Nucleation and Extinction in Nonequilibrium Statistical Field Theories

Many mathematical models address systems with so many degrees of freedom (such as the positions and velocities of molecules in a fluid) that they are best described by continuous fields (such as the density and velocity fields governing fluid flow). Often these fields undergo noisy dynamics. The noise represents random external forcing, or the effects of unresolved microscopic motion. Even when small, noise can have huge effects, carrying the system from one state to another through a region that is uncrossable without noise. An example is the nucleation of a liquid droplet within a vapour in thermal equilibrium. Here the nucleation rate is calculable from the free energy, expressed as a function of the droplet radius. The rate is exponentially small in the maximum free energy ('barrier height') along the 'reaction pathway', which is the pathway of lowest barrier height. This type of calculation for the rare-event rate is called 'classical nucleation theory' (CNT). However, there is currently no counterpart of CNT for many other nucleation problems: those arising far from equilibrium. For these, in general, no free energy can be defined. An example is the vapour-liquid transition in 'active' systems whose particles steadily convert chemical energy into motion. These include swarming micro-organisms such as bacteria, and also synthetic systems of micron-sized self-propelled particles. Other nonequilibrium examples of nucleation type include (a) the arrival of an invading species into a new environment: if a critical population size is reached it will grow further, otherwise the new colony will die out; (b) the noise-induced transition between different flow patterns in an externally pumped fluid. A closely related class of problems arise when a rare sequence of small random disturbances (the noise) leads to the extinction of an otherwise stable population, gene, or behavioural trait. Recent progress in the mathematical field of large deviation theory (LDT) has created tools whereby nonequilibrium rare-events of both nucleation and extinction type can be studied. However, these tools are yet to be developed for systems whose degrees of freedom are best described by continuous fields. We propose to create a new LDT methodology, analagous to CNT, to address a wide range of nonequilibrium problems involving continuous fields. To achieve this, we will develop a four-fold path. Step 1 is to identify a handful of reduced coordinates, that can track the progress of the rare event. Good coordinates may emerge from mechanistic insight, but if not we can alternatively identify them using machine learning (ML). Step 2 is to calculate barrier crossing rates numerically, both in the reduced coordinate space and in a larger one found by representing accurately the fields in a brute-force basis (e.g., Fourier modes). Comparing the outcomes will reveal whether the reduced representation needs improvement, either via better mechanistic insight, or via further ML. Once a good representation is found, step 3 is to calculate the 'quasipotential landscape' (an analogue of free energy far from equilibrium), and step 4 to reconstruct the full noisy dynamics of the dimensionally-reduced model. Our work will exploit innovative numerical methods that we have recently developed, allowing rapid and efficient study of rare events in (so far, modest-dimensional) noisy systems. The new approach will then be used to explore nucleation and extinction problems in active matter physics, quantitative biology, ecology, social models, and fluid dynamics.