Non-equilibrium transitions in multiscale systems with a bifurcating slow manifold

Noise-induced transitions between metastable fixed points in systems evolving on multiple time scales are analyzed in situations where the time scale separation gives rise to a slow manifold with bifurcation. This analysis is performed within the realm of large deviation theory. It is shown that these non-equilibrium transitions make use of a reaction channel created by the bifurcation structure of the slow manifold, leading to vastly increased transition rates. Several examples are used to illustrate these findings, including an insect outbreak model, a system modeling phase separation in the presence of evaporation, and a system modeling transitions in active matter self-assembly. The last example involves a spatially extended system modeled by a stochastic partial differential equation

Generalized Flows, Intrinsic Stochasticity, and Turbulent Transport

The study of passive scalar transport in a turbulent velocity field leads naturally to the notion of generalized flows which are families of probability distributions on the space of solutions to the associated ODEs, which no longer satisfy the uniqueness theorem for ODEs. Two most natural regularizations of this problem, namely the regularization via adding small molecular diffusion and the regularization via smoothing out the velocity field are considered. White-in-time random velocity fields are used as an example to examine the variety of phenomena that take place when the velocity field is not spatially regular. Three different regimes characterized by their degrees of compressibility are isolated in the parameter space. In the regime of intermediate compressibility, the two different regularizations give rise to two different scaling behavior for the structure functions of the passive scalar. Physically this means that the scaling depends on Prandtl number. In the other two regimes the two different regularizations give rise to the same generalized flows even though the sense of convergence can be very different. The ``one force, one solution'' principle and the existence and uniqueness of an invariant measure are established for the scalar field in the weakly compressible regime, and for the difference of the scalar in the strongly compressible regime.Comment: revised version, 16 pages, no figure