Let $$P_n$$ P n and $$Q_n$$ Q n be two probability measures representing two different probabilistic models of some system (e.g., an n-particle equilibrium system, a set of random graphs with n vertices, or a stochastic process evolving over a time n) and let $$M_n$$ M n be a random variable representing a “macrostate” or “global observable” of that system. We provide sufficient conditions, based on the Radon–Nikodym derivative of $$P_n$$ P n and $$Q_n$$ Q n , for the set of typical values of $$M_n$$ M n obtained relative to $$P_n$$ P n to be the same as the set of typical values obtained relative to $$Q_n$$ Q n in the limit $$n\rightarrow \infty $$ n → ∞ . This extends to general probability measures and stochastic processes the well-known thermodynamic-limit equivalence of the microcanonical and canonical ensembles, related mathematically to the asymptotic equivalence of conditional and exponentially-tilted measures. In this more general sense, two probability measures that are asymptotically equivalent predict the same typical or macroscopic properties of the system they are meant to model.
Journal of Statistical Physics – Springer Journals
Published: Feb 1, 2018
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