A Monte-Carlo approach to Poisson–Boltzmann like free-energy functionals

A simple technique is proposed for numerically determining equilibrium ion distribution functions belonging to free energies of the Poisson–Boltzmann type. The central idea is to perform a conventional Monte-Carlo simulation using the free energy as the “Hamiltonian” entering the Metropolis criterion and the spatially discretized density as degrees of freedom. This approach is complementary to the possibility of numerically solving the differential equations corresponding to the variational problem, but it is much easier to implement and to generalize. Its utility is demonstrated in two examples: valence mixtures and hard core interactions of ions surrounding a charged rod.