Young Researchers Workshop:
Kinetic and macroscopic models for complex systems

Averaged energy dynamics for linear FPEs under periodic friction and noise

Stephan Martin

Imperial College London

In my talk I will first summarize some of our recent works on swarming models. In particular, we show that nonlinearly stability of flocks as a family of states in a second-order model is inherited from first-order stability of the aggregation equation. I will also present advances on the recently introduced Quasi-Morse potentials and their explicit solvability.

In the second part of my talk I will introduce a stochastic averaging method for a class of linear Fokker-Planck equations. If a Hamiltonian system is under additional periodic forcing, say friction, energy dynamics and equilibria will change. Classical stochastic averaging cannot capture these small deviations. Instead, we use a higher-order averaging method based on an asymptotic expansion of the associated Fokker-Planck equation around the unperturbed equilibrium, which is able to (numerically) generate averages in the subtle limit of small noise and small friction. The influence of periodic forcing hence becomes visible in second order and new perturbed equilibria can be approximated. I will present the method and apply it to a highly nonlinear example inherited from the industrial application of fiber spinning processes.

Joint works with (1) J. A. Carrillo, Y. Huang, V. Panferov ; (2) L. L. Bonilla, A. Klar