Finite-size localization scenarios in condensation transitions

Abstract

We consider the phenomenon of condensation of a globally conserved quantity $H={\sum }_{i=1}^N{\epsilon }_i$ distributed on $N$ sites, occurring when the density $h=H/N$ exceeds a critical density $h_c$. We numerically study the dependence of the participation ratio $Y_2=⟨{\epsilon }_i^2⟩/(N{h}^2)$ on the size $N$ of the system and on the control parameter $\delta =(h-h_c)$, for various models: (i)~a model with two conservation laws, derived from the Discrete NonLinear Schrödinger equation; (ii)~the continuous version of the Zero Range Process class, for different forms of the function $f(\epsilon )$ defining the factorized steady state. Our results show that various localization scenarios may appear for finite $N$ and close to the transition point. These scenarios are characterized by the presence or the absence of a minimum of $Y_2$ when plotted against $N$ and by an exponent $\gamma \ge 2$ defined through the relation $N^{\ast }\simeq {\delta }^{-\gamma }$, where $N^{\ast }$ separates the delocalized region ($N\ll N^{\ast }$, $Y_2$ vanishes with increasing $N$) from the localized region ($N\gg N^{\ast }$, $Y_2$ is approximately constant). We finally compare our results with the structure of the condensate obtained through the single-site marginal distribution.

arXiv commentThe Introduction has been rewritten. Accepted for publication in Physical Review E

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• Localization in boundary-driven lattice models
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• Relaxation dynamics and finite-size effects in a simple model of condensation
• First-order condensation transition in the position distribution of a run-and-tumble particle in one dimension
• Localization and "classical entanglement'' in the Discrete Non-Linear Schrödinger Equation
• Infinite-temperature thermostats by energy localization in a nonequilibrium setup