Unraveling stochastic transport in complex systems: Analytical and computational frameworks

Abstract

Movement is a fundamental aspect of life, occurring across all scales from microscopic to macroscopic levels. At the cellular level, essential processes such as gene expression rely on the movement of RNA polymerases along DNA strands to produce messenger RNA (mRNA), followed by ribosomes traversing the mRNA to synthesize proteins. On a larger scale, vehicular flow in urban environments exemplifies a ubiquitous transport process that impacts daily life, facilitating access to workplaces, services, and institutions. These di verse examples represent non-equilibrium complex systems, unified by their inherent com plexity and categorized as driven-diffusive systems. This classification is crucial because non-equilibrium transport processes lack a unified theory for describing their steady-state properties. Grouping similar complex transport phenomena enhances our understanding and aids in developing solution strategies, making it a significant area of research across multiple disciplines, including mathematics, biology, and physics. To effectively study these systems, it is necessary to develop appropriate mathematical and computational models that can accurately analyze particle flow dynamics. The Totally Asymmetric Simple Exclusion Process (TASEP) has emerged as a preeminent model for studying driven diffusive systems. Over time, this exclusion model has established itself as a paradigmatic framework, offering a streamlined mathematical approach to capture the intricate stochastic transport dynamics on a one-dimensional discrete lattice. This lattice effectively represents pathways for the unidirectional flow of particles, which can be anal ogous to vehicles in traffic flow scenarios. The TASEP model’s strength lies in its ability to distill complex transport phenomena into a tractable form, enabling researchers to gain insights into fundamental principles governing non-equilibrium systems. Building upon this foundation, our research contributes to a more comprehensive under standing of the collective behavior of particles in various single-lane TASEP model adapta tions. Inspired by the presence of obstacles in vehicular and molecular motor transport, we investigate the impact of stochastic defects on system inhomogeneity. This research pro poses a TASEP model where particle entry and exit on an inhomogeneous lattice are gov erned by the occupancy of a finite reservoir connected to both ends, reflecting real-world resource limitations. We examine the collective effects of these dynamics on system proper ties. Furthermore, we extend our analysis to non-conserving TASEP models, exploring the non-trivial effects of defect dynamics and non-conserving kinetics on density profiles and phase diagrams. Our research also delves into biological and physical systems exhibiting stochastic local resetting phenomena. We scrutinize the stationary properties of systems where entities enter the lattice from a limited resource pool and either move horizontally or reset to specific lattice positions. Advancing our investigation, we study the impact of local stochastic resetting in bidirectional TASEP models, where particles of distinct species move in opposite directions. Additionally, motivated by entity flow in narrow channels and the goal of separating colloidal particles of different sizes, we examine a geometrically adapted TASEP model. In this model, each lattice site connects to a pocket-like structure with a defined particle capacity, mimicking the asymmetric geometry of narrow channels. We ana lyze the system’s stationary characteristics under both finite and infinite particle availability. In essence, our research employs mathematical modeling to elucidate previously unex plored complexities in transport processes. These findings, corroborated by simulations, underscore the importance of understanding the collective dynamics of moving entities in various contexts.