Chemistry - 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, $ρ(Γ, t)$ .

$\frac{d\rho }{dt}=\frac{\partial \rho }{\partial t}+\sum_{i=1}^{N}\left[ \frac{\partial \rho }{\partial q_{i}}\dot{q}_{i}+\frac{\partial \rho }{\partial p_{i}}\dot{p}_{i}\right] =0.$

(See Liouville's theorem (Hamiltonian) for further discussion of this step.)

Then we replace the velocities $\dot{q}_{i}$ and forces $\dot{p}_{i}$ 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

${\hat{L}}=\sum_{i=1}^{N}\left[ \frac{\partial H}{\partial p_{i}} \frac{\partial }{\partial q_{i}}-\frac{\partial H}{\partial q_{i}}\frac{\partial }{\partial p_{i}}\right].$

Another way to write down the Liouville Equation is

$\frac{\partial}{\partial t}\rho=-\{\,\rho,H\,\}$

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.

Categories: Statistical mechanics | Equations