When deriving the energy fluctuations in the canonical ensemble, a step is made to approximate Cv≈N, why is this?
In the Statistical Mechanics Mark E. Tuckerman 4.4.7
statistical mechanicsstatisticsthermodynamics
Related Solutions
If somebody tells you what the entropy is as a function of energy, volume, and number of particles, you have all the information you need (for a standard plain vanilla system). It is not necessary to define any other ensemble, but it is convenient. If your system for instance is in contact with a big other system ("reservoir") with which it can exchange energy, then you can either describe system plus reservoir microcanonically, or you describe only your system canonically. The latter is clearly more convenient, since you need not bother about the internal "workings" of the reservoir. For the purpose of your problem the entire reservoir is perfectly well characterized by a single number: its temperature.
The mathematical machinery of Legendre transforms provides a neat way to change from a thermodynamic potential (such as the entropy) to other potentials in which derivatives of the original thermodynamic potential become the new variables, and this transformation is being done without losing information. So, at the end of the day, this is just mathematical convenience: represent the necessary thermodynamic information in ways that are easier to handle in a given situation characterized by a particular set of constraints.
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 $N\sim\mathcal N_A≈ 6·10^{23}$.
Best Answer
The author indicates that $C_V$ is an extensive quantity, that is, it is proportional to the amount or quantity of material. The latter quantity is indeed $N$, the number of particles in the system.
The constant of proportionality is not mentioned and a clearer way to write the relation could be $C_V = O(N)$.