Analyses of two-channel asymmetric simple exclusion processes

Abstract

Inspired by the growing scientific needs of connectivity in today’s world, this thesis has primarily focused on understanding the complexities in two-channel transport systems. A large number of the real-world processes comprise of directed motion of particles along more than one channel. For example, highway traffic involves vehicles moving in several lanes with interchange between lanes. Even the microtubules1 , the protofilaments inside our cells, form a multi-channel system, which may involve different hopping and boundary rates. More recently, multilane systems have attracted the attention of the scientists working in the field of statistical physics. A discrete lattice gas model, namely asymmetric simple exclusion process (ASEP), which is the simplest paradigmatic stochastic model, has been adopted as a base camp model. Keeping in mind the numerous occurrences of systems with open boundaries, our aim is to analyze the collective properties of driven diffusive two-channel systems, in which each lattice is connected with boundary reservoirs at both the ends and particles obey hard-core exclusion principle. We adopt mean-field theory, which ignore particle-particle correlations, and its variants as the fundamental theoretical approximation. A decent agreement of mean-field results with numerical Monte Carlo simulations is seen, which ultimately validates the theoretically predicted phase behavior. The analysis allows for fruitful insights into the mechanisms of non-equilibrium physical phenomena, which have been identified for different models studied in this thesis. In the first part of this thesis, we examine a two-channel partially asymmetric simple exclusion process (PASEP), which has been studied to fill the gaps in the literature of two-channel ASEPs. With the modified and realistic dynamical rules, we try to derive a complete and comprehensive description about the steady-state behavior of two-channel exclusion processes. We employ vertical cluster mean-field theory (VCMFT) to find the steady-state phase diagrams and density profiles. It is found that the topology of phase diagram varies qualitatively as well as quantitatively under symmetric, fully asymmetric and partially asymmetric coupling conditions. The second part deals with a two-channel TASEP with Langmuir kinetics (LK), in which the additional dynamics of particle attachment and detachment are also included to mimic the situation in intracellular transport. This challenging problem under asymmetric coupling conditions has been solved using continuum mean-field approximation along with singular perturbation technique. The stationary phase diagrams, also describing the nature of boundary layers, have been derived and many non-equilibrium phenomena such as phase coexistence, the presence of a localized shock, formation of a kink in boundary layer and synchronization of shocks in both the lanes at higher magnitudes of lane-changing rates are thoroughly investigated. In the last part of this thesis, the two-channel model attempts to capture the effect of an inhomogeneity in an otherwise homogeneous two-channel transport system. Such situations may arise frequently in vehicular traffic due to certain reasons such as an accident, construction lane, road conditions etc., in protein synthesis where slow moving codons affect the overall ribosome translation rate. The steady-state phase diagrams, due to the effect of bottleneck, show very interesting non-equilibrium features such as mixed phases and the presence of a bottleneck-induced shock. We find that an increase in symmetric lane-changing rate weakens the bottleneck effect. However, the case of fully asymmetric coupling, the effect of coupling strength is found to be dependent on the type of asymmetricity in lane-changing rules. Moreover, we also identify the turning effect in the position of bottleneck-induced shock, which eventually turns out to be a finite-size effect.