# Stochastic fast-slow systems

The evolution of a physical system goverened by a nonlinear differential equation often occurs on two time scales when the parameters affecting the dynamics is small or large. This separation of time scales vastly simplifies their study and is a fundamental requirement to all sorts of perturbation techniques in the parameter space that one could employ. The subclass of linear systems on the hand can be readily generalized to the case of linear partial differential equations like the Fokker-Planck or forward Kolmogorov equations, Euler/Burgers equation using Mori-Zwanzig's projection operator formalism which has the effect integrating out the fast degrees of freedom of the system (e.g. solvent) and projecting this averaging effect onto the slow variables (e.g. time evolution of the spatial coordinate of a protein molecule) whose dyamics now acquires a memory kernel plus noise. The appearance of the memory kernel makes the overall dynamics non-Markovian. The projection operator formalism is a very powerful technique which leads to reduced order modelling of the system where one only tracks (or models) the dynamics of the slow variables with clear separation of two time scales. More details on this formalism can be found in the following excellent presentation:

{%youtube e8QFNh5u_1U %}

For a detailed introduction to slow-fast dynamical systems take a look at [Nils Berglund, Barbara Gentz - Noise-Induced Phenomena in Slow-Fast Dynamical Systems: A Sample-Paths Approach](https://link.springer.com/book/10.1007/1-84628-186-5).

Many real systems, however, possess a continuum of time scales, with no clear separation. In the multiscale analysis of such systems one is concerned with the bridging of disparate scales by merging different mathematical models appropriate at different scales (such as quantum, molecular, and continuum). Finally one looks for a phenomenological description of the phenomena under study that is valid at all scales using the more powerful renormalization group theory.

This paper discusses all these powerful theoretical concepts beautifully.

https://www.pnas.org/doi/10.1073/pnas.2102266118