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# Disclaimer

Peeter’s lecture notes from class. May not be entirely coherent.

# Liouville’s theorem

We’ve looked at the continuity equation of phase space density

which with

led us to *Liouville’s theorem*

We define *Ergodic*, meaning that with time, as you wait for , all *available* phase space will be covered. Not all systems are necessarily ergodic, but the hope is that all sufficiently complicated systems will be so.

We hope that

In particular for , we see that our continuity equation 1.2.1 results in 1.2.2.

For example in a SHO system with a cyclic phase space, as in (Fig 1).

Fig 1: Phase space volume trajectory

or equivalently with an *ensemble average*, imagining that we are averaging over a number of different systems

If we say that

so that

then what is this constant. We fix this by the constraint

So, is the allowed “volume” of phase space, the number of states that the system can take that is consistent with conservation of energy.

What’s the probability for a given configuration. We’ll have to enumerate all the possible configurations. For a coin toss example, we can also ask how many configurations exist where the sum of “coin tosses” are fixed.

# A worked example: Ideal gas calculation of

- gas atoms at phase space points
- constrained to volume
- Energy fixed at .

With defined implicitly by

so that with Heavyside theta as in (Fig 2).

Fig 2: Heavyside theta

we have

In three dimensions , the dimension of momentum part of the phase space is 3. In general the dimension of the space is . Here

is the volume of a “sphere” in – dimensions, which we found in the problem set to be

Since we have

the radius is

This gives

and

This result is almost correct, and we have to correct in 2 ways. We have to fix the counting since we need an assumption that all the particles are indistinguishable.

- Indistinguishability. We must divide by .
- is not dimensionless. We need to divide by , where is Plank’s constant.

In the real world we have to consider this as a quantum mechanical system. Imagine a two dimensional phase space. The allowed points are illustrated in (Fig 3).

Fig 3: Phase space volume adjustment for the uncertainty principle

Since , the question of how many boxes there are, we calculate the total volume, and then divide by the volume of each box. This sort of handwaving wouldn’t be required if we did a proper quantum mechanical treatment.

The corrected result is

# To come

We’ll look at entropy