Abstract
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.
Highlights
Additivity can be defined in very simple terms for physical systems
We have seen that ensemble inequivalence is connected with the occurrence of negative response functions, and that these anomalous responses are in turn associated with anomalous concavity properties of the thermodynamic functions
We note that all these response functions concern the variation of a constraint variable (E, V or N) with respect to the respective conjugate thermodynamic variable (T, P and μ, respectively)
Summary
Additivity can be defined in very simple terms for physical systems. a system is said to be additive if, thought as the union of several parts, the energy of interaction between the parts is negligible with respect to the total energy [1,2]. In the statistical mechanics formalism, this is expressed by the equivalence of the ensembles This equivalence is in general absent in nonadditive systems, and physically this implies that there are equilibrium states defined by given control parameters that do not exist if one chooses another set of control parameters. We have reminded above that ensemble inequivalence is associated with the fact that, for nonadditive systems, the possible equilibrium configurations depend on the specific control parameters used to define its state. Α/d)W, showing that the replica energy vanishes when α → d, which corresponds to the limit of an additive system Another remarkable example is the extended Thirring model [22], representing the physics of self-gravitating systems, for which the potential energy is.
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