Binless estimation of the potential of mean force

For a system in thermal equilibrium, described by classical statistical mechanics, we derive an unbiased estimator for the marginal probability distribution of a coordinate of interest, rho( x). This result provides a "binless" method for estimating the potential of mean force, Phi = -beta (-1) ln rho, eliminating the need to construct histograms or perform numerical thermodynamic integration. In our method, the distribution that we seek to compute is expressed as the sum of a reference distribution, rho 0(x)essentially an initial guess or estimate of rho( x)and a correction term. While the method is valid for arbitrary rho 0, we speculate that an accurate choice of the reference distribution improves the convergence of the method. Using a model molecule, simulated both in vacuum and in solvent, we validate our proposed approach and compare its performance with the histogram and thermodynamic integration methods. We also discuss and validate an extension in which our approach is used in combination with a biasing force, meant to improve uniform sampling of the coordinate of interest.