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Entropy 2017, 19(10), 517; https://doi.org/10.3390/e19100517

Participation Ratio for Constraint-Driven Condensation with Superextensive Mass

Laboratoire Interdisciplinaire de Physique (LIPHY), Université Grenoble Alpes and CNRS, F-38000 Grenoble, France
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Received: 30 August 2017 / Revised: 19 September 2017 / Accepted: 22 September 2017 / Published: 26 September 2017
(This article belongs to the Special Issue Thermodynamics and Statistical Mechanics of Small Systems)
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Abstract

Broadly distributed random variables with a power-law distribution f ( m ) m - ( 1 + α ) are known to generate condensation effects. This means that, when the exponent α lies in a certain interval, the largest variable in a sum of N (independent and identically distributed) terms is for large N of the same order as the sum itself. In particular, when the distribution has infinite mean ( 0 < α < 1 ) one finds unconstrained condensation, whereas for α > 1 constrained condensation takes places fixing the total mass to a large enough value M = i = 1 N m i > M c . In both cases, a standard indicator of the condensation phenomenon is the participation ratio Y k = i m i k / ( i m i ) k ( k > 1 ), which takes a finite value for N when condensation occurs. To better understand the connection between constrained and unconstrained condensation, we study here the situation when the total mass is fixed to a superextensive value M N 1 + δ ( δ > 0 ), hence interpolating between the unconstrained condensation case (where the typical value of the total mass scales as M N 1 / α for α < 1 ) and the extensive constrained mass. In particular we show that for exponents α < 1 a condensate phase for values δ > δ c = 1 / α - 1 is separated from a homogeneous phase at δ < δ c from a transition line, δ = δ c , where a weak condensation phenomenon takes place. We focus on the evaluation of the participation ratio as a generic indicator of condensation, also recalling or presenting results in the standard cases of unconstrained mass and of fixed extensive mass. View Full-Text
Keywords: large deviations; condensation phenomenon; ensemble inequivalence; canonical ensemble large deviations; condensation phenomenon; ensemble inequivalence; canonical ensemble
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Gradenigo, G.; Bertin, E. Participation Ratio for Constraint-Driven Condensation with Superextensive Mass. Entropy 2017, 19, 517.

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