A Nonlinear Fluctuation-Dissipation Test for Markovian Systems

Fluctuation-dissipation relations (FDRs) connect the internal spontaneous fluctuations of a system with its response to an external perturbation. In this work we propose a nonlinear generalized FDR (NL FDR) as a test for Markovianity of the considered nonequilibrium system; i.e., the violation of the NL FDR indicates a non-Markovian process. Previously suggested FDRs are based on linear response and require a significant number of measurements. However, the nonlinear relation holds for systems out of equilibrium and for strong perturbations. Therefore, its verification requires significantly less data than the standard linear relation. We test the NL FDR for two theoretical model systems: a particle in a tilted periodic potential and a harmonically bound particle, each driven either by white noise (leading to Markovian test cases, which should obey the NL FDR) or by colored noise (resulting in non-Markovian systems, which may not obey the relation). The degree of violation is systematically explored for the non-Markovian variants of our theoretical models. For the particle in the harmonically bound potential, all statistical measures entering the NL FDR can be calculated explicitly and can be used to elucidate why the relation is violated in the non-Markovian case. In addition, we apply our formalism and test for Markovianity in an inherently out-of-equilibrium experimental system, a tracer particle, embedded in an active bath of self-propelled agents (bristlebots) and subject to a force applied by an external air stream. An experimental violation of the NL FDR is witnessed by introducing an additional timescale to the process, when using bristlebots with two metastable speed states.