How can a system with a given temperature, volume and number of particles have many possible microstates of different internal energies? I am learning about the canonical ensemble where a system in contact with a heat reservior is considered. The objective is to find the probabilty of the system having the microstate marked by a particular energy (say, E). This implies that many microstates having different energies are possible for the same temperature (since temperature is the measure of only translational kinetic part of the internal energy), i would like to know in exactly what form might these states have the rest of energy that differentiates them from one another when their average kinetic energies (temperature) are the same?
I guess it might be some sort of potential energy, but how would the system actually gain potential energy from the reservior while its temperature remains the same, is a little puzzling and hard to imagine. I am missing a good picture for the microstates for this system.
 A: Essentially the question is, how is it possible for a system to exchange energy under constant temperature. Before I answer we need to understand that we are discussing a thought experiment in statistical mechanics, not an actual experiment in the lab (abut more about this below). In statistical mechanics the bath fixes temperature and through it the probability to observe a system, in thermal contact with the bath, at microstate $i$:
$$\text{(prob that system is in microstate $i$)} = \frac{e^{-\beta E_i}}Q{}$$
where $E_i$ is the energy of the microstate, $\beta=1/k T$ and $Q$ is the canonical partition function. In principle the system can exchange huge amounts of energy with the bath but the probability for such exchanges is minuscule. The most probable exchange involves amounts of energy that are just very very small.
The last observation brings to the analogous  experimental situation: a system in thermal equilibrium with the bath continuously exchanges heat with the bath as 'hot'  molecules collide with the wall that separates the system from the bath transfer their energy to the other side in either direction. These fluctuations are minuscule in magnitude and their probability is equal to the canonical probability in Eq (1).
In summary, even though the canonical ensemble allows the system to exchange any amount of energy with the bath, the probability  for such exchanges is very low, in agreement with the experimental conclusion.
A: The key point is that the temperature of the thermostat is fixed. The kinetic energy of the system is not. It can fluctuate according to the energy exchanges between the thermostat and the system. It is a simple exercise to evaluate the variance of the system's kinetic energy in the canonical ensemble, finding that it is proportional to $T^2$.
Similarly to the kinetic energy of the system, also the internal energy and the potential energy of the system fluctuate.
A: Because the microstates are determined by statistics on the huge number of degrees of freedom of the microscopic (molecular) description of the medium, in a statistical mechanics approach.
When you think at volume, temperature (1 value only of temperature) and mass, you're thinking at the macroscopic description of the system in a classical thermodynamics approach, working with average values or integral values of physical quantities of the microscopic view: roughly speaking, average kinetic energy goes to temperature, number of molecules goes to mass, and so on.
