can define a Hamiltonian vector field generated by a function as
Specifically, when the generating function is the Hamiltonian itself, , we get
where we utilized Hamilton's equations of motion and the definition of the chain rule.
In this formalism, Liouville's Theorem states that the Lie derivative of the volume form is zero along the flow generated by . That is, for a 2n-dimensional symplectic manifold,
In fact, the symplectic structure itself is preserved, not only its top exterior power. That is, Liouville's Theorem also gives