Nonequilibrium steady states from a random-walk perspective

Abstract

It is well known that at thermal equilibrium (whereby a system has settled into a steady state with no energy or mass being exchanged with the environment), the microstates of a system are exponentially weighted by their energies, giving a Boltzmann distribution. All macroscopic quantities, such as the free energy and entropy, can be in principle computed given knowledge of the partition function. In a nonequilibrium steady state, on the other hand, the system has settled into a stationary state, but some currents of heat or mass persist. In the presence of these currents, there is no unified approach to solve for the microstate distribution. This motivates the central theme of this work, where I frame and solve problems in nonequilibrium statistical physics in terms of random walk and diffusion problems. The system that is the focus of Chapters 2, 3, and 4 is the (Totally) Asymmetric Simple Exclusion Process, or (T)ASEP. This is a system of hard-core particles making jumps through an open, one-dimensional lattice. This is a paradigmatic example of a nonequilibrium steady state that exhibits phase transitions. Furthermore, the probability of an arbitrary configuration of particles is exactly calculable, by a matrix product formalism that lends a natural association between the ASEP and a family of random walk problems. In Chapter 2 I present a unified description of the various combinatorial interpretations and mappings of steady-state configurations of the ASEP. As well as deriving new results, I bring together and unify results and observations that have otherwise been scattered in the combinatorics and physics literature. I show that particular particle configurations of the ASEP have a one-to-many mapping to a set of more abstract paths, which themselves have a one-to-many mapping to permutations of numbers. One observation from this wider literature has been that this mapped space can be interpreted as a larger set of configurations in some equilibrium system. This naturally gives an interpretation of ASEP configuration probabilities as summations