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Liouville EquationThe Liouville equation is the most important equation of Statistical Mechanics. It describes the evolution of the probability distribution, , for a given microscopic system in the 6N-dim phase space, where N is the number of particles. Informal derivation We write down the total derivative with respect to time of the probability distribution, . -
\frac{d\rho }{dt}=\frac{\partial \rho }{\partial t}+\sum_{i=1}^{N}\left\rho }{\partial q_{i}}\dot{q}_{i}+\frac{\partial \rho }{\partial p_{i}}\dot{p}_{i}\right =0. (See Liouville's theorem (Hamiltonian) for further discussion of this step.) Then we replace the velocities and forces by the Hamiltonian equations where H is the Hamiltonian of the system and we arrive at -
\frac{\partial \rho }{\partial t}+{\hat{L}}\rho =0 where we have introduced the Liouvillian of the system -
{\hat{L}}=\sum_{i=1}^{N}\leftH}{\partial p_{i}} \frac{\partial }{\partial q_{i}}-\frac{\partial H}{\partial q_{i}}\frac{\partial }{\partial p_{i}}\right. Another way to write down the Liouville Equation is -
where the curly braces denote a Poisson bracket. Interpretation The Liouville Equation is a continuity equation for the probability distribution, . In other words no probability is created or destroyed, our degree of belief is conserved. See also: Liouville's theorem (Hamiltonian), Liouville equation (differential geometry), Sturm-Liouville equation.
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