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theorems:liouvilles_theorem

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theorems:liouvilles_theorem [2018/07/04 12:35]
jakobadmin [Concrete]
theorems:liouvilles_theorem [2018/07/04 15:20]
jakobadmin [Concrete]
Line 114: Line 114:
  
 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. $\frac{\partial \rho }{\partial t}= 0$.
 +
 +----
 +
 +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> ​
theorems/liouvilles_theorem.txt · Last modified: 2019/03/05 15:07 by 129.13.36.189