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In statistical physics, the challenging combinatorial enumeration of theconfigurations of a system subject to hard constraints (microcanonicalensemble) is mapped to a mathematically easier calculation where theconstraints are softened (canonical ensemble). However, the mapping is exactonly when the size of the system is infinite and if the property of ensembleequivalence (EE), i.e. the asymptotic identity of canonical and microcanonicallarge deviations, holds. For finite systems, or when EE breaks down,statistical physics is currently believed to provide no answer to thecombinatorial problem. In contrast with this expectation, here we establishexact relationships connecting conjugate ensembles in full generality, even forfinite system size and when EE does not hold. We also show that in thethermodynamic limit the ensembles are directly related through the matrix ofcanonical (co)variances of the constraints, plus a correction term thatsurvives only if this matrix has an infinite number of finite eigenvalues.These new relationships restore the possibility of enumerating microcanonicalconfigurations via canonical probabilities, thus reconnecting statisticalphysics and combinatorics in realms where they were believed to be no longer inmutual correspondence.

Reconnecting statistical physics and combinatorics beyond ensemble equivalence