Tricritical Behavior in Dynamical Phase Transitions.

We identify a new scenario for dynamical phase transitions associated with time-integrated observables occurring in diffusive systems described by the macroscopic fluctuation theory. It is characterized by the pairwise meeting of first- and second-order bias-induced phase transition curves at two tricritical points. We formulate a simple, general criterion for its appearance and derive an exact Landau theory for the tricritical behavior. The scenario is demonstrated in three examples: the simple symmetric exclusion process biased by an activity-related structural observable; the Katz-Lebowitz-Spohn lattice gas model biased by its current; and in an active lattice gas biased by its entropy production.

Extinctions of coupled populations, and rare event dynamics under non-Gaussian noise.

The survival of natural populations may be greatly affected by environmental conditions that vary in space and time. We look at a population residing in two locations (patches) coupled by migration, in which the local conditions fluctuate in time. We report on two findings. First, we find that, unlike rare events in many other systems, here the histories leading to a rare extinction event are not dominated by a single path. We develop the appropriate framework, which turns out to be a hybrid of the standard saddle-point method, and the Donsker-Varadhan formalism which treats rare events of atypical averages over a long time. It provides a detailed description of the statistics of histories leading to the rare event and the mean time to extinction. The framework applies to rare events in a broad class of systems driven by non-Gaussian noise. Second, applying this framework to the population-dynamics model, we find a phase transition in its extinction behavior. Strikingly, a patch which is a sink (where individuals die more than are born) can nonetheless reduce the probability of extinction, even if it lowers the average population's size and growth rate.

Occupation-time statistics of a gas of interacting diffusing particles.

The time that a diffusing particle spends in a certain region of space is known as the occupation time, or the residence time. Recently, the joint occupation-time statistics of an ensemble of noninteracting particles was addressed using the single-particle statistics. Here we employ the macroscopic fluctuation theory (MFT) to study the occupation-time statistics of many interacting particles. We find that interactions can significantly change the statistics and, in some models, even cause a singularity of the large-deviation function describing these statistics. This singularity can be interpreted as a dynamical phase transition. We also point out to a close relation between the MFT description of the occupation-time statistics of noninteracting particles and the level 2 large deviation formalism which describes the occupation-time statistics of a single particle.

Narrow Escape of Interacting Diffusing Particles.

The narrow escape problem deals with the calculation of the mean escape time (MET) of a Brownian particle from a bounded domain through a small hole on the domain's boundary. Here we develop a formalism which allows us to evaluate the nonescape probability of a gas of diffusing particles that may interact with each other. In some cases the nonescape probability allows us to evaluate the MET of the first particle. The formalism is based on the fluctuating hydrodynamics and the recently developed macroscopic fluctuation theory. We also uncover an unexpected connection between the narrow escape of interacting particles and thermal runaway in chemical reactors.

Fluctuations of absorption of interacting diffusing particles by multiple absorbers.

We study fluctuations of particle absorption by a three-dimensional domain with multiple absorbing patches. The domain is in contact with a gas of interacting diffusing particles. This problem is motivated by living cell sensing via multiple receptors distributed over the cell surface. Employing the macroscopic fluctuation theory, we calculate the covariance matrix of the particle absorption by different patches, extending previous works which addressed fluctuations of a single current. We find a condition when the sign of correlations between different patches is fully determined by the transport coefficients of the gas and is independent of the problem's geometry. We show that the fluctuating particle flux field typically develops vorticity. We establish a simple connection between the statistics of particle absorption by all the patches combined and the statistics of current in a nonequilibrium steady state in one dimension. We also discuss connections between the absorption statistics and (i) statistics of electric currents in multiterminal diffusive conductors and (ii) statistics of wave transmission through disordered media with multiple absorbers.

Survival of interacting diffusing particles inside a domain with absorbing boundary.

Suppose that a d-dimensional domain is filled with a gas of (in general, interacting) diffusive particles with density n_{0}. A particle is absorbed whenever it reaches the domain boundary. Employing macroscopic fluctuation theory, we evaluate the probability P that no particles are absorbed during a long time T. We argue that the most likely gas density profile, conditional on this event, is stationary throughout most of the time T. As a result, P decays exponentially with T for a whole class of interacting diffusive gases in any dimension. For d=1 the stationary gas density profile and P can be found analytically. In higher dimensions we focus on the simple symmetric exclusion process (SSEP) and show that -lnP≃D_{0}TL^{d-2}s(n_{0}), where D_{0} is the gas diffusivity, and L is the linear size of the system. We calculate the rescaled action s(n_{0}) for d=1, for rectangular domains in d=2, and for spherical domains. Near close packing of the SSEP s(n_{0}) can be found analytically for domains of any shape and in any dimension.