Analysis of nonequilibrium steady-states

Abstract

Non-equilibrium is the state of the almost all systems in the universe. Unlike equilibrium systems, they interfere with their surroundings which results in never ceasing uxes. There is no unified theory to understand these systems, since their complexity have no bounds. However, there is a restricted subset of them, namely a steady state, in which system maintains constant uxes and its macroscopic observables are not changing in time. Majority of the non-equilibrium problems that the scientific community is interested in comprise systems at steady states or the way such systems relax to steady states, due to their relative ease of analysis. Steady states of Totally Asymmetric Simple Exclusion Processes (TASEPs) are the main focus of this dissertation. We analyze them through Monte Carlo (MC) simulations. The technique is basically a computational experiment done by utilizing random numbers. Performing a computational experiment is a natural way to study these systems since most of the time they are still too complex to have analytical solutions. We present MC simulation results of our studies on the response of TASEP steady states to sinusoidal boundary oscillations. Typically over-damped systems, such as TASEPs, give monotonous frequency response to sinusoidal driving. However, there are exceptions to these all which draw significant attention from the community, e.g., stochastic resonance. We report a novel resonance phenomena on over-damped systems. We present our results in two different but related works. In our first work, we study the motion of shock profiles of TASEP with single class of particles under oscillatory boundary conditions using MC analysis. We also model its dynamics as a Fokker-Planck (FP) system, which incorporates a retarded-oscillatory force with a static single well potential. We solve the FP system by numerical integration. We showed that amplitudes of statistical quantities in both of these systems, (e.g., average position), display resonant effects and their results are qualitatively very similar. In our second work, we showed that by periodically manipulating the boundary conditions of TASEP with two classes of particles, we can achieve otherwise unreachable states of the system by the same parameters. We also report the hysteresis behavior in the same system, existence of which leads to the identifi- cation of typical velocity of the system. All these phenomena are the results of resonant response of the particle number density of the system.