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 '19 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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The ensemble of a system (all possible states under the constraint of a certain set of macroscopic state variables) represent a number of points in the phase space. The time-dependent evolution of the system then represents a number of trajectories in the phase space. Time-reversal symmetry requires that these trajectories do not intersect (or state number conserves), corresponding to the continuity equation for the number of state points. Hamiltonian equation further renders the evolution of the system incompressible in terms of its state points in the phase space, which is the Liouville theorem. Therefore, the Liouville theorem always holds for any constraint of macroscopic state variables (i.e. types of ensemble), since the theorem is just about the property of the density of states. This also means that the Lioiuville equation for the density of states of a system always holds for any type of ensemble.

However, the Liouville equation as a mathematical form can be about any variable. Particularly, the relation between the probability density function and the density of states of a system is different under different types of ensembles.

In general, we can only safely say that the dynamical equation for the probability density function of a system is jointly determined by two equation:

  1. The Liouville equation for the density of states.
  2. The relation between the probability density function and the density of states for the particular type of ensemble.

The postulation of a priori probability ensures that the second equation always take the form where the probability density function is proportional to the density of states. So, the Liouville equation for the probability density function always holds as well.

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