Fixed-configuration inversion of interaction potentials from equilibrium configurations at fixed state points

Recovering interaction potentials from structural information is a fundamental inverse problem in statistical physics and soft-matter research. We investigate a force-sampling implementation of fixed-configuration inversion for particle-resolved equilibrium configurations. A distance-histogram estimate from 10 000 configurations provides a fixed, potential-independent target, whereas forces recalculated from the current trial potential on fixed coordinate subsets generate smooth, potential-dependent responses. Disjoint training and validation subsets of 200 configurations each enable validation-controlled checkpoint selection and termination without generating new equilibrium trajectories. We test the method for a three-dimensional Lennard-Jones system at reduced inverse temperature β*=ε/kBT=1.0 and reduced number densities ρ*=ρσ3 of 0.01, 0.70, 0.75, 0.80, 0.85, and 0.90, where ρ is the number density and ε and σ are the energy and length scales. Over the reliable radial interval, the inversion recovers the principal features of the reference potential and reproduces the target structure within statistical resolution. A trajectory block bootstrap that repeats the complete inversion quantifies uncertainty in the target radial distribution function, validation response, and recovered potential, identifying sparse low-density pair statistics and limited pair occupancy near the repulsive core as important sources. Independent forward molecular dynamics simulations at ρ*=0.80 and 0.90 confirm reproduction under renewed sampling. At ρ*=0.80, a matched comparison showed an ∼204-fold reduction in wall-clock cost relative to conventional iterative Boltzmann inversion under the present implementations and validation-controlled stopping protocol. These results establish force sampling as an efficient, statistically controlled response estimator for fixed-configuration pair-potential inversion.