I am following a course on statistical mechanics. My instructor presented us the following Liouville theorem in two (claimed) equivalent ways:
- Differential statement: The probability distribution $\rho(p,q)$ characterizing a statistical ensemble over the microstates of a system described by a Hamiltonian $H(p,q)$ is such that $\frac{d}{dt}\rho(p,q)=0$
- Integral statement: the Hamiltonian flow $g^t$ preserves phase space volume, hence for any t $|g^t(V)|=|V|$. (A consequence of this is the Poincaré recurrence theorem).
On the other hand we assumed the following Ergodic assumption, fundamental to all the discussion:
- Time averages $\langle O\rangle_t$ equal ensemble averages $\langle O\rangle_E$, where for time averages one intends $\frac{1}{\tau}\int_{0}^{\tau} O(p(t),q(t))d\tau$ with $p(t),q(t)$ being the solution of Hamilton Equations.
Moreover we assumed the Equal Apriori Probability Postulate, stating that any microstate leading to the same macrostate should be equally probable, for an isolated system.
Now my questions are:
- Why it holds 1 iff 2?
In the proof of 1, we showed that the Hamiltonian velocities vector $v=(p',q')$ has zero divergence and then resorted to a continuity equation $\partial_t\rho+div(\rho)=0$ for $\rho$: what assures that this continuity equation should hold?
Moreover we wrote $\rho=Jv$ linking hence the density $\rho$ and the Hamiltonian velocities $v$: what assures we can do this? I understand that the ergodic assumption must involve some kind of compatibility between the Hamiltonian flow and $\rho$ but why this kind of compatibility? How can this idea of compatibility be rigorously formalized?
- Are the Ergodic assumption and the Equal Apriori Postulate linked or independent?
