Participation Ratio for Constraint-Driven Condensation with Superextensive Mass

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 o...

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Bibliographic Details
Published in:Entropy (Basel, Switzerland) Switzerland), 2017-10, Vol.19 (10), p.517
Main Authors: Gradenigo, Giacomo, Bertin, Eric
Format: Article
Language:English
Subjects:
canonical ensemble
Condensation
condensation phenomenon
ensemble inequivalence
large deviations
Random variables
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Summary: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 ( 0Mc . In both cases, a standard indicator of the condensation phenomenon is the participation ratio Yk=⟨∑imki/(∑imi)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∼N1+δ ( δ>0 ), hence interpolating between the unconstrained condensation case (where the typical value of the total mass scales as M∼N1/α for α
ISSN:1099-4300
1099-4300
DOI:10.3390/e19100517