Asymmetric simple exclusion process (ASEP) is a driven stochastic lattice model of particles that move preferentially in one direction. If particles move only in one direction, the model is known as totally asymmetric process. Conventionally, preferred direction of motion is chosen to be to the right. Particles interact through the hard core exclusion rule, meaning that no more than one particle is allowed to occupy one lattice site. In this work following ASEP models are presented. First we study square diagonal lattice with particles that occupy one lattice site and move along the square diagonals. Mean-field theory was developed for this model. The results that were obtained are the dependency of the current on density of the particles, spatial density distribution along the horizontal direction and the phase diagram of the system. Mean-field theory results were compared to simulations. Next model was lattice with extended particles, i.e. particles that occupy more than one lattice site. Unlike the first model, in this system the particle-hole symmetry is broken. Results for current flow, density distribution and phase diagrams were obtained both by mean-field theory and Monte-Carlo (MC) simulations. Another system was the lattice with vertical particle drift. Now particles that occupy one lattice site jump not only in one preferred horizontal directions but there is also one preferred vertical direction for particle flow. Both mean-field theory and simulations were studied for this system and results were compared. Also we explore the system with immovable obstacle. Obstacle is one or several particles located at fixed positions. In this model we observe increase in particle density in front of the obstacle and "shadow" behind it. It is expected that the shape and size of those formations are symmetrical in transverse direction.

Over the last few decades the interests of statistical physicists have broadened to include the detailed quantitative study of many systems - chemical, biological and even social - that were not traditionally part of the discipline. These systems can feature rich and complex spatiotemporal behaviour, often due to continued interaction with the environment and characterised by the dissipation of flows of energy and/or mass. This has led to vigorous research aimed at extending the established theoretical framework and adapting analytical methods that originate in the study of systems at thermodynamic equilibrium to deal with out-of-equilibrium situations, which are much more prevalent in nature. This thesis focuses on a microscopic model known as the asymmetric exclusion process, or ASEP, which describes the stochastic motion of particles on a one-dimensional lattice. Though in the first instance a model of a lattice gas, it is sufficiently general to have served as the basis to model a wide variety of phenomena. That, as well as substantial progress made in analysing its stationary behaviour, including the locations and nature of phase transitions, have led to it becoming a paradigmatic model of an exactly solvable nonequilibrium system. Recently an exact solution for the dynamics found a somewhat enigmatic transition, which has not been well understood. This thesis is an attempt to verify and better understand the nature of that dynamical transition, including its relation, if any, to the static phase transitions. I begin in Chapter 2 by reviewing known results for the ASEP, in particular the totally asymmetric variant (TASEP), driven at the boundaries. I present the exact dynamical transition as it was first derived, and a reduced description of the dynamics known as domain wall theory (DWT), which locates the transition at a different place. In Chapter 3, I investigate solutions of a nonlinear PDE that constitutes a mean-field, continuum approximation of the ASEP, namely the Burgers equation, and find that a similar dynamical transition occurs there at the same place as predicted by DWT but in disagreement with the exact result. Next, in Chapter 4 I report on efforts to observe and measure the dynamical transition through Monte Carlo simulation. No directly obvious physical manifestation of the transition was observed. The relaxation of three different observables was measured and found to agree well with each other but only slightly better with the exact transition than with DWT. In Chapter 5 I apply a numerical renormalisation scheme known as the Density Matrix Renormalisation Group (DMRG) method and find that it confirms the exact dynamical transition, ruling out the behaviour predicted by DWT. Finally in Chapter 6 I demonstrate that a perturbative calculation, involving the crossing of eigenvalues, allows us to rederive the location of the dynamical transition found exactly, thereby offering some insight into the nature of the transition.

In this thesis I study a model of self propelled particles exhibiting run-and tumble dynamics on lattice. This non-Brownian diffusion is characterised by a random walk with a finite persistence length between changes of direction, and is inspired by the motion of bacteria such as Escherichia coli. By defining a class of models with multiple species of particle and transmutation between species we can recreate such dynamics. These models admit exact analytical results whilst also forming a counterpart to previous continuum models of run-and- tumble dynamics. I solve the externally driven non-interacting and zero-range versions of the model exactly and utilise a field theoretic approach to derive the continuum fluctuating hydrodynamics for more general interactions. I make contact with prior approaches to run-and-tumble dynamics of lattice and determine the steady state and linear stability for a class of crowding interactions, where the jump rate decreases as density increases. In addition to its interest from the perspective of nonequilibrium statistical mechanics, this lattice model constitutes an efficient tool to simulate a class of interacting run-and-tumble models relevant to bacterial motion. Pattern formation in bacterial colonies is confirmed to be able to stem solely from the interplay between a diffusivity that depends on the local bacterial density and regulated division of the cells, in particular without the need for any explicit chemotaxis. This simple and generic mechanism thus provides a null hypothesis for pattern formation in bacterial colonies which has to be falsified before appealing to more elaborate alternatives. Most of the literature on bacterial motility relies on models with instantaneous tumbles. As I show, however, the finite tumble duration can play a major role in the patterning process. Finally a connection is made to some real experimental results and the population ecology of multiple species of bacteria competing for the same resources is considered.

In this thesis, we study the physics of the pulled polymer loops motivated by a biological problem of chromosome alignment during meiosis in fission yeast. During prophase I of meiotic fission yeast, the chromosomes form a loop structure by binding their telomeres to the Spindle Pole Body (SPB). SPB nucleates the growth of microtubules in the cytoplasm. Molecular motors attached to the cell membrane can exert the force on the microtubules and thus pull the whole nucleus. The nucleus performs oscillatory motion from one to the other end of the elongated zygote cell. Experimental evidence suggests that these oscillations facilitate homologous chromosome alignment which is required for the gene recombination. Our goal is to understand the physical mechanism of this alignment. We thus propose a model of pulled polymer loops to represent the chromosomal motion during oscillations. Using a freely-jointed bead-rod model for the pulled polymer loop, we solve the equilibrium statistics of the polymer configurations both in 1D and 3D. In 1D, we find a peculiar mapping of the bead-rod system to a system of particles on a lattice. Utilizing the wealth of tools of the particle system, we solve exactly the 1D stationary measure and map it back to the polymer system. To address the looping geometry, the Brownian Bridge technique is employed. The mean and variance of beads position along the loop are discussed in detail both in 1D and 3D. We then can calculate the three-dimensional statistics of the distance between corresponding beads from a pair of loops in order to discuss the pairing problem of homologous chromosomes. The steady-state shape of a three-dimensional pulled polymer loop is quantified using the descriptors based on the gyration tensor. Beyond the steady state statistics, the relaxation dynamics of the pinned polymer loop in a constant external force field is discussed. In 1D we show the mapping of polymer dynamics to the well-known Asymmetric Simple Exclusion Process (ASEP) model. Our pinned polymer loop is mapped to a half-filled ASEP with reflecting boundaries. We solve the ASEP model exactly by using the generalized Bethe ansatz method. Thus with the help of the ASEP theory, the relaxation time of the polymer problem can be calculated analytically. To test our theoretical predictions, extensive simulations are performed. We find that our theory of relaxation time fit very well to the relaxation time of a 3D polymer in the direction of the external force field. Finally, we discuss the relevance of our findings to the problem of chromosome alignment in fission yeast.

Cette thèse porte sur l’interprétation combinatoire des probabilitésde l’état stationnaire de l’ASEP par les tableaux escaliers, sur les arbresnon-ambigus et sur les polyominos parallélogrammes périodiques.Dans une première partie, nous étudions l’ansatz matriciel de Derrida,Evans, Hakim et Pasquier. Toute solution de ce système d’équation permet decalculer les probabilités stationnaires de l’ASEP. Nos travaux définissent denouvelles récurrences équivalentes à celles de l’ansatz matriciel. En définissantun algorithme d’insertion sur les tableaux escaliers, nous montrons combinatoirementet simplement qu’ils les satisfont. Nous faisons de même pour l’ASEPà deux particules. Enfin, nous énumérons les coins dans les tableaux associésà l’ASEP, nous permettant ainsi de donner le nombre moyen de transitionspossibles depuis un état de l’ASEP.Dans une deuxième partie, nous calculons de jolies formules pour les sériesgénératrices des arbres non-ambigus, desquelles nous déduisons des formulesd’énumérations. Puis, nous interprétons bijectivement certains de ces résultats.Enfin, nous généralisons les arbres non-ambigus à toutes les dimensions finies.Dans la dernière partie, nous construisons une structure arborescente surles polyominos parallélogrammes périodiques, inspirée des travaux de Boussicault,Rinaldi et Socci. Cela nous permet de calculer facilement leur sériegénératrice selon la hauteur et la largeur ainsi que deux nouvelles statistiques :la largeur intrinsèque et la hauteur de recollement intrinsèque. Enfin, nousétudions l’ultime périodicité de leur série génératrice selon l’aire. / This thesis deals with a combinatorial interpretation of the stationnarydistribution of the ASEP given by staircase tableaux and studiestwo combinatorial objects : non-ambiguous trees and periodic parallelogrampolyominoes.In the first part, we study the matrix ansatz introduced by Derrida, Evans,Hakim and Pasquier. Any solution of this equation system can be used tocompute the stationnary probabilities of the ASEP. Our work defines newrecurrences equivalent to the matrix ansatz. By defining an insertion algorithmfor staircase tableaux, we prove combinatorially and easily that they satisfyour new recurrences. We do the same for the ASEP with two types of particles.Finally, we enumerate the corners of the tableaux related to the ASEP, whichgives the average number of transitions from a state of the ASEP.In the second part, we compute nice formulas for the generating functionsof non-ambiguous trees, from which we deduce enumeration formulas. Then, wegive a combinatorial interpretation of some of our results. Lastly, we generalisenon-ambiguous trees to every finite dimension.In the last part, we define a tree structure in periodic parallelogram polyominoes,motivated by the work of Boussicault, Rinaldi and Socci. It allowsus to compute easily the generating function with respect to the height andthe width as well as two new statistics : the intrinsic width and the intrinsicgluing height. Finally, we investigate the ultimate periodicity of the generatingfunction with respect to the area.

In this thesis, we study the physics of the pulled polymer loops motivated by a biological problem of chromosome alignment during meiosis in fission yeast. During prophase I of meiotic fission yeast, the chromosomes form a loop structure by binding their telomeres to the Spindle Pole Body (SPB). SPB nucleates the growth of microtubules in the cytoplasm. Molecular motors attached to the cell membrane can exert the force on the microtubules and thus pull the whole nucleus. The nucleus performs oscillatory motion from one to the other end of the elongated zygote cell. Experimental evidence suggests that these oscillations facilitate homologous chromosome alignment which is required for the gene recombination. Our goal is to understand the physical mechanism of this alignment. We thus propose a model of pulled polymer loops to represent the chromosomal motion during oscillations. Using a freely-jointed bead-rod model for the pulled polymer loop, we solve the equilibrium statistics of the polymer configurations both in 1D and 3D. In 1D, we find a peculiar mapping of the bead-rod system to a system of particles on a lattice. Utilizing the wealth of tools of the particle system, we solve exactly the 1D stationary measure and map it back to the polymer system. To address the looping geometry, the Brownian Bridge technique is employed. The mean and variance of beads position along the loop are discussed in detail both in 1D and 3D. We then can calculate the three-dimensional statistics of the distance between corresponding beads from a pair of loops in order to discuss the pairing problem of homologous chromosomes. The steady-state shape of a three-dimensional pulled polymer loop is quantified using the descriptors based on the gyration tensor. Beyond the steady state statistics, the relaxation dynamics of the pinned polymer loop in a constant external force field is discussed. In 1D we show the mapping of polymer dynamics to the well-known Asymmetric Simple Exclusion Process (ASEP) model. Our pinned polymer loop is mapped to a half-filled ASEP with reflecting boundaries. We solve the ASEP model exactly by using the generalized Bethe ansatz method. Thus with the help of the ASEP theory, the relaxation time of the polymer problem can be calculated analytically. To test our theoretical predictions, extensive simulations are performed. We find that our theory of relaxation time fit very well to the relaxation time of a 3D polymer in the direction of the external force field. Finally, we discuss the relevance of our findings to the problem of chromosome alignment in fission yeast.