Debye screening in spatially inhomogeneous systems of charged particles. I. Model of spherical insulator

In a medium with permittivity $\varepsilon$ there is a spherical insulator $\Omega_0$ of radius $R_0$ with permittivity $\varepsilon_0<\varepsilon$. A system of ions represented by charged impermeable spheres of radius $r_0$ whose distribution around the sphere $\Omega_0$ satisfies the Brydges–Federbush neutrality condition is considered. Initially, the system is in a finite volume $\Lambda$ (sphere of radius $R\gg R_0$), and the interaction satisfies a Dirichlet condition on $\partial\Lambda$. For sufficiently high values of the temperature convergence of the cluster expansions and existence of the distribution functions in the limit $R\to\infty$ ($\Lambda\nearrow\mathbb R^3$) are proved. It is established that there is exponential clustering of the distribution functions along the radial directions of the sphere $\Omega_0$ with a power-law decrease along the surface $\partial\Omega_0$.