Characterization of detailed balance violation in noise-driven nonlinear dynamical systems

Understanding the spatio-temporal structure of most probable fluctuation pathways to rarely occurring states is a central problem in the study of noise-driven, far-from-equilibrium dynamical systems. When the underlying system does not possess detailed balance, the optimal fluctuation pathway to a particular state and relaxation pathway from that state may combine to form a loop-like structure in the system phase space called a fluctuation loop. Previous work on linear systems has demonstrated the utility of a time-dependent area tensor: at long times, the off-diagonal components grow linearly in time with a coefficient that relates to detailed balance violation [1]. Here, we show that a suitably averaged form of this result applies to general nonlinear systems. Furthermore, for systems with multiple stable fixed points and sufficiently weak noise amplitude, the area tensor exhibits a piecewise linear time dependence as the system rarely makes noise-induced hops from one basin of attraction to another, and is most often found in the neighborhoods of its fixed points. These results are demonstrated using both piecewise linear and continuous nonlinear model systems in two dimensions.
[1] A. Ghanta, J. C. Neu, and S. Teitsworth, Phys. Rev. E 95, 032128 (2017).