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Suppose we have some very imprecise knowledge of classical particle's coordinates and momentum: what we can only tell is the probability density to find it in some point of phase space. This is (almost?) all what is usually known by quantum state function.

For quantum particle, there's an equation, which governs such initial state — it's Schrödinger equation.

Is there any known equation, which would similarly govern evolution of classical particle in some external potential, given initial probability density in phase space?

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up vote 6 down vote accepted

It seems to me that you're looking for the Boltzmann transport equation:

$$ \frac{\partial f}{\partial t}+\frac{\mathbf p}{m}\cdot\nabla f+\mathbf F\cdot\frac{\partial f}{\partial\mathbf p}=Q+\left(\frac{df}{dt}\right)_{\rm coll} $$ with $f$ the distribution in phase-space, $\mathbf p$ the particle momentum, $Q$ some source term, and the RHS an interaction term based on collisions. Here we can use $\mathbf F=-\nabla\Phi$ for some potential $\Phi$.

There is also a Fokker-Planck equation that uses the Lorentz force for $\mathbf F$ but is otherwise the same. The Fokker-Planck equation without the collision term is called the Vlasov equation.

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Great. Even more interesting appears to be Liouville equation as an explicitly many-paticle version (without collision term though, but I didn't need it). – Ruslan Apr 14 '14 at 15:32

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