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, i.e., in the space S2, with applications to numerical simulations of two-dimensional (2D) 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, 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 S2. 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 S2. We notably obtain the expression of the static dielectric constant of a uniform isotropic polar fluid living in S2 in terms of the polarization fluctuations of subdomains of S2. 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.

A new dipolar potential for numerical simulations of polar fluids on the 4D hypersphere.

We present a new method for Monte Carlo or Molecular Dynamics numerical simulations of three-dimensional polar fluids. The simulation cell is defined to be the surface of the northern hemisphere of a four-dimensional (hyper)sphere. The point dipoles are constrained to remain tangent to the sphere and their interactions are derived from the basic laws of electrostatics in this geometry. The dipole-dipole potential has two singularities which correspond to the following boundary conditions: when a dipole leaves the northern hemisphere at some point of the equator, it reappears at the antipodal point bearing the same dipole moment. We derive all the formal expressions needed to obtain the thermodynamic and structural properties of a polar liquid at thermal equilibrium in actual numerical simulation. We notably establish the expression of the static dielectric constant of the fluid as well as the behavior of the pair correlation at large distances. We report and discuss the results of extensive numerical Monte Carlo simulations for two reference states of a fluid of dipolar hard spheres and compare these results with previous methods with a special emphasis on finite size effects.

Maxwell field E in a two-dimensional polar fluid in the presence of an external field E: a Monte-Carlo study.

We study a two-dimensional system of dipolar hard disks in the presence of a uniform external electric field [Formula: see text] by Monte Carlo simulations in a square with periodic boundary conditions. The study is performed in both the fluid at high temperature and the phase of living polymers at low temperature. In the considered geometry the macroscopic Maxwell field [Formula: see text] is computed and found to be equal to the external field [Formula: see text] in both phases. The dielectric properties of the system in the liquid phase as well as in the polymeric phase are investigated.

Ionic fluids: charge and density correlations near gas-liquid criticality.

The correlation functions of an ionic fluid with charge and size asymmetry are studied within the framework of the random phase approximation. The results obtained for the charge-charge correlation function demonstrate that the second-moment Stillinger-Lovett (SL) rule is satisfied away from the gas-liquid critical point (CP) but not, in general, at the CP. However, in the special case of a model without size asymmetry the SL rules are satisfied even at the CP. The expressions for the density-density and charge-density correlation functions valid far from and close to the CP are obtained explicitly.

Liquid-vapor transition of systems with mean field universality class.

We have considered a system where the interaction, v(r)=v(IS)(r)+xi(2)v(MF)(r), is given as a linear combination of two potentials, each of which being characterized with a well-defined critical behavior: for v(IS)(r) we have chosen the potential of the restricted primitive model which is known to belong to the three-dimensional Ising universality class, while for v(MF)(r) we have considered a long-range interaction in the Kac [J. Math. Phys. 4, 216 (1963)] limit, displaying mean field (MF) behavior. We study the performance of two theoretical approaches and of computer simulations in the critical region for this particular system and give a detailed comparison between theories and simulation of the critical region and the location of the critical point. Having shown by theoretical arguments that the system belongs to the MF universality class for any positive value of xi and shows nonclassical behavior only for xi=0, we examine to which extent theoretical approximations and simulation can reproduce this behavior. While in this limiting case theoretical approaches are known to fail, we find good agreement for the critical properties between the theoretical approaches and the simulations for xi(2) larger than 0.05.

Transport coefficients of the Yukawa one-component plasma.

We present equilibrium molecular-dynamics computations of the thermal conductivity and the two viscosities of the Yukawa one-component plasma. The simulations were performed within periodic boundary conditions, and Ewald sums were implemented for the potentials, the forces, and for all the currents which enter the Kubo formulas. For large values of the screening parameter, our estimates of the shear viscosity and the thermal conductivity are in good agreement with the predictions of the Chapman-Enskog theory.