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Despite more than 40 years of research in condensed-matter physics,state-of-the-art approaches for simulating the radial distribution function(RDF) g(r) still rely on binning pair-separations into a histogram. Suchmethods suffer from undesirable properties, including subjectivity, highuncertainty, and slow rates of convergence. Moreover, such problems goundetected by the metrics often used to assess RDFs. To address these issues,we propose (I) a spectral Monte Carlo (SMC) method that yields g(r) as ananalytical series expansion; and (II) a Sobolev norm that assesses the qualityof RDFs by quantifying their fluctuations. Using the latter, we show that,relative to histogram-based approaches, SMC reduces by orders of magnitude boththe noise in g(r) and the number of pair separations needed for acceptableconvergence. Moreover, SMC reduces subjectivity and yields simple,differentiable formulas for the RDF, which are useful for tasks such ascoarse-grained force-field calibration via iterative Boltzmann inversion.

Beyond histograms: efficiently estimating radial distribution functions via spectral Monte Carlo