## 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

# Disclaimer

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).

# Liouville’s theorem

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.

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