Abstract
We study the equivalence of microcanonical and canonical ensembles in continuous systems, in the sense of the convergence of the corresponding Gibbs measures and the first order corrections. We are particularly interested in extensive observables, like the total kinetic energy. This result is obtained by proving an Edgeworth expansion for the local central limit theorem for the energy in the canonical measure, and a corresponding local large deviations expansion. As an application we prove a formula due to Lebowitz–Percus–Verlet that express the asymptotic microcanonical variance of the kinetic energy in terms of the heat capacity.
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Notes
For a complex number z such that \(|z|<1\), we define \(\log (1+z) = \sum _n\frac{(-z)^n}{n}\).
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Acknowledgements
We thank Joel Lebowitz for pointing our attention to the microcanonical fluctuation formula of reference [8], that motivated the present work. This work has been partially supported by the European Advanced Grant Macroscopic Laws and Dynamical Systems (MALADY) (ERC AdG 246953) and by ANR grant LSD (ANR-15-CE40-0020-01). N.C. thanks Ceremade, Université Paris Dauphine for the kind hospitality.
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Cancrini, N., Olla, S. Ensemble Dependence of Fluctuations: Canonical Microcanonical Equivalence of Ensembles. J Stat Phys 168, 707–730 (2017). https://doi.org/10.1007/s10955-017-1830-y
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DOI: https://doi.org/10.1007/s10955-017-1830-y