Driven lattice Lorentz gases in confinement

Many material properties are connected to the fundamental question how particles move in their environment. Understanding such transport processes on the microscopic scale is therefore crucial to predict the macroscopic behavior of liquids, crystals or non-crystalline solids, for example glasses. In non-crystalline systems the environment does not display regular structure rather it appears random, therefore such systems are referred to as disordered media. Insight in the dynamics of disordered media has been gained mostly by experiments and computer simulations while analytical solutions of even simple theoretical models are rare. This projects aims at shedding light on questions related to the effects of confinement on the dynamics in the presence of disordered media. The theoretical framework is the one of the analytically tractable lattice Lorentz gas in restricted geometry at low obstacle density. The Lorentz gas is a simple model in which a particle performs a random walk on a square lattice in the presence of a frozen landscape of immobile impurities. In a nutshell, the questions we plan to answer revolve around the characterization of the dynamics in the presence of a strong driving, a typical situation where perturbative approaches are not informative because of the breakdown of the linear response regime. In particular, we intend to characterize how geometrical confinement, space dimensionality, and randomness affect the dynamics, the breaking of the linear regime, and the onset of the non-linear one. Addressing such a type of questions in the field of non-equilibrium statistical mechanics through analytical methods is the innovative aspect of the project. We intend to build upon recently obtained exact analytical solutions for the dynamics of a driven tracer particle in unbounded systems. This objective can be achieved by generalizing the currently existing exact solution for the unbounded case to the restricted one, a task that most likely can be carried out analytically. Besides transport- related properties, this project aims at studying first-passage times in confined and disordered systems, something that we plan to carry out by complementing the theory with stochastic simulations.

How does confinement affect the movement of particles in a disordered environment? This research explores the impact of spatial restrictions on the behavior of particles moving through a complex landscape of immobile obstacles-a problem relevant in physics, biology, and material sciences. Using the well-established lattice Lorentz gas model, where a particle moves randomly in a grid scattered with fixed impurities, the study investigates how strong external forces disrupt conventional transport properties. Unlike mild disturbances, where systems follow linear response regime, high-intensity forces often push systems into unexplored non-linear regimes. Understanding these transitions is crucial for developing better models of real-world scenarios, from cellular transport mechanisms to fluid dynamics in porous materials. This work breaks new ground by providing exact analytical solutions for particle motion under confinement, extending prior findings for unbounded systems. The approach combines rigorous mathematical methods with stochastic simulations, leading to precise descriptions of transport behaviors. By shedding light on fundamental aspects of non-equilibrium statistical mechanics, this project contributes to our understanding of transport phenomena in confined, disordered environments. Potential applications span biophysics, material science, and even technological advancements, wherever controlling particle dynamics is essential.