Particle-number fluctuations near the critical point of nuclear matter

The equation of state with quantum statistics corrections is used for particle number fluctuations $\omega$ of isotopically symmetric nuclear matter with interparticle van der Waals and Skyrme local density interactions. The fluctuations, $\omega\propto 1/\mathcal{K}$, are analytically derived through the isothermal incompressibility $\mathcal{K}$ at first order over a small quantum-statistics parameter. Our approximate analytical results appear to be in good agreement with the results of accurate numerical calculations. These results are also close to those obtained by using more accurate Tolman and Rowlinson expansions of the incompressibility $\mathcal{K}$ near the critical point. A more general formula for fluctuations $\omega$, improved at the critical point, was obtained for a finite particle-number average $\langle N \rangle$ by neglecting, for simplicity, small quantum statistics effects. It is shown that for a large dimensionless parameter, $\alpha \propto \mathcal{K}^2\langle N \rangle/\mathcal{K}^{\prime\prime}$, where $\mathcal{K}^{\prime\prime}$ is the second derivative of the incompressibility $\mathcal{K}$ as function of the average particle density $n$, far from the critical point ($\alpha \gg 1$), one finds the traditional asymptote, $\omega\propto 1/\mathcal{K}$, for the fluctuations $\omega$. For a small parameter, $\alpha \ll 1$, near the critical point, where $\mathcal{K}=0$ and $\alpha=0$, one obtains another asymptote of $\omega$. These fluctuations, having a maximum near the critical point as function of the average density $n$, for finite values of $\langle N \rangle$ are finite and relatively small, in contrast to the results of the traditional calculations.