Electrostatics on the sphere with applications to Monte Carlo simulations of two dimensional polar fluids

We present two methods for solving the electrostatics of point charges and multipoles on the surface of a sphere, \textit{i.e.} in the space $\mathcal{S}_{2}$, with applications to numerical simulations of two-dimensional polar fluids. In the first approach, point charges are associated with uniform neutralizing backgrounds to form neutral pseudo-charges, while, in the second, one instead considers bi-charges, \textit{i.e.} dumbells of antipodal point charges of opposite signs. We establish the expressions of the electric potentials of pseudo- and bi-charges as isotropic solutions of the Laplace-Beltrami equation in $\mathcal{S}_{2}$. A multipolar expansion of pseudo- and bi-charge potentials leads to the electric potentials of mono- and bi-multipoles respectively. These potentials constitute non-isotropic solutions of the Laplace-Beltrami equation the general solution of which in spherical coordinates is recast under a new appealing form. We then focus on the case of mono- and bi-dipoles and build the theory of dielectric media in $\mathcal{S}_{2}$. We notably obtain the expression of the static dielectric constant of a uniform isotropic polar fluid living in $\mathcal{S}_{2}$ in term of the polarization fluctuations of subdomains of $\mathcal{S}_{2}$. We also derive the long range behavior of the equilibrium pair correlation function under the assumption that it is governed by macroscopic electrostatics. These theoretical developments find their application in Monte Carlo simulations of the $2D$ fluid of dipolar hard spheres. Some preliminary numerical experiments are discussed with a special emphasis on finite size effects, a careful study of the thermodynamic limit, and a check of the theoretical predictions for the asymptotic behavior of the pair correlation function.