Question

Derivation of the Liouville theorem in statistical mechanics

I was trying to follow the derivation of the Liouville theorem here: http://en.wikiversity.org/wiki/Topic:Advanced_Classical_Mechanics/Liouville%27s_theorem
They say that "Taking the limit that the length, L, of the hypercube vanishes, we have the 2N dimensional continuity equation". Why is that true?

Answer 1

I mean this part of the derivation: \[\frac{ dQ }{ dt } = -\int\limits_{U}\nabla \cdot J dV_{2N} =\oint_{\partial U}J \cdot n dS_{2N-1}\]

Answer 2

From where it follows that\[0=\frac{\partial \rho }{ \partial t }+ \nabla \cdot J\]

Answer 3

Without getting into the nitty gritty of the derivation of which I am a long way removed, It looks like the Divergence Theorem which leads to the continuity equation as in E/M and Hydrodynamics if\[\int\limits_{S}^{}J.d \sigma=-\int\limits_{V}^{} \frac{ \partial \rho }{ \partial t }d \tau \]
using the Divergence Theorem you get the continuity equation.