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theorems:liouvilles_theorem [2018/05/06 14:04] ida [Intuitive] |
theorems:liouvilles_theorem [2018/07/04 15:20] jakobadmin [Concrete] |
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- | 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: |
$$\frac{\partial \rho }{\partial t}= -\sum_{i}\left(\frac{\partial \rho}{\partial q_i}\,\dot{q_i}+\frac{\partial\rho}{\partial p_i}\,\dot p_i\right).$$ | $$\frac{\partial \rho }{\partial t}= -\sum_{i}\left(\frac{\partial \rho}{\partial q_i}\,\dot{q_i}+\frac{\partial\rho}{\partial p_i}\,\dot p_i\right).$$ | ||
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. $\frac{\partial \rho }{\partial t}= 0$. | ||
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+ | ---- | ||
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+ | 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> | <tabbox Abstract> |