Renan Cabrera L. PhD: Conservation of Shannon information in classical mechanics

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.

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Renan Cabrera L. PhD

Sunday, February 12, 2012

Conservation of Shannon information in classical mechanics

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