An introduction to the thermodynamic and macrostate levels of nonequivalent ensembles

Touchette, Hugo

doi:10.1016/s0378-4371(04)00408-x

Cited by 5 publications

(14 citation statements)

References 23 publications

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“…By Lemma 2.2 and Proposition 2.3, R c ∈ C (0, ∞) 2 , D ρ so the transition is continuous (second order), which is directly related to the fact that p ∈ C 1 (D µ , R). This is in contrast to the case for systems with bounded local state space, where we would have D µ = R 2 and non-differentiability of p would lead to a first-order phase transition with discontinuous order parameter [13,14].…”

Section: Equivalence Of Ensemblesmentioning

confidence: 97%

“…we have partial equivalence of ensembles in the sense of [14]. By M every condensed phase region is mapped on ∂ D µ ∩ D µ .…”

Section: Phase Diagrammentioning

confidence: 99%

“…Since both components of R(µ * ) − ρ are negative, the half-space E * contains the cone ∆(µ * ), as given in (14). The boundary ∂ E * is a straight line.…”

Section: Proofs Of Sectionmentioning

confidence: 99%

“…A common feature of these models is a bounded Hamiltonian, which corresponds to D µ = R 2 in the above setting. In this case, phase separation is a consequence of long-range correlations, leading to non-differentiability of the Gibbs free energy (or non-convex canonical entropy) and a first-order transition [13,14]. In our case there are no spatial correlations, but typically D µ R 2 due to the unbounded local state space, and condensation is a result of large deviation properties of ν 1 µ on the boundary of D µ , where it turns out to be subexponential.…”

Section: Introductionmentioning

confidence: 99%

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Equivalence of ensembles for two-species zero-range invariant measures

Großkinsky

2008

Stochastic Processes and their Applications

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We study the equivalence of ensembles for stationary measures of interacting particle systems with two conserved quantities and unbounded local state space. The main motivation is a condensation transition in the zero-range process which has recently attracted attention. Establishing the equivalence of ensembles via convergence in specific relative entropy, we derive the phase diagram for the condensation transition, which can be understood in terms of the domain of grand-canonical measures. Of particular interest, also from a mathematical point of view, are the convergence properties of the Gibbs free energy on the boundary of that domain, involving large deviations and multivariate local limit theorems of subexponential distributions.

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Section: Equivalence Of Ensemblesmentioning

confidence: 97%

“…we have partial equivalence of ensembles in the sense of [14]. By M every condensed phase region is mapped on ∂ D µ ∩ D µ .…”

Section: Phase Diagrammentioning

confidence: 99%

“…Since both components of R(µ * ) − ρ are negative, the half-space E * contains the cone ∆(µ * ), as given in (14). The boundary ∂ E * is a straight line.…”

Section: Proofs Of Sectionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Equivalence of ensembles for two-species zero-range invariant measures

Großkinsky

2008

Stochastic Processes and their Applications

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“…In this and in the next section, we analyze the relation between ensemble inequivalence and the occurrence of negative response functions. Ensemble inequivalence can be studied with the help of the properties of the Legendre–Fenchel transformation; this approach, already well documented for constrained ensembles [ 24 , 25 , 26 ], can be extended to the case of the unconstrained ensemble. It is the Legendre–Fenchel transformation that allows one to connect ensemble inequivalence and negative response functions.…”

Section: Response Functions and Ensemble Inequivalencementioning

confidence: 99%

Concavity, Response Functions and Replica Energy

Campa

Casetti

Latella

et al. 2018

Entropy

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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 to 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, physically meaningful only for nonadditive systems. In fact, its partition function is associated to the replica energy, a thermodynamic function that identically vanishes when additivity holds, but that contains relevant information in nonadditive systems.

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Ensemble Equivalence for Mean Field Models and Plurisubharmonicity

Berman¹

2022

Arch Rational Mech Anal

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We show that entropy is globally concave with respect to energy for a rich class of mean field interactions, including regularizations of the point vortex model in the plane, plasmas and self-gravitating matter in 2D, as well as the higher-dimensional logarithmic interactions appearing in conformal geometry and power laws. The proofs are based on a corresponding “microscopic” concavity result at finite N, shown by leveraging an unexpected link to Kähler geometry and plurisubharmonic functions. Under more restrictive homogeneity assumptions, strict concavity is obtained using a uniqueness result for free energy minimizers, established in a companion paper. The results imply that thermodynamic equivalence of ensembles holds for this class of mean field models. As an application, it is shown that the critical inverse negative temperatures—in the macroscopic as well as the microscopic setting—coincide with the asymptotic slope of the corresponding microcanonical entropies. Along the way, we also extend previous results on the thermodynamic equivalence of ensembles for continuous weakly positive definite interactions, concerning positive temperature states, to the general non-continuous case. In particular, singular situations are exhibited where, somewhat surprisingly, thermodynamic equivalence of ensembles fails at energy levels sufficiently close to the minimum energy level.

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An introduction to the thermodynamic and macrostate levels of nonequivalent ensembles

Cited by 5 publications

References 23 publications

Equivalence of ensembles for two-species zero-range invariant measures

Equivalence of ensembles for two-species zero-range invariant measures

Concavity, Response Functions and Replica Energy

Ensemble Equivalence for Mean Field Models and Plurisubharmonicity

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