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Concavity, Response Functions and Replica Energy

National Center for Radiation Protection and Computational Physics, Istituto Superiore di Sanità, Viale Regina Elena 299, 00161 Roma, Italy
Dipartimento di Fisica e Astronomia and CSDC, Università di Firenze, and INFN, Sezione di Firenze, via G. Sansone 1, 50019 Sesto Fiorentino, Italy
INAF-Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, 50125 Firenze, Italy
Department of Mechanical Engineering, Université de Sherbrooke, Sherbrooke, QC J1K 2R1, Canada
Departament de Física de la Matèria Condensada, Facultat de Física, Universitat de Barcelona, Martí i Franquès 1, 08028 Barcelona, Spain
SISSA, INFN and ISC-CNR, via Bonomea 265, 34136 Trieste, Italy
Author to whom correspondence should be addressed.
Entropy 2018, 20(12), 907;
Received: 28 October 2018 / Revised: 21 November 2018 / Accepted: 21 November 2018 / Published: 28 November 2018
(This article belongs to the Special Issue Applications of Statistical Thermodynamics)
PDF [394 KB, uploaded 28 November 2018]


In nonadditive systems, like small systems or like long-range interacting systems even in the thermodynamic limit, ensemble inequivalence can be related to the occurrence of negative response functions, this in turn being connected with anomalous concavity properties of the thermodynamic potentials associated with the various ensembles. We show how the type and number of negative response functions depend on which of the quantities E, V and N (energy, volume and number of particles) are constrained in the ensemble. In particular, we consider the unconstrained ensemble in which E, V and N fluctuate, which is physically meaningful only for nonadditive systems. In fact, its partition function is associated with the replica energy, a thermodynamic function that identically vanishes when additivity holds, but that contains relevant information in nonadditive systems. View Full-Text
Keywords: long-range interactions; non-additive systems; ensemble inequivalence long-range interactions; non-additive systems; ensemble inequivalence

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Campa, A.; Casetti, L.; Latella, I.; Pérez-Madrid, A.; Ruffo, S. Concavity, Response Functions and Replica Energy. Entropy 2018, 20, 907.

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