The relationship between the structure of a fluid and the underlying pair potential between its particles is at the heart of liquid-state physics. The structure of fluids is often characterised by the pair distribution function, which in the limit of infinite dilution is directly related to the pair potential. However, at concentrations beyond this limit, this ‘inverse problem’ represents a long-standing challenge. While it has been shown for pairwise additive systems that a given pair distribution function arises from a unique pair potential [1], the practical step of inverting structural data to obtain this pair potential remains challenging.

Here, we solve the inverse problem by matching the pair distribution function obtained by our recently developed method [2] based on test-particle insertion [3, 4] with that from the traditional distance-histogram method. Crucially, the test-particle insertion method requires exact knowledge of the pair potential, and therefore provides an elegant and model-free method for measuring the pair potential from representative configurations of the fluid. We first demonstrate the accuracy and versatility of our method in simulations using a wide range of pair potentials, before applying it experimentally to a colloidal fluid composed of superparamagnetic particles, obtaining the anticipated dipolar repulsion [5]. As well as providing more quantitative comparisons between colloidal experiments and results from liquid-state theory, the method will enable experimental investigations into many-body interactions and allow for coarse-graining of interactions from simulations.

[1] R. Henderson, Phys. Lett. A 49, 197 (1974)

[2] A. E. Stones, R. P. A. Dullens, and D. G. A. L. Aarts, J. Chem. Phys. 148, 241102 (2018)

[3] B. Widom, J. Chem. Phys. 39, 2808 (1963)

[4] J. R. Henderson, Mol. Phys. 48, 389 (1983)

[5] A. E. Stones, R. P. A. Dullens and D. G. A. L. Aarts, (under review) arXiv:1901.04960 [cond-mat.soft] (2019)