Generalized Fluctuation-Dissipation relations holding in non-equilibrium dynamics

Abstract

We derive generalized Fluctuation-Dissipation Relations (FDR) holding for a general stochastic dynamics that includes as subcases both equilibrium models for passive colloids and nonequilibrium models used to describe active particles. The relations reported here differ from previous formulations of the FDR because of their simplicity: they require only the microscopic knowledge of the dynamics instead of the whole expression of the steady-state probability distribution function that, except for linear interactions, is unknown for systems displaying non-vanishing currents. From the response function, we can extrapolate generalized versions of the Mesoscopic Virial equation and the equipartition theorem, which still holds far from equilibrium. Our results are tested in the case of equilibrium colloids described by underdamped or overdamped Langevin equations and for models describing the non-equilibrium behavior of active particles. Both the Active Brownian Particle and the Active Ornstein-Uhlenbeck particle models are compared in the case of a single particle confined in an external potential.

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

“…In particular, in the case of Gaussian additive noises the generalized FDR can be further simplified exploiting the stationarity of the process and it can be expressed as a time-correlation of the dynamical variables only (i.e., without involving their time-derivatives). In this way one recovers the relation already derived by one of us in [26] (see also Ref. [53] for an application in turbulent systems):…”

“…The broad interest in the FDR emerged also in other areas of science ranging from equilibrium and non-equilibrium colloids [18,19], to granular particles [20][21][22], and even to biological systems, usually classified as active matter [23][24][25]. In these cases, generalized FDRs have been derived using nearequilibrium approximations [26] and have been employed to provide expressions for the transport coefficients [27], the effective temperature [28][29][30] and the effect of mild shear [31,32]. Exact relations for a colloid in an active environment have been recently investigated [33].…”

“…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.…”

Nonequilibrium statistical mechanics for stationary turbulent dispersion

“…The ergodicity of the stochastic systems (38) and (43) guarantees the unique NESS state can be achieved using a long-time simulation of ( 38) and (43), and the ensemble average can be replaced by the time average. The calculation formula we are lead to is the previously announced Eqn (10). On the other hand, using the proof we have shown in Appendix B, it is straightforward to obtain response formulas corresponding to other types of perturbations.…”

“…(III) The above analogies allow a heuristic derivation of the generalized fluctuation-dissipation relations (FDRs) for observables in the fluid system, in particular, for the Lagrangian particle. To this end, we use the result which are recently developed for the nonequilibrium molecular systems [4,5,6,10,40,64,1,39,38,64]. The heuristic derivation is later justified with rigorous mathematical proofs.…”