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--- theorems:liouvilles_theorem [2018/04/08 10:47]
jakobadmin [Why is it interesting?]
+++ 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 12: / Line 17: @@
 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>
+But all physical measurements have a definite limitation on how
+accurately they can be performed, and can only give information about a
+finite number of decimal places. [...] What this means,
+in terms of phase space, is that each of our 'discrete' alternatives must
+correspond to a region in [[basic_tools:phase_space|phase space]], so that different phase-space points
+lying in the same region would correspond to the same one of these
+alternatives for our device (Fig. 5.12)
+{{ :basic_notions:phasespaceregionpenroseemprerornewmind.png?nolink&600 |}}
+Now suppose that the device starts off with its phase-space point in some
+region Ro corresponding to a pal ticular one of these alternatives. We think ~f
+Ro as being dragged along the Hamiltonian vector field as time proceeds, until
+at time t the region becomes $R_t$. In picturing this, we are imagining
+simultaneously, the time-evolution of our system for all possible starting
+states corresponding to this same alternative. (See Fig. 5.13.) The question of
+stability (in the sense we are interested in here) is whether, as t increases, the
+region $R_t$ remains localized or whether it begins to spread out over the pase
+space. If such regions remain localized as time progresses, then we have a measure of stability for our system. Points of phase space which are close together (so that they correspond to detailed physical states of the system
+which closely resemble one another) will remain close together in phase
+space, and inaccuracies in their specification will not become magnified with
+time. Any undue spreading would entail an effective unpredictability in the
+behaviour of the system.
+{{ :basic_notions:phasespacepenroseemprerornewmind.png?nolink&600 |}}
+What can be said about Hamiltonian systems generally? Do regions in
+phase space tend to spread with time or do they not? It might seem that for a
+problem of such generality, very little could be said. However, it turns out
+that there is a very beautiful theorem, due to the distinguished French
+mathematician Joseph Liouville (1809-1882), that tells us that the volume of
+any region of the phase space must remain constant under any
+evolution. At first sight this would seems to answer our stability question in the affirmative. For the size - in the sense of this phase space volume - of our region cannot grow, so it would seem that our region cannot spread itself out over the phase space.  However this is deceptive, and on reflection we see that the very reverse is likely to be the case!
+{{ :basic_notions:phasespaceregionspread2penroseemprerornewmind.png?nolink&600 |}}
+[...]
+The
+volume indeed remains the same, but this same small volume can get very
+thinly spread out over huge regions of the phase space. For a somewhat
+analogous situation, think of a small drop of ink placed in a large container of
+water. Whereas the actual volume of material in the ink remains unchanged,
+it eventually becomes thinly spread over the entire contents of the container. [...] The trouble is that preservation of volume does not at all imply preservation of shape: small regions will tend to get distorted, and this distortion gets
+magnified over large distances.
+[...]
+In fact, far from being a
+'help', in keeping the region $R_t$ under control, Liouville's theorem actually
+presents us with a fundamental problem! Without Liouville's theorem, one
+might envisage that this undoubted tendency for a region to spread our in
+phase space could, in suitable circumstances, be compensated by a reduction
+in overall volume. However, the theorem tells us that this is impossible, and we have to face up to this striking implication - a universal feature of all classical dynamical (Hamiltonian) systems of normal type!
+We may ask, in view of this spreading throughout phase space, how is it possible at all to make predictions in classical mechanics? That is, in indeed, a good question. What this spreading tells us is that, no matter how accurately we know the initial state of a system (within some reasonable limits), the uncertainties will tend to grow in time and our initial information may become almost useless. Classical mechanics is, in this kind of sense, essentially unpredictable.
+How is it, then, that Newtonian dynamics has been seen to be so successful? In the case of celestial mechanics (o.e. the motion of heavenly bodies under gravity), the reasons seem to be, first, the one is concerned with a comparatively small number of coherent bodies (the sun, planets, and moons) which are greatly segregated with regard to mass- so that to a first approximation one can ignore the perturbing effect of the less massive bodies and treat the larger ones as just a few bodies acting under each other's influence- and, second, that the dynamical laws that apply to the individual particles which constitutes those bodies can be seen also to operate at the level of the bodies themselves- so that to a very good approximation, the sun, planets, and moons can themselves be actually treated as particles, and we do not have to worry about all the little detailed motions of the individual particles that actually compose these heavenly bodies. Again we get away with considering just a "few" bodies, and the spread in phase space is not important.
+[...]
+**This spreading effect in phase space has another remarkable implication. It tells us, in effect, that classical mechanics cannot actually be true of our world!**
+[...]
+As we now know, quantum theory is needed in
+order that the actual structure of solids can be properly understood. Somehow, quantum effects can prevent this phase-space spreading. This is an
+important issue to which we shall have to return later (see Chapters 8 and 9)
+<cite>page 174ff in "The Emperors new Mind" by R. Penrose</cite></blockquote>
 <blockquote>It turns out that the abstract [[basic_tools:phase_space|phase fluid]] has the same proper-
@@ Line 17: / 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 23: / 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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