A Note on Homogenization of Dynamics and Functionals of Generalized Langevin Systems

Soon Hoe Lim

Published 2019-11-18Version 1

We study homogenized dynamics of a class of multi-dimensional generalized Langevin systems and functionals along their trajectory in various limiting situations corresponding to different level of coarse graining. These are the situations where one or more of the inertial time scale(s), the memory time scale(s) and the noise correlation time scale(s) of the system are taken to zero. We find that, unless one restricts to special situations evoking symmetry, it is generally not possible to express the effective evolution of these functionals solely in terms of trajectory of the effective process describing the system dynamics via the Stratonovich convention. In fact, an anomalous term is often needed for a complete description, implying that convergence of these functionals needs more information than simply the limit of the dynamical process. We trace the origin of such impossibility to area anomaly, thereby linking symmetry breaking and area anomaly, and discuss its consequences for nonequilibrium systems. Moreover, our convergence results hold in a strong pathwise sense.