## Sunday, February 12, 2012

### Conservation of Shannon information in classical mechanics

There are many places where to find the proof of the conservation of the quantum Shannon information under unitary evolution. However, I just could not find the equivalent proof that everybody talks in classical mechanics, so here it goes.

$S = - \int dxdp\, \rho\, \ln \rho$

$\dot{S} = - \int dxdp\, \frac{\partial \rho}{\partial t}\, \ln \rho + \frac{\partial \rho}{\partial t}$

The Lioville equation can be expressed in terms of a self-adjoint operator as observed by Koopman and von Neumann, such that

$i \frac{\partial \rho}{\partial t} = \mathcal{L} \rho$

Given that the Liouvillian follows the Leibniz rule, which is true for conservative systems

$\mathcal{L} f(\rho) = f' \mathcal{L}\rho$

it is now easy to prove that

$\dot{S} = 0$

This fact is in complete contrast with respect to the thermodynamical entropy, which can be fundamentally irreversible.