  We study functionals, such as heat and work, along trajectories of a class of
multi-dimensional generalized Langevin systems in various limiting situations
that correspond to different level of homogenization. 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 systems are taken to zero. We find
that, unless one restricts to special situations that do not break symmetry of
the Onsager matrix associated with the fast dynamics, it is generally not
possible to express the effective evolution of these functionals solely in
terms of trajectory of the homogenized process describing the system dynamics
via the widely adopted 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 the symmetry breaking and area anomaly. This hold important
consequences for many nonequilibrium systems that can be modeled by generalized
Langevin equations. Our convergence results hold in a strong pathwise sense.
