2017/08/31 by Giacomo Gradenigo, Eric Bertin · 2 citations
Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Condensation #Distribution (mathematics) #Exponent #Independent and identically distributed random variables #Order (exchange) #Random variable #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #Value (mathematics) #Variable (mathematics) #cond-mat.stat-mech
paper · pdf · doi:10.3390/e19100517
published as Entropy 19, 517 (2017) · 11 pages, 2 figures, to appear in Entropy
openalex created_date 2017/09/15 · arxiv created 2017/09/22 · openalex publication_date 2017/09/26 · arxiv updated 2017/11/22 · openalex updated_date 2026/07/28
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.