Participation ratio for constraint-driven condensation with superextensive mass

Abstract

Broadly distributed random variables with a power-law distribution $f(m)\sim m^{-(1+\alpha )}$ are known to generate condensation effects, in the sense that, when the exponent $\alpha$ 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 ($0<\alpha <1$) one finds unconstrained condensation, whereas for $\alpha >1$ constrained condensation takes places fixing the total mass to a large enough value $M={\sum }_{i=1}^N{m}_i>M_c$. In both cases, a standard indicator of the condensation phenomenon is the participation ratio $Y_k=⟨{\sum }_i{m}_i^k/({\sum }_i{m}_i{)}^k⟩$ ($k>1$), which takes a finite value for $N\to \infty$ 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 $M\sim N^{1+\delta }$ ($\delta >0$), hence interpolating between the unconstrained condensation case (where the typical value of the total mass scales as $M\sim N^{1/\alpha }$ for $\alpha <1$) and the extensive constrained mass. In particular we show that for exponents $\alpha <1$ a condensate phase for values $\delta >{\delta }_c=1/\alpha -1$ is separated from a homogeneous phase at $\delta <{\delta }_c$ by a transition line, $\delta ={\delta }_c$, 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.

11 pages, 2 figures, to appear in Entropy

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• Probability distributions with singularities
• First-order condensation transition in the position distribution of a run-and-tumble particle in one dimension