Steady-State Analysis of Dynamically Disordered Asymmetric Simple Exclusion Process

Abstract

This thesis focuses primarily on comprehending the intricacies of stochastic transport systems, in motivation with the increasing scientific needs for optimized and efficient traffic. A significant number of real world processes include directed movement of particles along the lattice channels. Several randomly evolving analogous processes include traffic on highways which involve vehicle motion between different lanes and molecular motors moving along multi-lane microtubules. The motion in such processes may comprise different hopping rates and boundary conditions. The particular problem of congestion caused by dynamic defects in transport systems is gaining attraction of researchers and scientists in the growing field of statistical physics. Totally asymmetric simple exclusion process (TASEP) belongs to a special class of the driven lattice gas model and is considered to be a paradigmatic nonequilibrium model to study stochastic transport processes. We focus to analyze collective properties of diffusive systems with open boundaries in single-channel as well as two-channel, in which the lattice is connected to boundary reservoirs at either ends. The particles obey hard-core exclusion principle, in which no two particles can occupy same lattice site. There also exists dynamic defects in the system which can stochastically bind and unbind from a lattice site with certain rates. For theoretical investigations, we adopt simple mean-field approximation theory and its variant- vertical cluster mean-field theory to derive continuum master equations. The continuum equations are solved using finite difference numerical scheme. To further validate the theoretical findings, we run extensive Monte Carlo simulation which shows a good agreement with theoretical findings. The study provides valuable insights into the underlying mechanisms of physical phenomena in non-equilibrium and have been identified for the various models examined in this thesis. In the first model studied in this thesis, we investigate a two-channel asymmetric simple exclusion process with site-wise dynamic disorder under symmetric and asymmetric coupling environment. The study is an extension to literature on disordered systems by analyzing a general coupled TASEP model. The model presented here well explains the hindrance in biological process of gene transcription. We made an attempt to provide a comprehensive depiction of the stationary dynamics of a disordered two-channel exclusion process. The dynamic defects obey un-constrained dynamics, which allows a defect to bind to or unbind from a lattice site regardless of particle presence. It is found that the rich topology of steady-state phase diagram differs qualitatively as well as quantitatively subject to symmetric and fully asymmetric coupling conditions. The second model deals with a dynamically disordered TASEP model with LK. The inclusion of LK, inspite of adding more realism into the model, brings along many mathematical challenges as well. The analysis is explicitly categorized for two different values of binding constant. The competing interplay between dynamic disorder and LK discovers various non-equilibrium phenomena like phase coexistence and appearance of localised shock which otherwise are not found in ddTASEPs without LK. The shock dynamics are explored through varying defect density and defect strength. In the last part, we study a coupled two-channel exclusion process with dynamic disorder and LK. The model is inspired from real transport situation like periodic switching of traffic lights on roads and stochastic binding of motor proteins to microtubule. The complex problem considers two different couplings- symmetric and asymmetric environments. Under symmetric coupling conditions, the twochannel ddTASEP with LK behaves similar to its analogue in single-channel. Under fully asymmetric coupling environment, the system is studied for two different values of binding constant. The rich steady-state phase diagram, due to the effect of lane-changing, shows very interesting non-equilibrium feature of mixed phases and

phase reduction. The current-density relation plot infers that the maximum current reduces with an increase in defect density. Further, we observe that the effect of coupling weakens for higher values of attachment rate, signifying LK dominance.