The Reweighting Principle in Statistical Mechanics

Reweighting of probability measures provides a unifying perspective on conditioning, exponential tilting, and, more generally, ensemble transformations in statistical mechanics. We show that exponential tilting and conditioning arise as the minimum-relative-entropy updates associated with soft and hard constraints, respectively. Their relative entropies naturally inherit complementary thermodynamic structures: exponential tilting gives rise to the Legendre structure of the canonical ensemble and reduces to the Gibbs entropy for a uniform reference measure, while conditioning reduces to the Boltzmann entropy through the surprisal of the constrained macrostate. By introducing an enlarged probability space in which observables are treated as explicit variables, we further show that microcanonical and canonical ensembles arise as conditional and marginal distributions of a common structural prior after exponential reweighting. In the thermodynamic limit, described through large deviation theory, conditioning emerges from exponential tilting by concentration of measure, revealing ensemble equivalence as a consequence of entropy--bias competition. Finally, we outline how the same information-theoretic framework naturally extends to path space, suggesting a unified probabilistic description of equilibrium thermodynamics and conditioned stochastic dynamics.