Linear Response Theory and Green-Kubo Relations for Active Matter

We address the question of how interacting active systems in a nonequilibrium steady state respond to an external perturbation. We establish an extended fluctuation-dissipation theorem for active Brownian particles (ABP), which highlights the role played by the local violation of detailed balance due to activity. By making use of a Markovian approximation we derive closed Green-Kubo expressions for the diffusivity and mobility of ABP and quantify the deviations from the Stokes-Einstein relation. We compute the linear response function to an external force using unperturbed simulations of ABP and compare the results with the analytical predictions of the transport coefficients. Our results show the importance of the interplay between activity and interactions in the departure from equilibrium linear response.

Figure 1

Transport coefficients for ABP at $\varphi =0.01$, 0.1, and 0.2 as a function of Pe. Left: Diffusion coefficient obtained from the long-time behavior of the mean-squared displacement (MSD, with points) compared to the Green-Kubo expression Eq. (21) (GK, with line points). Right: Mobility obtained from simulations of unperturbed ABP using Malliavin weights (MW, with error bars) compared to the Green-Kubo prediction Eq. (19) (GK, with point lines).