I’m having some difficulties with my Statistical Mechanics course.

In particular, I have some doubts about Liouville’s theorem and the various ensembles. Consider, for instance, the Canonical ensemble. Does this theorem hold here? Or it doesn’t because the Hamiltonian is not conserved.

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    $\begingroup$ It seems to me that the statement of Liouville's theorem is true for any distribution (possibly with some smoothness conditions though), thus for any ensemble. $\endgroup$ – Raskolnikov Jan 24 at 10:01

An ensemble is nothing else that a probability density function (PDF) $\rho(p,q)$, where $q$ are the canonical coordinates and $p$ the momenta. There are several forms of this PDF, depending on the macroscopic conditions to which the system is subject:

  • Constant number of particles ($N$), volume ($V$) and energy ($E$): mircocanonical ensemble (*)

$$\rho(p,q)= \begin{cases} 1 & E<\mathcal H (p,q) < E+\Delta E\\ 0 & \text{otherwise} \end{cases} $$

  • Constant number of particles ($N$), volume ($V$) and temperature ($T$): canonical ensemble

$$\rho(p,q) = \frac{e^{-\mathcal H(p,q)}}{\int dp dq \ e^{-\mathcal H(p,q)}}$$

... and so on ($\mathcal H$ is the Hamiltonian). This PDF (or "ensemble") tells you how likely is it that, given the macroscopic conditions (the macrostate) you find your system in a certain microstate $(p,q)$.

Lioville's theorem is valid for any of these PDF, as long as they describe a system governed by an Hamiltonian $\mathcal H (p,q)$ which does not depend on any time derivative of $p$ and $q$.

This is because in such system a state point $(p,q)$ moves following Hamilton's equations of motion:

$$ \dot p = -\frac{\partial \mathcal H}{\partial q}; \ \ \ \ \dot q = \frac{\partial \mathcal H}{\partial p} $$

These equations are invariant under time reversal and uniquely determine the motion of a state point at all times. As a consequence, the trajectory of the system in phase space is either a closed curve or a curve that never intersects itself (otherwise, where would the system evolve if $t$ is reversed?). Furthermore, the trajectories of two different state points can never intersect (for the same reason as above).

As a consequence of this, the probability distribution of these state points, that is, the ensemble $\rho(p,q)$ must move like an incompressible fluid. Exactly like in fluid dynamics, we can express this fact mathematically as

$$ \frac{\partial \rho} {\partial t} + \nabla(\mathbf v \rho) = \frac{d \rho}{dt} = 0 $$

or, using Poisson's brackets:

$$ \frac{\partial \rho} {\partial t} + \{\rho,\mathcal H\} = 0 $$

Main source: K. Huang, Statistical Mechanics. For an online proof, see for example here.

(*) Some authors, like Landau, prefer $\rho(p,q)=\delta(\mathcal H - E)$ for the microcanonical ensemble.


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