PHY452H1S Basic Statistical Mechanics. Lecture 5: Motion in phase space. Liouville and Poincar’e theorems. Taught by Prof. Arun Paramekanti
Posted by peeterjoot on January 22, 2013
Peeter’s lecture notes from class. May not be entirely coherent.
Motion in phase space
Classical system: with dimensionality
Hamiltonian is the “energy function”
Expressed in terms of the Hamiltonian this is
In phase space we can have any number of possible trajectories as illustrated in (Fig1).
We are interested in asking the question of how the density of a region in phase space evolves as illustrated in (Fig2)
We define a phase space density
and seek to demonstrate Liouville’s theorem, that the phase space density does not change. To do so, consider the total time derivative of the phase space density
We’ve implicitly defined a current density above by comparing to the continuity equation
Here we have
Usually we have
but we don’t care about this diffusion relation, just the continuity equation equivalent
The implication is that
Flow in phase space is very similar to an “incompressible fluid”.
Time averages, and Poincar\’e recurrence theorem
We want to look at how various observables behave over time
We’d like to understand how such averages behave over long time intervals, looking at .
This long term behaviour is described by the Poincar\’e recurrence theorem. If we wait long enough a point in phase space will come arbitrarily close to its starting point, recurring or “closing the trajectory loops”.
A simple example of a recurrence is an undamped SHO, such as a pendulum. That pendulum bob when it hits the bottom of the cycle will have the same velocity (and position) each time it goes through a complete cycle. If we imagine a much more complicated system, such as harmonic oscillators, each with different periods, we can imagine that it will take infinitely long for this cycle or recurrence to occur, and the trajectory will end up sweeping through all of phase space.