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Liouville equation
The Liouville equation is the most important equation of Statistical Mechanics. It describes the evolution of the probability distribution, ρ(Γ,t), 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, ρ(Γ,t).
![\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.](a/../upload/math/1a517d7ecc89ae02d86330e8c3efba48.png)
(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
where we have introduced the Liouvillian of the system
![{\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].](a/../upload/math/3cac1ab33461ad7193bc17d217212c89.png)
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, ρ(Γ,t). 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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