Statistical mechanics of non-Markovian exclusion processes

Abstract

The Totally Asymmetric Simple Exclusion Process (TASEP) is often considered one of the fundamental models of non-equilibrium statistical mechanics, due to its well understood steady state and the fact that it can exhibit condensation, phase separation and phase transitions in one spatial dimension. As a minimal model of traffic flow it has enjoyed many applications, including the transcription of proteins by ribosomal motors moving along an mRNA track, the transport of cargo between cells and more human-scale traffic flow problems such as the dynamics of bus routes. It consists of a one-dimensional lattice of sites filled with a number of particles constrained to move in a particular direction, which move to adjacent sites probabilistically and interact by mutual exclusion. The study of non-Markovian interacting particle systems is in its infancy, due in part to a lack of a framework for addressing them analytically. In this thesis we extend the TASEP to allow the rate of transition between sites to depend on how long the particle in question has been stationary by using non-Poissonian waiting time distributions. We discover that if the waiting time distribution has infinite variance, a dynamic condensation effect occurs whereby every particle on the system comes to rest in a single traffic jam. As the lattice size increases, so do the characteristic condensate lifetimes and the probability that a condensate will interact with the preceding one by forming out of its remnants. This implies that the thermodynamic limit depends on the dynamics of such spatially complete condensates. As the characteristic condensate lifetimes increase, the standard continuous time Monte Carlo simulation method results in an increasingly large fraction of failed moves. This is computationally costly and led to a limit on the sizes of lattice we could simulate. We integrate out the failed moves to create a rejection-free algorithm which allows us to see the interacting condensates more clearly. We find that if condensates do not fully dissolve, the condensate lifetime ages and saturates to a particular value. An unforeseen consequence of this new technique, is that it also allowed us to gain a mathematical understanding of the ageing of condensates, and its dependence on system size. Using this we can see that the fraction of time spent in the spatially complete condensate tends to one in the thermodynamic limit. A random walker in a random force field has to escape potential wells of random depth, which gives rise to a power law waiting time distribution. We use the non-Markovian TASEP to investigate this model with a number of interacting particles. We find that if the potential well is re-sampled after every failed move, then this system is equivalent to the non-Markovian TASEP. If the potential well is only re-sampled after a successful move, then we restore particle-hole symmetry, allow condensates to completely dissolve, and the thermodynamic limit spends a finite fraction of time in the spatially complete state. We then generalised the non-Markovian TASEP to allow for particles to move in both directions. We find that the full condensation effect remains robust except for the case of perfect symmetry.