Question

Consider a system of fixed volume V in equilibrium with a reservoir of both heat and particles (hence we may describe the system using the Grand Canonical Ensamble).

While I was trying to derive the "Density Fluctuations Relation" I have stumbled upon the fact that I need to use the canonical expression for the boundary pressure (so that I could differentiate it twice by μ):
P=−(∂A∂V)N,T
however, I noticed that if I don't specify the value of N then I get an expression for σ2[N] which depends on N itself, which is impossible (the variance of a random variable can not depend on the random variable itself).
When I asked my teacher about this fact he told me that the expression above should be evaluated at ⟨N) because we are assuming the system to be at equilibrium and we are describing it using the Canonical Ensamble, thus in the thermodynamic limit it contains ⟨N⟩particles.

I would like to know if this reasoning is correct or if there is something missing. Also I would like to know if I am right in saying that books like Huang's are a little sloppy when dealing with these derivations (indeed, this specific derivation is carried out at pages 152-153 and Huang never mentions the problem, actually he just changes N to its mean value N¯ after "minor rewritings").

Answer 1

The reasoning that the canonical expression for pressure should be evaluated at the mean number of particles <N> is correct, since this mean is representative in the Grand Canonical Ensemble. Density fluctuations are derived by differentiating the expression of pressure with respect to the chemical potential. Textbooks sometimes assume the reader will interpret N as its mean value without explicitly stating so.

Step-by-step explanation:

Your teacher's reasoning appears to be correct; when dealing with density fluctuations in the context of the Grand Canonical Ensemble, it is standard to evaluate expressions such as pressure at the mean particle number, <N>, especially in the thermodynamic limit where this mean value becomes representative of the system.

In statistical mechanics, particularly for a system in equilibrium with a reservoir that allows for exchange of heat and particles, the Grand Canonical Ensemble is the appropriate framework. Here, fluctuations about this mean number are central to the physics of the system.

A canonical expression for pressure is given by P = -(∂A/∂V)_N,T, where A is the Helmholtz free energy, and this expression should be differentiated twice with respect to the chemical potential μ to derive the density fluctuations. However, throughout these derivations, the number of particles N appears variably and must eventually be replaced by its average <N> to correctly represent the equilibrium properties of the ensemble.

As for literature like Huang's, it is not uncommon for textbooks in statistical mechanics to gloss over some of the subtleties in the derivations, assuming that the reader will interpret N as the mean number of particles <N> in the context of ensemble averages. A rigorous approach would indeed highlight when approximations are made or when variables are to be taken as mean values and not as fluctuating quantities.