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Concept
Liouville's theorem (Hamiltonian)
Summary
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.
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.
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