Analytical description of some far from equilibrium complex processes

Abstract

Statistical mechanics is one of the pillars of science where the bulk behavior of a large number of identically distributed particles is predicted rather than the individual particles themselves. It also lays the descriptive framework for understanding of thermodynamic equilibrium and have opened up new prospectives of multidisciplinary research. But despite the continuous e orts of researchers over the last several decades, an equivalently specific groundwork of non-equilibrium statistical mechanics remains unexplored. In comparison to the equilibrium systems, no general theoretical foundation exists for non-equilibrium systems, resulting in the knowledge of both types of non-equilibrium processes 􀀀 systems evolving towards an equilibrium steady-state and systems far from equilibrium, to be in a much former stage. The latter class of non-equilibrium systems including biological tra c-like collective phenomena in living systems, vehicular tra c, pedestrian flow, along with epidemic spread, is looked upon in this thesis. Present life depends significantly on physical bidirectional vehicular tra c. Our day to day movements to o ces, markets, schools, our daily needs and requirements all depend on vehicular tra c. Interestingly, because of the close similarity between vehicular tra c and systems of interacting particles driven far from equilibrium, the study of tra c phenomenon has attracted researchers in the sense to develop and analyze models of transport. Moreover, nature has handled transport phenomena both at microscopic and macroscopic levels. The environment inside a biological cell resembles an urban bidirectional traffic system in many aspects. The majority of cellular processes are controlled by nano biological machines called motor proteins which collectively carry cargo in both the directions over long distances by moving along highly dynamic filamentary tracks, namely microtubules, that are made up of tubulin dimers. Microtubules display complex length dynamics and a small curvature resulting in narrow entrances. The malfunctioning of the biological transport system can lead to several diseases. Accordingly, a fundamental understanding of collective intracellular transport is necessary for smooth cell functioning and survival. Moreover, at the macroscopic level, in the transfer of infectious disease from an infected source to a susceptible host, either directly or indirectly, movement is one of the central attributes. To fight against their unpredictable and explosive impacts, epidemiologists are e ortlessly working in the direction to assist with tracking down the sources of infection along with control strategies to understand infectious disease. The thesis mainly focuses on the modeling of the bidirectional physical and biological transport system using a totally asymmetric simple exclusion process (TASEP) as a powerful tool, which describes the stochastic motion of particles on a one-dimensional lattice. The entities accomplishing the transport phenomena are represented as particles and the tracks on which these entities progress are mimicked as one-dimensional discrete lanes. We assimilated significant intracellular aspects in the single and multi-lane TASEP scheme to analyze the collective dynamics of molecular motors in real-time scenarios. Incorporating the finite progressive nature of the molecular motors in the cell, leading to shifts between phases of directed and di usive motion we analyzed a two-lane bidirectional non-conserving TASEP with narrow entrances. We explored the non-trivial role of neighboring particle interactions on the transport network with narrow entrances. Stepping ahead, we studied a more realistic two-channel transport system with bidirectional movement on each lane. Understanding the complexity involved in the system, we examined the evolving steady-state dynamics under the impact of narrow entrances. Besides we examined the role of neighboring particle interactions and finite supply of free tubulin dimers on the length regulation of microtubules. We carried out theoretical analysis based on the mean-field approach and its modifications, to predict analytical outcomes, including significant features in terms of phase segregation and phase transitions. The theoretical findings are explored by extensive Monte Carlo simulations (MCs). Further, we present a theoretical model, based on cellular automata technique, to feature epidemic spreading. The proposed model serves as a basis to simulate time delay based epidemics incorporating real data. We investigated the crucial impact and possible control strategies for COVID-19 pandemic within a heterogeneous population and geography. The studied model is a generic e ort to gain deep insight into the dynamics of brutal infection helping the concerned authorities to maintain vital resources and plan its control actions. Overall, the work is a mathematical attempt to gain a broad understanding into the steadystate dynamics of various non-equilibrium systems present in nature.