Thermal average, thermal fluctuations

I've a doubt concerning the physical meaning of "thermal average" and the "thermal fluctuation" in the canonical ensemble.

Let's consider a very simple thermodynamic system: N particles, at fixed temperature T, each of them can live in a discrete set of energy levels. Each level has an occupation probability proportional to the Boltzmann factor $\exp(-\beta E_s)$.

What is exactly the canonical ensemble?

• The system itself, because it's made up of many replica of the same particle?

• The ensemble made up of many replica of the same system, each of them made up of N particles?

• Both of them can be considered as canonical ensembles?

The single particle energy, E, is a random variable, linked to a mean value and to a variance,$\textrm{var} (E)$. We can define a single-particle average energy $\langle E\rangle$ by summing all the possible energy levels multiplied by the occupation probability. This is what people mean with "thermal average"?

The thermal fluctuation is the variance of the random variable "single particle energy E"?

The total average energy of the system will be $N\cdot \langle E\rangle$, i.e. the number of particles times the average energy of one particle.

Is it possible to say that, the absolute instantaneous energy of the system temporally varies around the mean value $N\cdot\langle E\rangle$, with a temporal standard deviation equal to $\sqrt N \textrm{var}(E)$ ?

Last but not least... The central limit theorem, guarantees that, if $N \rightarrow + \infty$ then the fluctuations around the mean value become negligible with respect to the mean value. In this sense statistical mechanics turns into thermodynamics?

• The canonical (or NVT) ensemble is the collection of all configurations (that is position and velocity) of $N$ particles confined to the given volume $V$ at a temperature $T$. The total energy of the system (not per-particle) averaged over the entire collection ('ensemble') is what is usually referred to when speaking of thermal averages... – lemon Mar 5 '16 at 16:54
• Ok! So each replica of the system will have a certain energy E and the Boltzmann distribution is, in principle, related with this energy? – AndreaPaco Mar 5 '16 at 17:52
• That's right. The probability of the total system being in that specific configuration is given by the Boltzmann distribution where $E$ is the total energy. I think a lot of people find this a little non-intuitive at first because they imagine a constant temperature system to have a constant energy, but that's not the case. – lemon Mar 6 '16 at 9:18

What is exactly the canonical ensemble?

Thermodynamic ensembles are ensembles in the mathematical sense, so your option no. 2 is the correct one. Consider a system of non-identical particles, this will appear much more clearly.

What do "thermal average" and "thermal fluctuation" mean?

"Average" is not something per se, one should speak about the thermal average of a quantity $A$. This is the average value taken by $A$ over all configurations of the ensemble, the average being weighted by the probability of each configuration (Boltzmann factor in the canonical ensemble).

The same remark holds for "fluctuation". The thermal fluctuation of a quantity $A$ is the weighted variance (or std. dev.) over the ensemble.

What about time evolution?

The time evolution of the system will reflect the ensemble statistics if the system is ergodic. Some systems are not; glasses are one notable example of non-ergodicity.

(Note: the std. dev. is $\sqrt{N\text{var}(E)}$.)

Is thermodynamics a limit case of statistical mechanics?

Yes, and the limit $N→+∞$ is appropriately called "thermodynamic limit". In practice any macroscopic system has negligible fluctuations, for $N\sim\mathcal N_A≈ 6·10^{23}$.