Modelling of transport processes: Theory and simulations

Weak Pinning and Long-Range Anticorrelated Motion of Phase Boundaries in Driven Diffusive Systems

We show that domain walls separating coexisting extremal current phases in driven diffusive systems exhibit complex stochastic dynamics with a subdiffusive temporal growth of position fluctuations due to long-range anticorrelated current fluctuations and a weak pinning at long times. This weak pinning manifests itself in a saturated width of the domain wall position fluctuations that increases sublinearly with the system size. As a function of time t and system size L, the width w(t,L) has a scaling behavior w(t,L)=L^{3/4}f(t/L^{9/4}), with f(u) constant for u≫1 and f(u)∼u^{1/3} for u≪1. An Orstein-Uhlenbeck process with long-range anticorrelated noise is shown to capture this scaling behavior. The exponent 9/4 is a new dynamical exponent for relaxation processes in driven diffusive systems.

Totally asymmetric simple exclusion process with local resetting in a resource-constrained environment

Inspired by the process of mRNA translation, in which the stochastic degradation of mRNA-ribosome machinery is modeled by the resetting dynamics, we study an open totally asymmetric simple exclusion process with local resetting at the entry site in a resource-constrained environment. The effect of constrained resources on the stationary properties of the system has been comprehended in the form of the filling factor. The mean-field approximations are utilized to obtain stationary state features, such as density profiles and phase diagrams. The phase diagram possesses pure phases as well as coexisting phases, including a low-density-high-density phase separation, which did not manifest under periodic boundary conditions despite the system being closed there as well. The role of the resetting rate has been investigated on the stationary properties of the system, depending on how the filling factor scales with the system size. In contrast to the resetting model for infinite resources, two distinct phase transitions are observed for the smaller values of the filling factor leading to a change in the topology of the phase diagram. The impact of the resetting rate along with the finite-size effect has also been examined on the shock dynamics. All the mean-field results are found in remarkable agreement with the Monte Carlo simulations.

Exclusion processes on a roundabout traffic model with constrained resources

Motivated by the vehicular traffic phenomenon at roundabouts, we examine how the limited availability of resources affects the movement of two distinct types of particles on bidirectional lanes connected by two bridges, with each bridge specifically designated for the transportation of one species. To provide a theoretical ground for our findings, we employ a mean-field framework and successfully validate them through dynamic Monte Carlo simulations. Based on the theoretical analysis, we analytically derive various stationary properties, such as the particle densities, phase boundaries, and particle currents, for all the possible symmetric as well as asymmetric phases. The qualitative as well as quantitative behavior of the system is significantly affected by the constraint on the number of resources. The complexity of the phase diagram shows a nonmonotonic behavior with an increasing number of particles in the system. Analytical arguments enable the identification of several critical values for the total number of particles, leading to a qualitative change in the phase diagrams. The interplay of the finite resources and the bidirectional transport yields unanticipated and unusual features such as back-and-forth transition, the presence of two congested phases where particle movement is halted, as well as shock phases induced by boundaries and the bulk of the system. Also, it is found that spontaneous symmetry-breaking phenomena are induced even for very few particles in the system. Moreover, we thoroughly examine the location of shocks by varying the parameters controlling the system's boundaries, providing insights into possible phase transitions.