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The Liouville equation is the most important equation of Statistical Mechanics.
It describes the evolution of the probability distribution, <math>\rho (\Gamma,t)<math> , 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, <math>\rho (\Gamma,t)<math>.
- <math>
\frac{d\rho }{dt}=\frac{\partial \rho }{\partial t}+\sum_{i=1}^{N}\left[
\frac{\partial \rho }{\partial q_{i}}\dot{q}_{i}+\frac{\partial \rho }{\partial p_{i}}\dot{p}_{i}\right] =0.
<math>
(See Liouville's theorem (Hamiltonian) for further discussion of this step.)
Then we replace the velocities <math>\dot{q}_{i}<math> and forces <math>\dot{p}_{i}<math> by the Hamiltonian equations where H is the Hamiltonian of the system and we arrive at
- <math>
\frac{\partial \rho }{\partial t}+{\hat{L}}\rho =0
<math>
where we have introduced the Liouvillian of the system
- <math>
{\hat{L}}=\sum_{i=1}^{N}\left[ \frac{\partial H}{\partial p_{i}}
\frac{\partial }{\partial q_{i}}-\frac{\partial H}{\partial q_{i}}\frac{\partial }{\partial p_{i}}\right].
<math>
Another way to write down the Liouville Equation is
- <math>\frac{\partial}{\partial t}\rho=-\{\,\rho ,H\,\}<math>
where the curly braces denote a Poisson bracket.
Interpretation
The Liouville Equation is a continuity equation for the probability distribution, <math>\rho (\Gamma,t)<math>.
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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