A non-equilibrium Monte Carlo approach to potential refinement in inverse problems

Abstract

The inverse problem for a disordered system involves determining the interparticle interaction parameters consistent with a given set of experimental data. Recently, Rutledge has shown (Phys. Rev. E63, 021111 (2001)) that such problems can be generally expressed in terms of a grand canonical ensemble of polydisperse particles. Within this framework, one identifies a polydisperse attribute (`pseudo-species') $σ$ corresponding to some appropriate generalized coordinate of the system to hand. Associated with this attribute is a composition distribution $\barρ(σ)$ measuring the number of particles of each species. Its form is controlled by a conjugate chemical potential distribution $μ(σ)$ which plays the role of the requisite interparticle interaction potential. Simulation approaches to the inverse problem involve determining the form of $μ(σ)$ for which $\barρ(σ)$ matches the available experimental data. The difficulty in doing so is that $μ(σ)$ is (in general) an unknown {\em functional} of $\barρ(σ)$ and must therefore be found by iteration. At high particle densities and for high degrees of polydispersity, strong cross coupling between $μ(σ)$ and $\barρ(σ)$ renders this process computationally problematic and laborious. Here we describe an efficient and robust {\em non-equilibrium} simulation scheme for finding the equilibrium form of $μ[\barρ(σ)]$. The utility of the method is demonstrated by calculating the chemical potential distribution conjugate to a specific log-normal distribution of particle sizes in a polydisperse fluid.