Asymptotic Behavior of a Sequence of Conditional Probability Distributions and the Canonical Ensemble

TL;DR

This paper establishes a limit law for conditional probability distributions of subsystems, explaining the canonical ensemble and extending it to strongly interacting systems, with implications for understanding temperature baths.

Contribution

It introduces a new theorem describing the asymptotic behavior of conditional distributions, applicable even with strong interactions, broadening the theoretical foundation of the canonical ensemble.

Findings

Limit law for conditional distributions involving exponential weighting

Extension of the canonical ensemble to strongly correlated systems

Probabilistic characterization of temperature baths

Abstract

The probability distribution of a function of a subsystem conditioned on the value of the function of the whole, in the limit when the ratio of their values goes to zero, has a limit law: It equals the unconditioned marginal probability distribution weighted by an exponential factor whose exponent is uniquely determined by the condition. We apply this theorem to explain the canonical equilibrium ensemble of a system in contact with a heat reservoir. Since the theorem only requires analysis at the level of the function of the subsystem and reservoir, it is applicable even without the knowledge of the composition of the reservoir itself, which extends the applicability of the canonical ensemble. Furthermore, we generalize our theorem to a model with strong interaction that contributes an additional term to the exponent, which is beyond the typical case of approximately additive functions. This result is new in both physics and mathematics, as a theory for the Gibbs conditioning principle for strongly correlated systems. A corollary provides a precise formulation of what a temperature bath is in probabilistic term