Lee-Yang Zeros And Particle Fluctuations

We consider classical particles in the continuum in the grand canonical ensemble, with a stable, tempered and lower-regular pair potential and boundary conditions of uniformly bounded density. We prove that if the Lee–Yang zeros of the grand canonical partition function in the complex fugacity plane $z=e^{\beta \mu }$ remain bounded away from a real point $z_0>0$ for all sufficiently large volumes, then along cubes the thermodynamic limit and differentiation commute at $z_0$: every derivative of the finite-volume pressure in the chemical potential converges, uniformly in a neighborhood of $z_0$, to the corresponding derivative of the limiting pressure. The limiting values of all derivatives are independent of the boundary condition; in particular, the density and the particle-number variance per unit volume converge to ${\beta }^{-1}{\mathrm{\partial }}_{\mu }p$ and ${\beta }^{-2}{\mathrm{\partial }}_{\mu }^{2}p$, respectively. The result extends to the unbounded boundary conditions of Procacci and Yuhjtman for super-stable potentials in addition to Ruelle’s tempered boundary conditions.