Relative Fluctuations in volume at thermodynamic limit

Question

Volume fluctuation of a pressure ensemble is given by $$\langle(\Delta V)^2\rangle = k_BT\langle V\rangle\kappa_T$$ where $\kappa_T$ is the isothermal compressibility.
In the book, it is said that in the thermodynamic limit, the relative fluctuations vanish in the thermodynamic limit, which I don't seem to understand why?

$$\frac{\sqrt{\langle(\Delta V)^2\rangle}}{\langle V\rangle} = \frac{\sqrt{k_BT\kappa_T}}{\sqrt{\langle V\rangle}}$$ Both $\kappa_T$ and $V$ are finite in the thermodynamic limit. So, how are the relative fluctuations vanishing?

Answer 1

The thermodynamic limit (TL) is a theoretical tool necessary to justify the connection between Statistical Mechanics and Thermodynamics. Indeed, for interacting finite systems, the logarithm of the partition functions in different ensembles fails to have some key properties like additivity, convexity, and the possible presence of non-analytic behavior (aka phase transitions). The thermodynamic limit restores the expected thermodynamic behavior of macroscopic systems.

However, the exact statement about how to perform the TL changes in each statistical ensemble. The general recipe is that the limit that should exist is the limit of the logarithm of the partition function divided by one extensive independent variable; all the extensive independent variables must go to infinity by keeping fixed their ratios.

To list a few well-known and some less well-known cases, the way to implement the TL in different ensembles is the following.

microcanonical ensemble $$ \lim_{{{E\rightarrow \infty} \atop {V \rightarrow \infty}} \atop {N \rightarrow \infty }} \frac{1}{V} \log W(E,V,N)~~~~~~~~~~~~~~~~~~~~~~{\mathrm {by~keeping~constant}}~~~~~~ \frac{E}{V}= e ~~~~ {\mathrm {and}}~~~~~~~~ \frac{N}{V}= \rho $$ canonical ensemble $$ \lim_{{ {V \rightarrow \infty}} \atop {N \rightarrow \infty }} \frac{1}{V} \log Z(\beta,V,N)~~~~~~~~~~~~~~~~~~~~~~{\mathrm {by~keeping~constant}}~~~~~~ ~~~~~ \frac{N}{V}= \rho $$ grand canonical ensemble $$ \lim_{V \rightarrow \infty} \frac{1}{V} \log \Xi(\beta,V,z=e^{\beta \mu}) $$ isothermal-isobaric ensemble $$ \lim_{N \rightarrow \infty} \frac{1}{N} \log X(\beta,\beta P,N) $$ Notice that there is only one extensive independent variable in the last two cases.

The previous TLs, if they exist, correspond respectively to

• entropy density $s(e,\rho)$ (microcanonical);
• $-\beta f(\beta,\rho)$, $f$ density of free energy;
• $-\beta P(\beta,z)$;
• $-\beta \mu(T,P)$

In the case of the isothermal-isobaric ensemble, the existence of TL implies the extensiveness of the Gibbs free energy $G= N \mu$. Therefore, $\langle V \rangle=\frac{\partial G}{\partial P}$ grows like $N$ and the relative fluctuation of the volume vanishes at the TL.