Response Theory: A Trajectory-Based Approach

Abstract

We collect recent results on deriving useful response relations also for non-equilibrium systems. The approach is based on dynamical ensembles, determined by an action on trajectory space. (Anti)Symmetry under time-reversal separates two complementary contributions in the response, one entropic the other frenetic. Under time-reversal invariance of the unperturbed reference process, only the entropic term is present in the response, giving the standard fluctuation-dissipation relations in equilibrium. For non-equilibrium reference ensembles, the frenetic term contributes essentially and is responsible for new phenomena. We discuss modifications in the Sutherland-Einstein relation, the occurence of negative differential mobilities and the saturation of response. We also indicate how the Einstein relation between noise and friction gets violated for probes coupled to a non-equilibrium environment. We end with some discussion on the situation for quantum phenomena, but the bulk of the text concerns classical mesoscopic (open) systems. The choice of many simple examples is trying to make the notes pedagogical, to introduce an important area of research in non-equilibrium statistical mechanics.

Fluctuation–Dissipation Relations in Active Matter Systems

“…However, in this approach, the FDR involves correlations with the noise and the physical meaning of these terms remain often difficult to catch. Other schemes in this class include the description in terms of "frenetic" contributions [14,15], focusing on the role of time-symmetric fluctuations out of equilibrium (see also in [16] for derivations of analogous FDR for discrete spin variables).…”

“…This approach has been also employed to calculate the transport coefficients, such as the mobility, in combination with an approximate method valid at low-density values or small persistence regimes [35,36]. An attempt to generalize FDR to active systems has been recently presented in [15,37,38]: in the latter case, a relation involving a double-time derivative of correlators appears, making the meaning of the formula and its numerical (and experimental) application not immediate.…”

Fluctuating Motion in an Active Environment

“…We will not use knowledge of the stationary distribution of the medium, even though many aspects are known for the simplest versions [37,44,45]. Instead, we use a trajectory-based response theory [46] that is both applicable to more complicated situations and useful for obtaining the resulting fluctuating dynamics in explicit measurable quantities.…”

“…(10) corresponds to the socalled frenetic contribution in linear response of the same order as the dissipative friction with the kernel, Eq. (12) (see [46,52]). The kernel, Eq.…”

“…The first correction around the quasistatic limit, Eqs. (4)- (5), is provided by the linear response of the medium (see [46] for an introduction). The response or backreaction of the AOU particles determines the fluctuating motion of the probe [27,28,57,58].…”