Study On The Complex Phase Transitions Of Asymmetric Simple Exclusion Processes

Since the first use in1968for describing the kinetics of biopolymerization, asymmetricsimple exclusion processes (ASEPs), which are discrete non-equilibrium models that describe the stochastic dynamics of multi-particle transport across one-dimensional lattices, have attracted great interests from physicists, biologists and chemists. This simple transport model provides the first crucial steps towards the modeling of realistic processes, such as protein synthesis, surface growth, the motion of motor proteins along cytoskeleton filaments, and vehicular traffic. On the other hand, ASEP is a paradigmatic example of nonequilibrium statistical mechanics. A model with possibly the simplest of rules, it nevertheless displays a rich variety of non-equilibrium statistical systems(NESSs), including boundary-induced phase transitions, shock formation, spontaneously symmetry breaking. ASEP is proved having great academic value in understanding of NESS. In this paper, our research work fouce on coupling ASEP model. We develop the basic one-dimensinal ASEP by two strategies:(i) Couple ASEP with other processes, such as LK or KLS model. Details about these work are presented in Chapter Two and Three;(ii) Investigate ASEP in two or multiple dimensions. We study both strong and weak coupling of multiple PASEPs in Chapter Four and Five. The contents of the paper are as follows:In Chapter two we present a one-dimensional totally asymmetric exclusion process with particles adsorption/desorption in the bulk, at the same time considering a global constraint on the total number of particles. In this model, the entry rate and attachment rate of particles into the lattice are influenced by the particle number available in the reservoir. The phase diagrams of model in different total particle numbers are shown. It is interesting to find that phase coexistence can be observed and phase diagrams are affected by constant supply of particles. A mean-field approach is used to interpret the numerical results obtained by Monte Carlo simulations.In Chapter three we study a periodic driven diffusive system, which separates into two equal-sized parts. Particles move with different dynamic rules:in part one particles move as normal TASEP model; in part two particles are govered by KLS model, in which hopping rate of particles depend on the particle occupation of the nearest and nextnearest-neighbor site. Competition of two different driven parts leads to various bulk-driven phase transitions, including shock and anti-shock. In addition, for the symmetric scenario, one can observe shock and anti-shock simultaneously in the system. We have explained the coexistence of shock and anti-shock via effective boundary reservoir density. Theoretical analysis has been carried out to characterize the emerging nonequilibrium steady states, which is in good agreement with Monte Carlo simulations.Chapter four focuses on asymmetric strong coupling of multiple PASEPs, in which particles on lane i could move forward and backward with different rate, p. and qi seperately. Particles could also jump fully asymmetrically from lane i to lane i+1. Firstly, we study asymmetric strong coupling effect in two parallel exclusion processes, which is a generalization of previous work of Kolomeisky group. It is shown that with different configurations of hopping rates, the two-lane system exhibits diverse phase diagrams and density profiles. Specifically, it shows how the phase diagram changes from having seven phases, via six phases, to three phases. Moreover, it shows that the phase diagram could have only one phase, which exhibits quadrilateral or triangular density profile. The vertical cluster mean-field approach is used to get the stationary-state bilk densities and phase diagrams, which explicitly takes into account the correlations inside the vertical cluster of lattice site. Then we consider the asymmetric strong coupling of three-lane PASEPs. With different magnitude order of value pi-qi, the geometric structures of phase diagrams are diverse. We analyze the details of each situation with both Monte Carlo simulations and mean-field approach. Based on our reaserch about two and three-lane PASEPs, we try to generalize the principles of phase diagrams to multiple PASEPs. However, we only succeed in concluding the general rule when Pi-qi=Pi-qi (pi-qi≠pj-qj) are satisfied for all random pair i, j.Besides of the strong coupling, we also study the weak coupling in multiple asymmetric exclusion process. Weak coupling means the lane-change rates ωA,ωB are are inversely proportional to size L. Firstly we review the weak coupling models in two-lane ASEPs, including both parallel and antiparallel. Similar Theoretical analysis is adopted:based on the main master equation, develop differential equations to describe the density profiles. With the proper boundary conditions, the density profiles and phase diagrams are constructed in the hydrodynamic limit, by numerically integrating the differential equations. We study the weak coupling in a three lane parallel ASEPs. The phase diagram is composed of ten areas, including phase LLL, phase LLS, phase LLH, phase LSS, phase HSL, phase SSH, phase LHH, phase SHH, phase HHH and region S. Normally shock enters the system from boundaries while in region S a shock is shaped in one site of lattce. And in region S we have two shocks which coexist simultaneously in one lane. The region S is new to our knowledge so far. For0.5point is a singular point when we integrate differential equations and we do the integrations separately from two boundaries to the singular point.