I have the following problem.
Given a finite system, let us say a box of side $L$ containing $N$ "ideal gas" non-interacting particles, properties such as the entropy can be defined, in this case, $S = - \log ((L^3)^N/N!)$ (I am neglecting the momentum contribution, just configurational entropy).
In the thermodynamic limit, as $L \to \infty$ and $N \to \infty$ such that the density $ \rho = \frac{L^3}{N}$ is constant, all works perfectly.
But how to handle a system where $N$ is constant, and $L \to \infty$ ?
Take the simplest case, $N = 1$. The entropy of one particle in an infinite box diverges due to the volume term in the logarithm, while the density $\to 0$.
This seems absurd to me: a syste with "infinite entropy" and nihil density. Furthermore, if anybody mentioned a zero density case, can this refer to an arbitrary, yet finite number of particles in an "infinite" volume?
I must be missing some point.