Signature of out-of-equilibrium dynamics in a partially observed system

There are numerous challenges in statistically testing whether recordings from a dynamical system show signatures of out-of-equilibrium dynamics. It is common to be able to monitor only some but not all relevant degrees of freedom of a dynamic system, which increases the difficulty of the already challenging inference task. For example, we can record the motion of a fluorescent particle in a cell but do not have direct access to the dynamics of structures with which it interacts. Inspired by the development of [Battle extit{et al.}, Science 352, 604--607 (2016)], we show that a statistical signature of out-of-equilibrium dynamics can be detected in a partially observed system. Focusing on the classic dumbbell with two coupled particles driven by Brownian noise at different temperatures, we show that a spectral Bayesian approach can reliably detect out-of-equilibrium dynamics from the trajectory of a single particle. We study the statistical properties of this inference and discuss future extensions.