Is the total energy of a canonical ensemble system of $N$ particles, with single-particle energy levels given by $\epsilon_i$ fixed?

Question

Is the total energy of a canonical ensemble system of $N$ particles, with single-particle energy levels given by $\epsilon_i$ fixed ?

We know the total energy of the system is given by : $$E=\sum_{i} n_i \epsilon_i$$

Here $n_i$ is the number of particles in the $\epsilon_i$ energy level.

However, we know the probability of a single particle having energy $\epsilon_j$ is given by :

$$P(\epsilon_j)=\frac{g_j e^{-\beta\epsilon_j}}{Z}$$

Here, $g_j$ is the degeneracy of that energy level, and $Z$ is the single particle partition function.

Moreover, we know that the probability of a single particle having energy $\epsilon_j$, is the total number of particles in that energy level, divided by the total number of particles - according to the definition of probability.

Hence, $$\frac{n_j}{N}=P(\epsilon_j)=\frac{g_j e^{-\beta\epsilon_j}}{Z}$$

This implies,

$$n_j=NP(\epsilon_j)=N\frac{g_j e^{-\beta\epsilon_j}}{Z}$$

Hence we can easily find $n_j$ for any $\epsilon_j$. So, if we know the number of particles in each of these energy levels, we can determine the exact total energy of the system $E$, in the first equation.

However, this seems problematic. If we find out the total energy of the system, and the number of particles in each energy level, we are restricting this entire system to one particular microstate. The probability of obtaining this particular microstate is $1$. The probability of obtaining any other microstate must be $0$.

However, shouldn't every possible microstate of the system have some finite probability i.e. every possible value of total energy have some finite probability?

I've asked a couple of related questions, and the amazing answers to those questions suggest that $n_j$ is not the actual number of particles in the $\epsilon_j$ level. Rather, it is the expected number of particles in that energy level. However, many answers over different websites and comments to one of my previous answers disagree and claim that $n_j$ is the exact actual number of particles in that level indeed.

Can anyone shed some light on this, and clear my doubt.

Answer 1

No, the energy of a canonical ensemble is not fixed. The three classic ensembles are

• Microcanonical: Fixed energy (think a gas in a perfectly insulated box): the system itself has been constrained to have a constant energy.
• Canonical: Fixed temperature (think of a system immersed in an infinite, temperature controlled water bath). Energy can fluctuate into and out of the system.
• Grand Canonical: Fixed temperature and fixed chemical potential. Energy and particles can fluctuate into and out of the system. The only difference here is that the microstate space includes all possible occupancy numbers.

The confusion comes from the notation where people write, e.g., $E$ as shorthand for $\langle E\rangle$ in the canonical ensemble. However, it is a physically different situation -- if you have the capability to measure thermal fluctuations, e.g. by measuring the heat capacity, you will plainly see that the energy of a canonical ensemble actually fluctuates randomly about its expectation value. The law of large numbers suppresses these fluctuations, so they're usually not important. A perfectly insulated microcanonical ensemble by definition does not have any energy fluctuations.

In all three ensembles, there is arguably no sensible way to talk about the "number of particles in microstate $i$". The point of thermodynamics is that we know nothing about the system, we can only make statements about their broad statistical properties and the probability distribution of microstates.

The $n_j$ you look at in the canonical ensemble are expectation values: at any given moment there may well be no particles in that microstate. Now bear in mind that for very large systems, it is extremely unlikely that the true occupancy of a given energy level is far from its expectation value, which is why some sources may suggest that $n_j$ is the "actual" number of particles in macrostate $i$.