Intrinsic Matching Frustration in Fluctuating Finite Systems

Abstract

We show that matching between interacting constituents is intrinsically incomplete in finite systems due to equilibrium fluctuations. While fluctuations in finite populations are a well-established feature of statistical mechanics, their direct impact on processes requiring simultaneous matching has not been explicitly formulated. Because matching is determined by the smaller of fluctuating populations rather than by their average values, equilibrium fluctuations systematically reduce the number of realizable matching events. We formalize this effect as intrinsic matching frustration (IMF), a universal statistical constraint that arises whenever discrete fluctuating populations must be paired. IMF generates a finite population of intact ("spectator") particles even at equilibrium and suppresses matching-dependent processes relative to expectations based on mean populations. For independent Poisson populations, IMF admits an exact solution based on the Skellam distribution, valid for all particle numbers. More generally, interactions and correlations may modify the detailed fluctuation statistics, but for sufficiently large populations the mismatch is determined solely by the variance of the population imbalance and exhibits universal scaling behavior. IMF therefore represents a generic equilibrium property of finite systems, independent of microscopic rates and interaction mechanisms. The effect provides a universal fluctuation-induced limitation on pairwise association, binding, and matching processes, with implications for chemical reactions, molecular binding, nanoscale quantum systems, and biological environments. By identifying a fundamental fluctuation-induced constraint on matching, IMF complements established descriptions of finite-size and equilibrium fluctuation effects in statistical mechanics.