Is particle number a problem for formulating statistical physics in a mathematically rigorous manner?

Question

Quantities like the chemical potential can be expressed as something like

$$\mu=-T\left(\tfrac{\partial S}{\partial N}\right)_{E,V}.$$

Now the entropy is the log some volume, which depends on the particle number $N$. As in this definition, we sum over natural numbers of particles, is there any good way of actually evaluating the derivative?

What one practically does, i.e. when dealing with an ideal gas, is computing the quantity $\Gamma_N$, which might turn out to be $\frac{\pi^{N/2}}{(n/2)!}$, and then one will get an expression $S(E,N,V)$ which can of course be treated as if it was a function over $\mathbb{R}^3$. Even if that assumes that one has a closed expression which is a function $N$. In principle I'd be fine with that - if one has a given function (Or at least the bunch of values for all $N$) over a grid and a procedure to introduce more and more grid points to get a finer mash, then there is a notion of convergence to a derivative. But here the N's are clearly always at least 1 value apart - no matter how many N s there are (thermodynamics limit), the mesh doesn't get finer between any two given points.

You might define the derivative as computing the average rate of change between two partcle numbers $n$ and $n+d$ and say $\tfrac{\partial S}{\partial N}$ evaluated at $N'$ gives the same value for all $N'$ in one of the $d$-Intervals, but then you would have to postulate how to come up with $d$ in every new situation. This might be overcome in very specific situations in coming up with a "reasonable" fraction of Avogadros number, but this is not quite mathematical and the values of different finite difference approximation schemes are always different.

In full generality, I feel there is no categorical understanding of what the fractional dimensional space (phase space in this case) has to be and so the procedure of evaluation of the derivative should be explicitly postulate.

Answer 1

The formula you write down is one of thermodynamics. In the statistical mechanics version it is valid in the grand canonical ensemble only if you interpret the extensive variables as expectation values. (See, e.g., Chapter 9 of my online book http://lanl.arxiv.org/pdf/0810.1019v2.pdf .) But expectation values are continuous even when the corresponding operator has a discrete spectrum.

If you work with instead with the canonical or microcanonical ensemble, the formula is valid only in the thermodynamic limit. In this limit, the thermodynamic variable that makes sense in the limit is not $N$ but $\overline N = N\overline V/V$, where $\overline V$ is a finite reference volume that in the thermodynamic formulas takes the place of $V$.Again, $\overline N$ takes in the limit a continuum of values.

Thus in each of the versions typically covered in statistical mechanics, everything is consistent, as there is no theoretical problem in taking the derivative. Of course, finding closed formulas for the thermodynamic limit is the hard part, tractable only in a few cases, but in these cases one can verify differentiability except at points of phase transition.