Stochasticity and fluctuations in non-equilibrium transport models

The transportation of mass is an inherently `non-equilibrium' process, relying on a current of mass between two or more locations. Life exists by necessity out of equilibrium and non-equilibrium transport processes are seen at all levels in living organisms, from DNA replication up to animal foraging. As such, biological processes are ideal candidates for modelling using non-equilibrium stochastic processes, but, unlike with equilibrium processes, there is as of yet no general framework for their analysis. In the absence of such a framework we must study specific models to learn more about the behaviours and bulk properties of systems that are out of equilibrium. In this work I present the analysis of three distinct models of non-equilibrium mass transport processes. Each transport process is conceptually distinct but all share close connections with each other through a set of fundamental nonequilibrium models, which are outlined in Chapter 2. In this thesis I endeavour to understand at a more fundamental level the role of stochasticity and fluctuations in non-equilibrium transport processes. In Chapter 3 I present a model of a diffusive search process with stochastic resetting of the searcher's position, and discuss the effects of an imperfection in the interaction between the searcher and its target. Diffusive search process are particularly relevant to the behaviour of searching proteins on strands of DNA, as well as more diverse applications such as animal foraging and computational search algorithms. The focus of this study was to calculate analytically the effects of the imperfection on the survival probability and the mean time to absorption at the target of the diffusive searcher. I find that the survival probability of the searcher decreases exponentially with time, with a decay constant which increases as the imperfection in the interaction decreases. This study also revealed the importance of the ratio of two length scales to the search process: the characteristic displacement of the searcher due to diffusion between reset events, and an effective attenuation depth related to the imperfection of the target. The second model, presented in Chapter 4, is a spatially discrete mass transport model of the same type as the well-known Zero-Range Process (ZRP). This model predicts a phase transition into a state where there is a macroscopically occupied `condensate' site. This condensate is static in the system, maintained by the balance of current of mass into and out of it. However in many physical contexts, such as traffic jams, gravitational clustering and droplet formation, the condensate is seen to be mobile rather than static. In this study I present a zero-range model which exhibits a moving condensate phase and analyse it's mechanism of formation. I find that, for certain parameter values in the mass `hopping' rate effectively all of the mass forms a single site condensate which propagates through the system followed closely by a short tail of small masses. This short tail is found to be crucial for maintaining the condensate, preventing it from falling apart. Finally, in Chapter 5, I present a model of an interface growing against an opposing, diffusive membrane. In lamellipodia in cells, the ratcheting effect of a growing interface of actin filaments against a membrane, which undergoes some thermal motion, allows the cell to extrude protrusions and move along a surface. The interface grows by way of polymerisation of actin monomers onto actin filaments which make up the structure that supports the interface. I model the growth of this interface by the stochastic polymerisation of monomers using a Kardar-Parisi-Zhang (KPZ) class interface against an obstructing wall that also performs a random walk. I find three phases in the dynamics of the membrane and interface as the bias in the membrane diffusion is varied from towards the interface to away from the interface. In the smooth phase, the interface is tightly bound to the wall and pushes it along at a velocity dependent on the membrane bias. In the rough phase the interface reaches its maximal growth velocity and pushes the membrane at this speed, independently of the membrane bias. The interface is rough, bound to the membrane at a subextensive number of contact points. Finally, in the unbound phase the membrane travels fast enough away from the interface for the two to become uncoupled, and the interface grows as a free KPZ interface. In all of these models stochasticity and fluctuations in the properties of the systems studied play important roles in the behaviours observed. We see modified search times, strong condensation and a dramatic change in interfacial properties, all of which are the consequence of just small modifications to the processes involved

On the Sublinear Regret of GP-UCB

In the kernelized bandit problem, a learner aims to sequentially compute the optimum of a function lying in a reproducing kernel Hilbert space given only noisy evaluations at sequentially chosen points. In particular, the learner aims to minimize regret, which is a measure of the suboptimality of the choices made. Arguably the most popular algorithm is the Gaussian Process Upper Confidence Bound (GP-UCB) algorithm, which involves acting based on a simple linear estimator of the unknown function. Despite its popularity, existing analyses of GP-UCB give a suboptimal regret rate, which fails to be sublinear for many commonly used kernels such as the Mat\'ern kernel. This has led to a longstanding open question: are existing regret analyses for GP-UCB tight, or can bounds be improved by using more sophisticated analytical techniques? In this work, we resolve this open question and show that GP-UCB enjoys nearly optimal regret. In particular, our results yield sublinear regret rates for the Mat\'ern kernel, improving over the state-of-the-art analyses and partially resolving a COLT open problem posed by Vakili et al. Our improvements rely on a key technical contribution -- regularizing kernel ridge estimators in proportion to the smoothness of the underlying kernel $k$. Applying this key idea together with a largely overlooked concentration result in separable Hilbert spaces (for which we provide an independent, simplified derivation), we are able to provide a tighter analysis of the GP-UCB algorithm.Comment: 20 pages, 0 figure

Time-Uniform Self-Normalized Concentration for Vector-Valued Processes

Self-normalized processes arise naturally in many statistical tasks. While self-normalized concentration has been extensively studied for scalar-valued processes, there is less work on multidimensional processes outside of the sub-Gaussian setting. In this work, we construct a general, self-normalized inequality for $\mathbb{R}^d$-valued processes that satisfy a simple yet broad "sub-$\psi$" tail condition, which generalizes assumptions based on cumulant generating functions. From this general inequality, we derive an upper law of the iterated logarithm for sub-$\psi$ vector-valued processes, which is tight up to small constants. We demonstrate applications in prototypical statistical tasks, such as parameter estimation in online linear regression and auto-regressive modeling, and bounded mean estimation via a new (multivariate) empirical Bernstein concentration inequality.Comment: 50 pages, 3 figure

Width Scaling of an Interface Constrained by a Membrane

We investigate the shape of a growing interface in the presence of an impenetrable moving membrane. The two distinct geometrical arrangements of the interface and membrane, obtained by placing the membrane behind or ahead of the interface, are not symmetrically related. On the basis of numerical results and an exact calculation, we argue that these two arrangements represent two distinct universality classes for interfacial growth: whilst the well-established Kardar-Parisi-Zhang (KPZ) growth is obtained in the `ahead' arrangement, we find an arrested KPZ growth with a smaller roughness exponent in the `behind' arrangement. This suggests that the surface properties of growing cell membranes and expanding bacterial colonies, for example, are fundamentally distinct.Comment: 6 pages, 6 figures; revised version contains a small amount of additional discussion and the supplementary figures. To appear in Phys. Rev. Let

Shared Negative Experiences Lead to Identity Fusion via Personal Reflection

Across three studies, we examined the role of shared negative experiences in the formation of strong social bonds--identity fusion--previously associated with individuals' willingness to self-sacrifice for the sake of their groups. Studies 1 and 2 were correlational studies conducted on two different populations. In Study 1, we found that the extent to which Northern Irish Republicans and Unionists experienced shared negative experiences was associated with levels of identity fusion, and that this relationship was mediated by their reflection on these experiences. In Study 2, we replicated this finding among Bostonians, looking at their experiences of the 2013 Boston Marathon Bombings. These correlational studies provide initial evidence for the plausibility of our causal model; however, an experiment was required for a more direct test. Thus, in Study 3, we experimentally manipulated the salience of the Boston Marathon Bombings, and found that this increased state levels of identity fusion among those who experienced it negatively. Taken together, these three studies provide evidence that shared negative experience leads to identity fusion, and that this process involves personal reflection

If Racism Vanished for a Day

This picture book was co-written and illustrated by 17 children from Bristol Schools. They think it's important to learn from children's experiences of racism so have posed simple sentences and illustrative responses for readers to critically reflect on

Effect of Partial Absorption on Diffusion with Resetting

The effect of partial absorption on a diffusive particle which stochastically resets its position with a finite rate $r$ is considered. The particle is absorbed by a target at the origin with absorption `velocity' $a$; as the velocity $a$ approaches $\infty$ the absorption property of the target approaches that of a perfectly-absorbing target. The effect of partial absorption on first-passage time problems is studied, in particular, it is shown that the mean time to absorption (MTA) is increased by an additive term proportional to $1/a$. The results are extended to multiparticle systems where independent searchers, initially uniformly distributed with a given density, look for a single immobile target. It is found that the average survival probability $P^{av}$ is modified by a multiplicative factor which is a function of $1/a$, whereas the decay rate of the typical survival probability $P^{typ}$ is decreased by an additive term proportional to $1/a$.Comment: 17 pages, 3 figure