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. 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 o
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