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
of Boltzmann weights. The nonequilibrium partition function of the ASEP is then a
summation over this equilibrium ensemble, however one encounters difficulties when
calculating more detailed measures of this state space, such as the entropy.
This motivates the work in Chapter 3. I calculate a quantity known as the Rényi
entropy, which is a measure of the partitioning of the state space, and a deformation of
the familiar Shannon entropy. The Rényi entropy is simple for an equilibrium system,
but has yet to be explored in a classical nonequilibrium steady state. I use insights
from Chapter 2 to frame one of these Rényi entropies | requiring the enumeration
of the squares of configuration weights | in terms of a two-dimensional random walk
with absorbing boundaries. I find the appropriate generating function across the full
phase diagram of the TASEP by generalising a mathematical technique known as
the obstinate kernel method. Importantly, this nonequilibrium Rényi entropy has a
different structural form to any equilibrium system, highlighting a clear distinction
between equilibrium and nonequilibrium distributions.
In Chapter 4 I continue to examine the Rényi entropy of the TASEP, but now
performing a time and space continuum limit of the random walk problem in Chapter 3.
The resultant problem is a two-dimensional dffusion problem with absorbing boundary
conditions, which once solved should recover TASEP dynamics about the point in the
phase diagram where the three dynamical phases meet. I derive a generating function,
sufficiently simple that its singularities can be analysed by hand. This calculation
entails a novel generalisation of the obstinate kernel method of Chapter 3: I find a
solution by exploiting a symmetry in the Laplace transform of the diffusion equation.
I finish in Chapter 5 by introducing and solving another nonequilibrium system, termed
the many-filament Brownian ratchet. This comprises an arbitrary number of filaments
that stochastically grow and contract, with the net effect of moving a drift-diffusing
membrane by purely from thermal fluctuations and steric interactions. These dynamics
draw parallels with those of actin filament networks at the leading edge of eukaryotic
cells, and this improves on previous 'pure ratchet' models by introducing interactions
and heterogeneity in the filaments. I find an N-dimensional diffusion equation for the
evolution of the N filament-membrane displacements. Several parameters can be varied
in this system: the drift and diffusion rates of each of the filaments and membrane, the
strength of a quadratic interaction between each filament with the membrane, and the
strength of a surface tension across the filaments. For several interesting physical cases
I find the steady-state distribution exactly, and calculate how the mean velocity of the
membrane varies as a function of these parameters
