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--- theorems:liouvilles_theorem [2018/05/06 14:04]
ida [Intuitive]
+++ theorems:liouvilles_theorem [2019/03/05 15:07] (current)
129.13.36.189 [Concrete]
@@ Line 109: / Line 109: @@
 $$
-Now, if $\rho$ is constant $\frac{\partial \rho }{\partial t}= 0$, then the left-hand side is $0$ and we get:
+Now, if $\rho$ is constant $\frac{d \rho }{d t}= 0$, then the left-hand side is $0$ and we get:
 $$\frac{\partial \rho }{\partial t}= -\sum_{i}\left(\frac{\partial \rho}{\partial q_i}\,\dot{q_i}+\frac{\partial\rho}{\partial p_i}\,\dot p_i\right).$$
-This is a dynamical equation for the time-evolution of $\rho(t,\vec p,\vec q)$ that follows when the flow of the probability density $\rho(t,\vec p,\vec q)$ is incompressible, i.e. $\frac{\partial \rho }{\partial t}= 0$.
+This is a dynamical equation for the time-evolution of $\rho(t,\vec p,\vec q)$ that follows when the flow of the probability density $\rho(t,\vec p,\vec q)$ is incompressible, i.e. $\frac{d \rho }{d t}= 0$.
+----
+Take note that Liouville's theorem can be violated by any of the following:
+  *  sources or sinks of particles;
+  *  existence of collisional, dissipative, or other forces causing $\nabla_{\mathbf{p}}$ $\cdot$ $\mathbf{F}$ $\neq$ 0;
+  *  boundaries which lead to particle trapping or exclusion, so that only parts of a distribution can be mapped from one point to another;
+  *  spatial inhomogeneities that lead to velocity filtering (e.g., particle drift velocities that prevent particles with smaller velocities from reaching the location they would have reached had they not drifted);
+  *  temporal variability at source or elsewhere which leads to non-simultaneous observation of oppositely-directed trajectories;
+  *  etc. [_Paschmann and Daly_ 1998].
 <tabbox Abstract>

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