Fluctuation–Dissipation Relations in Active Matter Systems

Abstract

We investigate the non-equilibrium character of self-propelled particles through the study of the linear response of the active Ornstein–Uhlenbeck particle (AOUP) model. We express the linear response in terms of correlations computed in the absence of perturbations, proposing a particularly compact and readable fluctuation–dissipation relation (FDR): such an expression explicitly separates equilibrium and non-equilibrium contributions due to self-propulsion. As a case study, we consider non-interacting AOUP confined in single-well and double-well potentials. In the former case, we also unveil the effect of dimensionality, studying one-, two-, and three-dimensional dynamics. We show that information about the distance from equilibrium can be deduced from the FDR, putting in evidence the roles of position and velocity variables in the non-equilibrium relaxation.

Generalized Fluctuation-Dissipation relations holding in non-equilibrium dynamics

“…For instance, this technique has been employed to numerically calculate i) the effective temperature of active systems [30][31][32][33][34], with a recent attention to phase-separation [35], and ii) the transport coefficients, such as the mobility, to test an approximated prediction valid at low-density values [36,37] iii) the response function due to a shear flow [38]. Finally, in recent studies based on path-integral approaches, generalized versions of the FDR holding also in far from equilibrium regimes have been reported in the specific case of athermal active particles [39,40].…”

Critical active dynamics is captured by a colored-noise driven field theory

“…The first is that computing χ(q, t) at different q-values requires a different simulation for each q. Secondly, to ensure that the field amplitude is small enough to avoid non-linear effects, one should repeat the simulations for various values of h. Fortunately, applying the field is not required for measuring χ(q, t) in AOUPs. Indeed, among the "family" of active models with exponentially correlated noise [52], AOUPs have the unique feature that the exact linear response of any dynamic variable can be computed in unperturbed simulations as demonstrated in Ref.s [39,40]. In the present work we employ the method developed by Szamel [40], which generalizes the Malliavin weights method [53] to systems driven by persistent Gaussian noise and efficiently combines it with parallel (GPU-based) simulations.…”

“…This approach has been widely employed to study the properties of several off-equilibrium systems such as glasses, gels and granulars [25][26][27][28][29][30][31][32][33][34]. Simultaneous measurements of response and correlation functions have also been used to reveal non-equilibrium fluctuations in active particle simulations [35][36][37][38][39][40] and in active experimental systems, such as living red-blood cell membranes [41] and suspensions of swimming bacteria probed by passive tracers [42,43].…”

A handy fluctuation-dissipation relation to approach generic noisy systems and chaotic dynamics

“…In gradient systems an easy computation shows that both correlations are O(D) (see for instance the results in Ref. [48]). On the contrary, when the dynamics is chaotic and dissipative, an extended attractor is present and the fluctuations are mainly ruled by the deterministic dynamics.…”

“…Examples were studied in the context of spin systems [41] and non-equilibrium colloidal particles, both in overdamped [42][43][44][45] and underdamped dynamics [46,47]. In these cases, FDRs have been expressed in terms of time-correlations involving the dynamical variables and their time-derivatives; recently, a version of the FDR only depending on time-correlations of dynamical variables have been derived for a specific active matter model [48] and, more generally, for additive non-equilibrium stochastic processes [26] also establishing an interesting connection with virial and equipartition theorems.…”