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Estimation of Local Microcanonical Averages in Two Lattice Mean-Field Models Using Coupling Techniques

Kalle Koskinen, Jani Lukkarinen

2020
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Journal of statistical physics
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We consider an application of probabilistic coupling techniques which provides explicit estimates for comparison of local expectation values between label permutation invariant states, for instance, between certain microcanonical, canonical, and grand canonical ensemble expectations. A particular goal is to obtain good bounds for how such errors will decay with increasing system size. As explicit examples, we focus on two well-studied meanfield models: the discrete model of a paramagnet and the
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... paramagnet and the mean-field spherical model of a continuum field, both of which are related to the Curie-Weiss model. The proof is based on a construction of suitable probabilistic couplings between the relevant states, using Wasserstein fluctuation distance to control the difference between the expectations in the thermodynamic limit. Keywords Curie-Weiss · Mean field models · Wasserstein distance · Microcanonical ensemble · Coupling in probability Introduction We consider a novel method of analysis of convergence of local expectation values in probability distributions associated with microcanonical ensembles. Our approach aims at answering the following question which would be natural, for example, to control expectations in states arising in ergodic theory: Assume that the system is in a microcanonical state with one or Communicated by Ivan Corwin. Joel L. Lebowitz has been one of the driving forces and main supporters of mathematical statistical physics for over half a century. It is a particular honour and a pleasure to dedicate this, in relation humble, update on foundations of statistical mechanics to him. Estimation of Local Microcanonical Averages... and, for ε = 0, we have where the microcanonical partition function Z MC (ε, ρ; N ) is defined by the specific microcanonical entropy s MC (ε, ρ) is defined by and the canonical partition function Z C (β, ρ; N ) by It follows that ψ β,ρ is strictly convex and obtains a unique global minimum when ψ β,ρ (ε) = 0. Computing it from the above, we see that ψ β,ρ (ε) = 0 ⇐⇒ ε = − J ρ 2 1 − 1 β J ρ . In particular, we see that for every ε ∈ − J ρ 2 , 0 there exists β ≥ 1 J ρ such that the given ε minimizes ψ β,ρ , and, conversely, for every β ≥ 1 ρ J there exists a minimizing value ε ∈ − ρ J 2 , 0 . Furthermore, if β < 1 ρ J , then ψ β,ρ is strictly negative on the entire interval, and, as a result ψ β,ρ is minimized for ε = 0. For the asymptotics, if β ≥ 1 J ρ , then the asymptotics are standard and we have

doi:10.1007/s10955-020-02612-1
fatcat:r7cvgdgawrhp7jc5nbyjtakq74