Differences

This shows you the differences between two versions of the page.

--- theorems:liouvilles_theorem [2017/10/22 13:27]
jakobadmin ↷ Page moved from basic_notions:phase_space:liouvilles_theorem to basic_tools:phase_space:liouvilles_theorem
+++ 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 Why is it interesting?>
+<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.
+Suppose you have a bowl, perhaps of some slightly-wonky shape, and a marble that can roll around in the bowl. You put the marble down somewhere and give it a push. (We'll call the this the initial state.) The marble does its thing. Using physics, you can predict where it will be and how fast it will be going 10 seconds from now. (We'll call this the final state.)
+However, you might not know the initial state perfectly. Instead there is some range of possible initial states. The size of the range of initial states represents your uncertainty.
+Due to the uncertainty, you can't calculate the exact final state. Instead, there is uncertainty about the final state. Liouville's theorem says that you have the same amount of uncertainty about the initial and final states.
+The size of the uncertainty is a measure of how much information you have, so Liouville's theorem says that you neither gain nor lose information, i.e. information is conserved. (Specifically, the information you have is the negative of the logarithm of the uncertainty.)
+<cite>http://qr.ae/TU1ODq</cite></blockquote>
 <blockquote>
@@ Line 81: / Line 95: @@
 <cite>page 174ff in "The Emperors new Mind" by R. Penrose</cite></blockquote>
-<tabbox Layman>
-<note tip>
+<blockquote>It turns out that the abstract [[basic_tools:phase_space|phase fluid]] has the same proper-
-Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party.
+ties 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>
-</note>
-<tabbox Student>
+<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 $\vec q$.
-<note tip>
-In this section things should be explained by analogy and with pictures and, if necessary, some formulas.
+In general, for a function $\rho(t,\vec p,\vec q)$ we have
-</note>
+$$
-<tabbox Researcher>
+\frac{\mathrm d}{\mathrm dt} \rho(t, \vec p(t), \vec q(t)) =\frac{\partial \rho }{\partial t}+ \sum_{i}\left(\frac{\partial \rho}{\partial q_i}\,\dot{q_i}+\frac{\partial\rho}{\partial p_i}\,\dot p_i\right) .
+$$
+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. $\frac{d \rho }{d t}= 0$.
+----
+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>
 <blockquote>The intuition for the Lagrangian principle comes specific applications of Newton's laws, especially reversible systems with constraints, like nonspherical particles rolling along complicated surfaces. Newton's formulation of Newton's laws was not the end of the story, because there was more structure in the solutions of these types of problems than that which Newton made obvious.
@@ Line 117: / Line 150: @@
 <cite>https://physics.stackexchange.com/a/15958/37286</cite></blockquote>
---> Common Question 1#
+<tabbox Why is it interesting?>
-<--
---> Common Question 2#
+<tabbox FAQ>
-<--
-<tabbox Examples>
---> Example1#
+-->What's the connection between Liouville's theorem and Heisenberg's uncertainty principle?#
+<blockquote>The central idea of Liouville’s theorem – that volume of phase space is constant –
+is somewhat reminiscent of quantum mechanics. Indeed, this is the first of several occasions
+where we shall see ideas of quantum physics creeping into the classical world.
+Suppose we have a system of particles distributed randomly within a square $\Delta q \Delta p$
+in
+phase space. Liouville’s theorem implies that if we evolve the system in any Hamiltonian
+manner, we can cut down the spread of positions of the particles only at the
+cost of increasing the spread of momentum. We’re reminded strongly of Heisenberg’s
+uncertainty relation, which is also written $\Delta q \Delta p=$ constant.
+While Liouville and Heisenberg seem to be talking the same language, there are very
+profound differences between them. The distribution in the classical picture reflects
+our ignorance of the system rather than any intrinsic uncertainty. This is perhaps best
+illustrated by the fact that we can evade Liouville’s theorem in a real system! The
+crucial point is that a system of classical particles is really described by collection of
+points in phase space rather than a continuous distribution $\rho(q, p)$  as we modelled it
+above. This means that if we’re clever we can evolve the system with a Hamiltonian
+so that the points get closer together, while the spaces between the points get pushed
+away. A method for achieving this is known as stochastic cooling and is an important
+part of particle collider technology. In 1984 van der Meer won the the Nobel prize for
+pioneering this method.<cite>http://www.damtp.cam.ac.uk/user/tong/dynamics/four.pdf</cite></blockquote>
 <--
---> Example2:#
+-->Does Liouville's theorem apply to real systems?#
+No!
+<blockquote> This is perhaps best
+illustrated by the fact that we can evade Liouville’s theorem in a real system! The
+crucial point is that a system of classical particles is really described by a collection of
+points in phase space rather than a continuous distribution $\rho(q, p)$ as we modeled it
+above. This means that if we’re clever we can evolve the system with a Hamiltonian
+so that the points get closer together, while the spaces between the points get pushed
+away. A method for achieving this is known as stochastic cooling and is an important
+part of particle collider technology. In 1984 van der Meer won the the Nobel prize for
+pioneering this method.<cite>http://www.damtp.cam.ac.uk/user/tong/dynamics/four.pdf</cite></blockquote>
 <--
-<tabbox History>
 </tabbox>

User Tools

Site Tools

Differences

Page Tools