Two subsystems in thermal contact in the microcanonical ensemble, confusion in the terminology of the microcanonical and canonical ensemble

Question

I am confused about the terminology used in an example illustrating the microcanonical ensemble. The example is found in chapter 4 (p.176) of Gould and Tobochnik's manual Statistical and Thermal Physics: With Computer Applications (which can be freely and legaly downloaded here : https://www.compadre.org/stp/).

TL;DR If two systems, $A$ and $B$, are allowed to exchange energy and that the composite system $A + B$ is isolated (fixed $E$, $N$ and $V$), we can model the composite system with the microcanonical ensemble, but system $A$ and $B$ cannot be modelled with the microcanonical ensemble nor the canonical ensemble unless one of them is a heat bath. Is that correct? Is there a name for the ensemble that models system $A$ in that situation?

In the example, we consider an isolated system divided in two isolated subsystems, A and B with fixed volumes, both of which are composed of two ($N=2$) Einstein solids. At first, the energy available to A is 5 and the energy available to B is 1. The composite system thus have an energy of

$$E_{tot} = E_A + E_B = 5 + 1 = 6.$$

At this point, we can state that we're in a situation where the composite system can be analysed with the microcanonical ensemble, right? By that, I mean that the probability distribution over all the possible microstates of the composite system is uniform.

1. Can we also say the same for each subsystem? if I only consider subsystem A, since it has fixed volume $V$, number of particles $N$ and energy $E_A$, it has a uniform distribution over its possible microstates. We're in the microcanonical ensemble.

Now, if we allow the exchange of energy between A and B, the composite system now has access to more microstates. The probability distribution of finding the composite system in a specific microstate is still uniform, but the probabilities for each state are lower than in the previous situation. It's still the microcanonical ensemble.

2. But what happens to subsystems A and B? Now they can exchange energy and we can compute the average energy of A (3) and the variance of the energy (also 3). This implies that the energy in A is not fixed and that I cannot model each possible microstate of A as having equal probabilities. These fluctuation in energy violate the principles of the microcanonical ensemble. Is that right?

3. This is the source of my confusion because it's as if we allow exchange of energy in the microcanonical ensemble, but that's not possible since energy should be fixed in that ensemble. Thus, in this situation with thermal exchange, is that statement correct : The composite system can be analysed with the microcanonical ensemble, but not the subsystems.

Then comes my other question. In the canonical ensemble, the composite system (Heat bath + system of interest, which we'll call A) respects the microcanonical ensemble. The composite system is isolated from its environment and this allows us to write an expression for the probability of finding system A at a specific energy, just like in the previous example.

4. So, in the first example, when thermal interactions are allowed, system A could be thought of as being in a canonical ensemble since its energy fluctuates BUT since system B is not a heat bath (subsystem B is drastically affected by when energy is exchanged) this is not true. However, subsystem A is also not in the microcanonical ensemble since its energy fluctuates, which makes me wonder : in which ensemble is the subsystem A in the example?

Answer 1

Regarding the bath:

• The canonical ensemble represents the microstates under constant $(T,V,N)$. If we place two systems in thermal contact with each other their temperature will settle to some value which we can calculate using the heat capacities. If instead we place a system into contact with a bath the final temperature of that system is the same as of the bath, no calculation required. By using the device of the bath we know the value of $T$ without having to calculate it for each system. In summary, if two equilibrium systems are in thermal contact they are both canonical regardless of whether one of them is a bath or not.

Regarding the questions:

1. Yes: as long the energy of subsystems A and B are fixed the subsystems are microcanonical

2. Once we allow exchange of energy between A and B the subsystems become canonical and their microstates are no longer equally probable but they are weighted by the Boltzmann factor $e^{-\beta E}$.

3. It correct to say that the composite system can be analyzed with the microcanonical ensemble, but not the subsystems.

4. The fact that we don't have a bath is not an issue. At equilibrium the subsystems will satisify $$ \tag{1} \frac{\partial\Omega_A(E_A)}{\partial E_A} = \frac{\partial\Omega_B(E_B)}{\partial E_B} = \beta $$ which is the condition of thermal equilibrium and defines the common temperature $T$. Then, the probability of microstate $i$ in subsystem A is $$ \tag{2} p_i = \frac{e^{-\beta E_{A,i}}}{Q} $$ with $\beta$ from Eq (1). The only requirement is that both subsystems are large enough to be in the thermodynamic limit.