The Reweighting Principle in Statistical Mechanics

Abstract

Reweighting of probability measures provides a unifying perspective on ensemble transformations in statistical mechanics. We distinguish two complementary classes of reweighting: soft constraints, which redistribute probability while preserving the support of the reference measure, and hard constraints, which impose support restrictions through conditioning. We show that exponential tilting and conditioning on an exact observable value arise as the minimum relative entropy updates associated with soft expectation constraints and hard exact-value constraints, respectively. Their relative entropies naturally inherit complementary thermodynamic structures: exponential tilting gives rise to the Legendre structure of the canonical ensemble and reduces, for a uniform reference measure, to Gibbs entropy, whereas conditioning reduces to Boltzmann entropy through the surprisal of the constrained macrostate. By introducing an enlarged probability space in which observables are treated as explicit random variables, we further show that canonical and microcanonical ensembles arise as marginal and conditional distributions of a common joint reference measure. In the thermodynamic limit, large-deviation concentration makes soft and hard constraints macroscopically equivalent, providing a probabilistic interpretation of canonical--microcanonical ensemble equivalence. Finally, we outline how the same information-theoretic framework naturally extends to path space, suggesting a unified probabilistic description of equilibrium statistical mechanics and conditioned stochastic dynamics.