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--- theorems:liouvilles_theorem [2018/04/08 10:47]
jakobadmin [Intuitive]
+++ theorems:liouvilles_theorem [2019/03/05 15:07] (current)
129.13.36.189 [Concrete]
@@ Line 1: / Line 1: @@
 ====== Liouville's theorem ======
+//also known as Liouville equation//
 <tabbox Intuitive>
+<blockquote>It turns out that the abstract phase fluid has the same properties as a real fluid that is incompressible. An incompressible real fluid is one in which any volume-element of fluid (any sample of neighbouring ‘particles’) cannot be compressed, that is, it keeps the same volume. Likewise, for the phase fluid we may examine any small 2n-dimensional volume-element within the fluid, and watch this volume as it is ‘carried along’ by the flow. Although the shape of the volume-element may become distorted yet its total volume always remains unchanged.
+<cite>The Lazy Universe by Coopersmith</cite></blockquote>
 <blockquote>Liouville's theorem can be thought of as conservation of information in classical mechanics.
@@ Line 95: / Line 100: @@
 <tabbox Concrete>
+Liouville's theorem tells us that the flow of the probability density $\rho(t,\vec p,\vec q)$ is incompressible. The probability density is a non-negative real quantity that tells us the probability that we'll find a particle with momentum $\vec p$ at the position $\vec q$.
 In general, for a function $\rho(t,\vec p,\vec q)$ we have
@@ Line 101: / 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).$$
+This is a dynamical equation for the time-evolution of $\rho(t,\vec p,\vec q)$ that follows when the flow of the probability density $\rho(t,\vec p,\vec q)$ is incompressible, i.e. $\frac{d \rho }{d t}= 0$.
+----
-$$\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)$$
+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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