Record Dynamics in the Parking Lot Model

TL;DR

This paper analyzes the parking lot model as a kinetically constrained system exhibiting aging, record fluctuations, and intermittent quakes, providing insights into granular relaxation and dense colloid dynamics.

Contribution

It establishes the parking lot model as a kinetically constrained model with aging behavior and characterizes its dynamics through record fluctuations and quakes.

Findings

Active cluster length grows logarithmically with time.

Number of active clusters decreases as system approaches equilibrium.

Quakes are uncorrelated and follow a Poisson process in logarithmic time.

Abstract

We present an analytical and numerical study of the parking lot model (PLM) of granular relaxation and make a connection to the aging dynamics of dense colloids. As we argue, the PLM is a Kinetically Constrained Model which features astronomically large equilibration times and displays a characteristic aging behavior on all observable time scales. The density of parked cars displays quasi-equilibrium Gaussian fluctuations interspersed by increasingly rare intermittent events, quakes, which can lead to an increase of the density to new record values. Defining active clusters as the shortest domains of parked cars which must be re-arranged to allow further insertions, we find that their typical length grows logarithmically with time for low enough temperatures and show how the number of active clusters on average gradually decreases as the system approaches equilibrium. We further characterize the aging process in terms of the statistics of the record sized fluctuations in the interstitial free volume which lead to quakes and show that quakes are uncorrelated and that they can be approximately described as a Poisson process in logarithmic time.