# Thermal Equilibrium and Canonical Ensemble

1 - Are two closed systems (with fixed volumes and of the same gas) in thermal equilibrium equivalent to two isolated systems at the same temperature?

2 - In the canonical ensemble, the "small system (1)" we wish to study is supposed to be in thermal equilibrium with a heat reservoir. If that means (?) they are at the same temperature and there are no heat exchanges between them, how can we say there are several possible energy states for system 1 (each with a certain probability)?

For your first question, the answer is "not really". If you imagine two systems that were in thermal equilibrium (e.g. two things touching one another), and you did something to change the temperature of one, the combined system would adjust to reach thermal equilibrium again. If you instead had each of the two systems isolated, you could make any arbitrary change to one system without affecting the other.

For the second question, in the canonical ensemble the small system is in thermal equilibrium with a large reservoir (i.e. they are at the same temperature). So to think about why there are multiple allowed energy states, let's think a bit about why it would be weird if there weren't.

To begin, temperature isn't a statement that says a system only has one energy, it is a measure of the average kinetic energy of individual particles. So in the large reservoir with lots of particles you will have constant collisions and interactions that give a wide spread in energy. In the canonical ensemble this reservoir is free to exchange energy with the "small system". Thermal equilibrium happens when the system is on average losing as much energy to the reservoir as it is gaining from it. Since the energy being transmitted to the system from the reservoir has a wide spectrum, the energy inside the system will have some spectrum as well, meaning in general many of the possible energy states of the system are accessible.

Like you say, the canonical ensemble is just a general way to assign probabilities to the different energy states available to the system given a certain temperature. Some energies will be very unlikely (e.g. a 5 MeV air molecule at room temperature), and some will be much more likely (e.g. a harmonic oscillator in its ground state very close to absolute zero).

• 1 - I understand your point, but if no change would occur in one of the systems, it seems both configurations could be regarded as equivalent, on which case there probably isn't much one can do with it anyway. 2 - It seems the main issue was the definition of thermal equilibrium. A lot of books/sites define it as "no exchange of heat" and since "heat" is usually defined as "the transfer of energy other than by work or transfer of matter", I concluded no energy should be flowing from one system to the other (fixed $V$'s and $N$'s). With that definition of T.E. it all makes sense. Thanks! – xihiro Dec 31 '14 at 15:54

In statistical mechanics and thermodynamics, thermal equilibrium between two systems doesn't require persistent contact between two systems, it merely means that if an energy-transmitting path is opened between them, then nothing will change in their macroscopic properties.

In statistical mechanics this is a bit subtle. While it is true that opening the path changes the future microscopic behaviour, and the amount of energy on each system fluctuates as they swap energy back and forth, it doesn't matter since we don't know what the initial energies were to start with. It also means that for two systems to be in thermal equilibrium, their energy probability distributions need to be set up very carefully so that they are insensitive to the process of energy exchange. But this sort of "careful" probability distribution is guaranteed if both have a Boltzmann distribution with the same parameter! And so, in statistical mechanics when we say something has temperature $T$, what we really mean is it has Boltzmann distribution with parameter $T$.

In thermodynamics when we speak of the system's energy we are always referring to average energy. Heat exchanges always refer to average energy exchanges. So heat exchange is strictly zero in thermal equilibrium. But in statistical mechanics the term "energy" could refer to either the real energy (which is a "microscopic" variable-we cannot actually know it) or the average energy. In statistical mechanics it's good to be precise about the distinction.

So the answer to 1 is yes, two isolated systems can indeed be in thermal equilibrium with each other even if they never met, provided they have somehow been given the same temperature (both having Boltzmann distributions with same $T$).

Answering 2, note what I said about heat exchange above... even though heat exchange is zero, the systems do randomly exchange energy.

• That settles the issue I had with the definition of thermal equilibrium. The "no exchange of heat" definition actually means "no macroscopically detectable transfer of energy other than by work or transfer of matter", which still allows a microscopic exchange of energy between particles, which averages to zero ($\langle Q \rangle = 0$). Thanks! – xihiro Dec 31 '14 at 16:17