Markovian systems, in which the statistics of future states depend only on the present state of the system, are an important class of stochastic models that can describe systems both in and out of thermal equilibrium. Determining if some time-series data are Markovian is a crucial first step in model building for nonequilibrium systems. One approach relies on generalized fluctuation-dissipation relations, which connect the internal spontaneous fluctuations of a system with its response to an external perturbation. However, in many specific applications, the typical limited amount of experimental data is insufficient to conduct a proper test. Here, we show how to overcome this challenge and enhance the ability to detect non-Markovian behavior in time-series data.

Specifically, we present a generalized nonlinear fluctuation-dissipation relation. Previously suggested fluctuation-dissipation relations are based on the linear response of a system and require many measurements. A nonlinear relation holds for systems that are out of equilibrium and for strong external perturbations. Therefore, its verification requires less data than the standard linear relation. Using this nonlinear relation, we test for Markovian dynamics with as little as half the amount of data typically required in a couple of model theoretical systems and in an experimental system consisting of a small swarm of active walkers pushing around a Styrofoam ball.

Our ideas can be extended to other stochastic nonequilbrium systems and to multidimensional systems. This work lays the foundation for a simple, model-free, experimental classification of diverse nonequilibrium systems in terms of their Markovian dynamics.