Using tools of statistical mechanics, it is routine to average over the distribution of microscopic configurations to obtain equilibrium free energies. These free energies teach us about the most likely molecular arrangements and the probability of observing deviations from the norm. Frequently, it is necessary to interrogate the probability not just of static arrangements, but of dynamical events, in which case analogous statistical mechanical tools may be applied to study the distribution of molecular trajectories. Numerical study of these trajectory spaces requires algorithms which efficiently sample the possible trajectories. We study in detail one such Monte Carlo algorithm, transition path sampling, and use a non- equilibrium statistical mechanical perspective to illuminate why the algorithm cannot easily be adapted to study some problems involving long-timescale dynamics. Algorithmically generating highly-correlated trajectories, a necessity for transition path sampling, grows exponentially more challenging for longer trajectories unless the dynamics is strongly-guided by the “noise history,” the sequence of random numbers representing the noise terms in the stochastic dynamics. Langevin dynamics of Weeks-Chandler-Andersen (WCA) particles in two dimensions lacks this strong noise guidance, so it is challenging to use transition path sampling to study rare dynamical events in long trajectories of WCA particles. The spin flip dynamics of a two-dimensional Ising model, on the other hand, can be guided by the noise history to achieve efficient path sampling. For systems that can be efficiently sampled with path sampling, we show that it is possible to simultaneously sample both the paths and the (potentially vast) space of non-equilibrium protocols to efficiently learn how rate constants vary with protocols and to identify low-dissipation protocols.

When high-dimensional molecular dynamics can be coarse-grained and represented by a simplified dynamics on a low-dimensional state space, the trajectory space may also be analytically studied using methods of large deviation theory. We review these methods and introduce a simple class of dynamical models whose dynamical fluctuations we compute exactly. The simplest such model is an asymmetric random walker on a one-dimensional ring with a single heterogeneous link connecting two sites of the ring. We derive conditions for the existence of a dynamic phase transition, which separates two dynamical phases—one localized and the other delocalized. The presence of distinct classes trajectories results in profoundly non-Gaussian fluctuations in dynamical quantities. We discuss the implications of such large dynamical fluctuations in the context of simple stochastic models for biological growth.

Systems driven out of equilibrium display a rich variety of patterns and surprising response behaviors. There exist different types of non-equilibrium processes, for instance a system that has been prepared in a non-Boltzmann initial state and is relaxing back to equilibrium, or a system that adopts a non-equilibrium steady state distribution when it is driven by an external field. In these different cases, the main characteristic that distinguishes these systems as non-equilibrium is that they are constantly dissipating heat, or likewise producing entropy. This entropy production is often the starting point for developing a systematic theory to describe such non-equilibrium processes.

Entropy production can be related to the irreversible processes occurring within a system. Particularly strong statements can be made about non-equilibrium systems when a local equilibrium assumption can be made, that is, when smaller subsets of a large system can be considered to be in equilibrium. This turns out to be justified for a wide variety of systems under different conditions. When this holds, the entropy production can be written as a generalized thermodynamic force (often the gradient of some intensive variable of the system) multiplied by a flux. When the thermodynamic force is small, the fluxes can be written as linear combinations of the thermodynamic forces, connected by response coefficients–this is known as linear irreversible thermodynamics. The full extension of equilibrium thermodynamic concepts to dissipative processes beyond this linear regime, including the development of microscopic principles justifying irreversible thermodynamic theories (as equilibrium statistical mechanics justifies equilibrium thermodynamics), is still a work in progress.

In this thesis, we work towards advancing the thermodynamic theory of non-equilibrium phenomena by studying models of driven-diffusive systems, growth processes, and active matter. We use developments from stochastic thermodynamics, large deviation theory, and irreversible thermodynamics to characterize the non-equilibrium phases and properties exhibited by these systems. We question to what extent equilibrium approximations are valid for predicting pattern formation in these systems and whether there exist general unifying features describing these non-equilibriums processes. In the process we develop trajectory sampling methods to investigate the statistics of dynamical order parameters distinguishing these phases. We show how the first and second laws of thermodynamics, including consistent expressions for entropy production, can be extended to active systems, where microscopic reversibility is broken at the level of individual particles. Additionally we derive fluctuation relations, exact analytical results for the fluctuations of entropy production in the form of equalities, for the entropy production in active systems. We also extend the Irving-Kirkwood procedure to active systems, deriving the balance laws of mass, momentum, and energy. Consequently we obtain expressions for the stress and couple stress tensors in the system as functions of the microscopic variables. This provides a foundation to extend the framework of irreversible thermodynamics to active systems.

Some of the most important and interesting phenomena in physical chemistry, such as heterogeneous catalysis, semi-conduction, and self-assembly depend crucially upon the surface properties of the material under consideration. This is particularly relevant for nanoscopic objects, whose surface-to-volume ratio is much higher than macroscopic materials. Thus, it is often necessary to carefully engineer nanoparticle surfaces so as to prevent them rom coalescing or reacting with their environment. This is achieved by using passivating ligands that stabilize nanoparticle surfaces and consequently, modify the chemical, optical, and electrical properties of nanocrystals and modulate inter-nanoparticle interactions. As a result, gaining an understanding of ligand behavior is essential to synthesizing new nanomaterials with useful technological applications; particularly because probing ligand structure is experimentally difficult.

We approach this problem by performing atomistic computer simulations of alkyl ligands on a semiconducting nanocrystal facet to elucidate their phase behavior at different temperatures and solvent conditions. These simulations provide a detailed description of the structure of the ligand molecules, specifically providing insight into the order-disorder transition they undergo as the temperature is varied. This phase transition changes the arrangement of the surface ligands, affecting how a nanoparticle interacts with solvent and other nanoscale objects in its environment. We proceed to map the observed statistics of ligand orientation onto a coarse-grained field theoretic model of the ordering transition, which is parametrized by physical properties obtained from simulation data. By extracting the underlying physics of the transition and removing irrelevant atomistic details, this coarse-grained model considerably reduces computational costs, while still describing the collective behavior of ligand molecules on a nanoparticle surface. This new understanding can be leveraged to describe ligand ordering when multiple nanoparticle surfaces are close to each other and its effect on the phase behavior of ligand passivated nanocrystals.

Living systems, even at the scale of single molecules, are constantly adapting to changing environmental conditions. The physical response of a nanoscale system to external gradients or changing thermodynamic conditions can be chaotic, nonlinear, and hence difficult to control or predict. Nevertheless, biology has evolved systems that reliably carry out the cell’s vital functions efficiently enough to ensure survival. Moreover, the development of new experimental techniques to monitor and manipulate single biological molecules has provided a natural testbed for theoretical investigations of nonequilibrium dynamics. This work focuses on developing paradigms for both understanding the principles of nonequilibrium dynamics and also for controlling such systems in the presence of thermal fluctuations.

Throughout this work, I rely on a perspective based on two central ideas in nonequilibrium statistical mechanics: large deviation theory, which provides a formalism akin to thermodynamics for nonequilibrium systems, and the fluctuation theorems which identify time symmetry breaking with entropy production. I use the tools of large deviation theory to explore concepts like efficiency and optimal coarse-graining in microscopic dynamical systems. The results point to the extreme importance of rare events in nonequilibrium dynamics. In the context of rare dynamical events, I outline a formal approach to predict efficient control protocols for nonequilibrium systems and develop computational tools to solve the resulting high dimensional optimization problems. The final chapters of this work focus on applications to self-assembly dynamics. I show that the yield of desired structures can be enhanced by driving a system away from equilibrium, using analysis inspired by the theory of the hydrophobic effect. Finally, I demonstrate that nanoscale, protein shells can be modeled and controlled to robustly produce monodisperse, nonequilibrium structures strikingly similar to the microcompartments observed in a variety of bacteria.

In this thesis we conduct a thorough study of the forces that act on ions when they are near air-water interfaces. These forces are important because they produce behavior which is very ion specific. That is, certain ions have a strong propensity for air-water interfaces and other ions avoid them completely. We will see that the dominant forces that allow ion adsorption to surfaces are fairly general and exist in a very broad class of liquids, so that even ions in a very simple model of a polar fluid exhibit a preference for the surface. In models of water, however, there are also forces which are very ion specific. In particular, the degree to which an ion is surface enhanced or surface repelled is very dependent on the sign of the charge. We will conduct a thorough study of this charge asymmetry in a simulated model of water and find that it is sensitive to various model details like ion size, the magnitude of the charge and polarizability. We will also study the way that solvent polarizability renormalizes the interactions between a pair of ions in solution, and a pair of ions at the interface and we will find that a simple effective model is fairly good at capturing the effects of polarizability. Finally, we will discuss attempts to improve dielectric continuum theory so that it is more useful for studying problems that involve solutes at interfaces.

Proteins in photosynthetic membranes can organize into patterned arrays that span the membrane's lateral size. Attractions between proteins in different layers of a membrane stack play a key role in this ordering, as has been demonstrated by both empirical and computational methods. The architecture of thylakoid membranes, depending on physiological conditions, also may create circumstances for inter-layer interactions that instead disfavor the high protein densities of ordered arrangements. This dissertation introduces several statistical mechanical models for exploring the interplay between these opposing forces and for characterizing phases that reflect the periodic geometry of stacked thylakoid membrane discs. First, we propose a lattice model that roughly accounts for proteins' attraction within a layer and across the stromal gap, steric repulsion across the lumenal gap, and regulation of protein density by exchange with the stroma lamellae. Mean field analysis and computer simulation reveal a broken-symmetry striped phase disrupted at both high and low extremes of density. We expect that the widely varying light and stress conditions in higher plants explore the space of protein density and interaction strength broadly. The phase transitions we identify should thus lie within or near the range of naturally occurring conditions. Second, we expand upon this lattice description, allowing the thickness of each thylakoid's lumenal gap to fluctuate. This fluctuating-gap model introduces the possibility of mechanical control of photosynthetic function. We monitor how changing gap thickness affects mean protein occupation on both sides of the discs. Via mean field analysis and computer simulation we find even richer phase behavior for this model, featuring transitions that originate in long-ranged protein interactions mediated by lumenal gap fluctuations. These results suggest that compression or expansion of lumenal gaps could lead to sudden and dramatic changes in the population and spatial patterning of photosynthetic proteins. Taken together, the lattice models we have constructed and explored provide a framework for minimalistic modeling of the physics underlying structure and function of photosynthetic membranes.