Generalized fluctuation-dissipation theorem as a test of the Markovianity of a system

¹ Department of Physics, Humboldt Universität zu Berlin - Newtonstr 15, 12489 Berlin, Germany
² Raymond and Beverly Sackler School of Chemistry Tel Aviv University - Tel Aviv 6997801, Israel
³ Bernstein Center for Computational Neuroscience - Haus 2, Philippstr. 13, 10115 Berlin, Germany

Received: 23 March 2017
Accepted: 30 May 2017

Abstract

We study how well a generalized fluctuation-dissipation theorem (GFDT) is suited to test whether a stochastic system is not Markovian. To this end, we simulate a stochastic non-equilibrium model of the mechanosensory hair bundle from the inner ear organ and analyze its spontaneous activity and response to external stimulation. We demonstrate that this two-dimensional Markovian system indeed obeys the GFDT, as long as i) the averaging ensemble is sufficiently large and ii) finite-size effects in estimating the conjugated variable and its susceptibility can be neglected. Furthermore, we test the GFDT also by looking only at a one-dimensional projection of the system, the experimentally accessible position variable. This reduced system is certainly non-Markovian and the GFDT is somewhat violated but not as drastically as for the equilibrium fluctuation-dissipation theorem. We explore suitable measures to quantify the violation of the theorem and demonstrate that for a set of limited experimental data it might be difficult to decide whether the system is Markovian or not.