theorems:liouvilles_theorem
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theorems:liouvilles_theorem [2018/05/05 12:40] jakobadmin ↷ Page moved from basic_tools:phase_space:liouvilles_theorem to theorems:liouvilles_theorem |
theorems:liouvilles_theorem [2019/03/05 15:07] (current) 129.13.36.189 [Concrete] |
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| <tabbox Intuitive> | <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. | ||
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| + | <cite>The Lazy Universe by Coopersmith</cite></blockquote> | ||
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| <blockquote>Liouville's theorem can be thought of as conservation of information in classical mechanics. | <blockquote>Liouville's theorem can be thought of as conservation of information in classical mechanics. | ||
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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. $\frac{d \rho }{d t}= 0$. |
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| + | ---- | ||
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| + | Take note that Liouville's theorem can be violated by any of the following: | ||
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| + | * 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.1525516813.txt.gz · Last modified: 2018/05/05 10:40 (external edit)
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