Canonical ensemble, ergodicity and Liouville’s equation

Question

I understand that in Statistical Mechanics Liouville’s equation applies to the probability density of ensembles where microstates’ trajectories are governed by Hamiltonian dynamics. However I’m confused about its application to microstates in the canonical ensemble where microstates can have different energies. If we assume ergodicity, a microstate in the canonical ensemble would visit configurations with different energies and thus its energy would not be conserved along its trajectory. Is ergodicity not applicable to the canonical ensemble or is Liouville’s equation more general and can apply to non-conserving energy trajectories?

Answer 1

The canonical ensemble is best understood as a small subsystem of a larger microcanonical ensemble. Specifically, the canonical ensemble is assumed to have a fixed volume and particle number, but it is allowed to exchange energy with the rest of the larger microcanonical ensemble.

You can picture this as a number of particles in a small box with conductive walls immersed in a much larger reservoir of gas. The particles in the wall of the small box pass energy between the two gases. The total energy of the entire system (small box + reservoir) is still conserved but the small system can obviously acquire various values of total internal energy through this coupling. What one then assumes is that the microstate of the entire system evolves ergodically, and that the microstate of the small system acquires various parts of the total internal energy of the system accordingly. This leads to the statistics of the canonical ensemble.

You can read more about this in Part II of the 1985 textbook of Callen Thermodynamics and an Introduction to Thermostatistics.

Answer 2

Short version

To be precise, the idea of an ensemble is that to consider the same system to exist in multiple configurations (microstates), with each microstate associated with a probability. Further, it is necessary to point out that existence in the microstates are not completely arbitrary but are fixed through specific constraints (like constant energy, constant temperature etc.). Each of these constraints lead to different ensembles. While you're referring to the trajectory of a single phase-space point, the Liouville's equation describes the behaviour of the collection of phase-space points.

Therefore, according to Liouville's equation, the collection of points still conserve the energy, as each individual point in the collection follow an energy conserved path. When considering a collection with individual phase-space points and respective probabilities, it doesn't make sense to conserve the total energy of all the points, as all the points don't by themselves exist, they're merely statistical constructs (ensemble). Whereas, it is notable that in the canonical case, the average energy of the system, which is again a statistical quantity, is completely conserved.

More details

The idea of Statistical Mechanics is connect the phenomenological observation of thermodynamics to underlying microscopic laws of molecular motion.

There are two major ingredients to this approach.

1. The phase-space trajectory, $\Gamma(t) = \{\mathbf{p}(t), \mathbf{q}(t)\}$, of the system is treated as a stochastic process.

The implication of this consideration is that the system randomly moves from one point to another in the phase-space, with certain associated laws of probability evolution. This evolution law is the Liouville's equation of motion. If the probability density of the phase-space variable is, $p(\Gamma,t)$, the Liouville's equation follows,

$$ \frac{\partial p(\Gamma,t)}{\partial t} = \{ H, p(\Gamma,t) \} $$ where $\{\}$ refer to the Poisson brackets. This equation ensures that the probability evolution respects the Hamiltonian ($H$) evolution laws of the classical system.

2. Thermodynamic states are equilibrium states, (more precisely steady state solutions) wherein the probability density has no explicit time-dependence, $p_{eq}(\Gamma, t) = p_{eq}(\Gamma)$

*This consideration takes into account that only steady state solutions of Liouville's equation can lead to thermodynamic observations. This raises the question of about the arbitrariness in $p_{eq} (\Gamma)$.

From Liouville's equation we can show that $p_{eq} (\Gamma) = p_{eq}\left(H(\Gamma)\right)$*

In this framework, thermodynamic variables ($T \text{ (Temperature)}, U \text{(Internal Energy)}, ..$) are related to average values of the probability distribution in Statistical Mechanics.

In understanding ergodicity, we reflect on the fact that the stochastic process $\Gamma(t)$, traverses most regions of the phase space, effectively filling it over time. Throughout its evolution, $\Gamma(t)$ moves according to the laws governed by Liouville's equation, causing the probability distribution $p(\Gamma,t)$ to evolve as dictated by this equation.

After a sufficient time (relaxation to equilibrium), the stochastic process reaches the steady state, wherein the system can be said to achieved a thermodynamic state. In this steady state, the probability density doesn't change anymore and has a value $p_{eq}(\Gamma)$. But do remember that, this steady is only achieved by the system by traversing through most of the phase-space during it's evolution towards the equilibrium. Therefore, even in the canonical ensemble the condition of ergodicity is implied.

As to your question of how the system moves through different phase-space points with different energy (ergodicity) - this happens as the system is approaching steady state (equilibrium). During this time evolution towards the steady state, there is substantial transfer of energy between bath and system, in the probability density function also changes with time $p(\Gamma, t)$, as given by Liouville's theorem.

PS: The evolution of the system towards equilibrium is an area of study in non-equilibrium statistical mechanics.