# Difference between microcanonical and canonical ensemble

I feel a little unclarity about this. Canonical ensemble is pictured as many systems in heat reservoir of infinite capacity having $N$ (number of particles), $V$ (volume) and $T$ (temperature) constant whereas microcanonical ensemble is the analogous system having $E$ (energy) instead of $T$ fixed. From my simple viewpoint of introductory pchem, it seems obvious to me that heat capacity is constant hence how is it possible for canonical ensemble to have different energies while having same temperature. Feeling temporarily frustrated I grab McQuarrie's pchem it reads on the topic:

"Each system has the same values of N, V, and T but is likely to be in a different quantum state, consistent with the values of N and V"

A light shines into my lower mood state. So we talk about very little changes where the concept of heat capacity must be taken with reserve? That is, it is possible that system sucks heat from this infinite heat reservoir which is used to promote a particle in the system to higher rotational state for instance, which causes the system to acquire a tiny bit of energy while temperature will not change at all? Temperature is understood as describing translational motion of particles, right? Is my idea of the difference between these to ensembles correct?

• In a microcanonical ensemble the energy (sum of kinetic and potential E) is conserved. In a canonical ensemble the temperature (dependent on kinetic E) is conserved via coupling to an infinite heat bath. What exactly is the problem you're having? – tschoppi Jan 27 '16 at 22:28
• Rotational and vibrational states are understood as form of potential energy? – wuschi Jan 27 '16 at 22:32

Typically (at least how I learned this after three years of chemistry) the canonical NVT ensemble is created by considering many microcanonical NVE "ensembles" as microstates in this NVT ensemble. I quote from my lecture script for statistical thermodynamics:

An ensemble with a constant number $N$ of particles in a constant volume $V$ and at thermal equilibrium with a heat bath at constant temperature $T$ can be considered as an ensemble of microcanonical subensembles with different energies $\epsilon_i$. The energy dependence of [the] probability density conforms to the Boltzmann distribution. Such an ensemble is called a canonical ensemble.

So your NVT ensemble is many NVE ensembles at different energies. Their statistical weights (the probability of finding a microstate in that particular NVE state) are Boltzmann distributed. The way this distribution of NVE subensembles looks like then defines the temperature of the overarching NVT ensemble.

To illustrate this I have written a small Python program that calculates this distribution for different temperatures of the NVT ensemble. The probability density as a function of the energy of the 48 possible (this number was arbitrarily defined, in principle it could be infinite) NVE subensembles is given below: As you can see, the probability of finding NVE microstates with low energies drops drastically as the temperature increases. You can think of this as the thermostat taking a NVE microstate at higher energy, removing some of that energy, and then putting it in a lower energy state. Or the other way around if you want to "heat up" a system.

• I'm not sure whether I adequately addressed the problem you have. Please indicate whether this cleared up your confusion. If not, maybe try to clarify in the question post. – tschoppi Jan 27 '16 at 22:39
• Concrete example: Suppose you have microcanonical ensemble where systems are composed of many vials of water. How do you manage to change their energy while keeping the temperature constant i.e. how do you increase only potential energy of this system? – wuschi Jan 27 '16 at 22:50
• I'll get back to this after I had a good night's sleep. – tschoppi Jan 27 '16 at 23:03
• I meant canonical ensemble (sorry). – wuschi Jan 27 '16 at 23:05
• @wuschi I amended the answer. Did it become clearer? You don't change the energy of the subensembles/microstates. You change the way they are distributed in your ensemble. – tschoppi Jan 28 '16 at 10:03

One should distinguish finite systems and systems in the thermodynamic limit. In the latter limit there are no fluctuations and all ensembles become equivalent. In this case, one can calculate e.g. the energy $E$ of the system from $N$, $V$ and $T$. This is the realm of classical thermodynamics.

For finite systems however, there will be fluctuations of energy when you regard a system with constant $N$ and $V$ which is in contact with a heat bath of temperature $T$. Note that in this case you cannot calculate the energy from knowledge of these three quantities. And vice versa, you cannot deduce the temperature from knowledge of a particular microstate of the canonical ensemble. So one should regard temperature not as a property of the individual microstates comprising this ensemble, but rather as a property of the heat bath or of the ensemble as a whole.

Regarding the heat capacity: I'm no expert in this and I guess that there exist definitions for it in the different ensembles. But to be on the safe side, I would recommend you to apply the concept only for macroscopic systems.

Regarding the speculation of OP: there should not be made a distinction between rotational and translational motion, both kinds are relevant for temperature. Furthermore, the heat bath will always excite all kinds of motion in your system. As I said, one should simply abandon the idea that two different microstates from a canonical ensemble with temperature $T$ both "have" this temperature individually.