On the fluctuation-dissipation relation in non-equilibrium and non-Hamiltonian systems

Generalized Langevin equation with shear flow and its fluctuation-dissipation theorems derived from a Caldeira-Leggett Hamiltonian

We provide a first-principles derivation of the Langevin equation with shear flow and its corresponding fluctuation-dissipation theorems. Shear flow of simple fluids has been widely investigated by numerical simulations. Most studies postulate a Markovian Langevin equation with a simple shear drag term in the manner of Stokes. However, this choice has never been justified from first principles. We start from a particle-bath system described by a classical Caldeira-Leggett Hamiltonian modified by adding a term proportional to the strain-rate tensor according to Hoover's DOLLS method, and we derive a generalized Langevin equation for the sheared system. We then compute, analytically, the noise time-correlation functions in different regimes. Based on the intensity of the shear rate, we can distinguish between close-to-equilibrium and far-from-equilibrium states. According to the results presented here, the standard, simple, and Markovian form of the Langevin equation with shear flow postulated in the literature is valid only in the limit of extremely weak shear rates compared to the effective vibrational temperature of the bath. For even marginally higher shear rates, the (generalized) Langevin equation is strongly non-Markovian, and nontrivial fluctuation-dissipation theorems are derived.

Inertial and geometrical effects of self-propelled elliptical Brownian particles

Active particles that self-propel by transforming energy into mechanical motion represent a growing area of research in mathematics, physics, and chemistry. Here we investigate the dynamics of nonspherical inertial active particles moving in a harmonic potential, introducing geometric parameters which take into account the role of eccentricity for nonspherical particles. A comparison between the overdamped and underdamped models for elliptical particles is performed. The model of overdamped active Brownian motion has been used to describe most of the basic aspects of micrometer-sized particles moving in a liquid ("microswimmers"). We consider active particles by extending the active Brownian motion model to incorporate translation and rotation inertia and account for the role of eccentricity. We show how the overdamped and the underdamped models behave in the same way for small values of activity (Brownian case) if eccentricity is equal to zero, but increasing eccentricity leads the two dynamics to substantially depart from each other-in particular, the action of a torque induced by external forces, induced a marked difference close to the walls of the domain if eccentricity is high. Effects induced by inertia include an inertial delay time of the self-propulsion direction from the particle velocity, and the differences between the overdamped and underdamped systems are particularly evident in the first and second moments of the particle velocities. Comparison with the experimental results of vibrated granular particles shows good agreement and corroborates the notion that self-propelling massive particles moving in gaseous media are dominated by inertial effects.

Time-reversal symmetries and equilibriumlike Langevin equations

Graham has shown [Z. Phys. B 26, 397 (1977)0340-224X10.1007/BF01570750] that a fluctuation-dissipation relation can be imposed on a class of nonequilibrium Markovian Langevin equations that admit a stationary solution of the corresponding Fokker-Planck equation. The resulting equilibrium form of the Langevin equation is associated with a nonequilibrium Hamiltonian. Here we provide some explicit insight into how this Hamiltonian may lose its time-reversal invariance and how the "reactive" and "dissipative" fluxes loose their distinct time-reversal symmetries. The antisymmetric coupling matrix between forces and fluxes no longer originates from Poisson brackets and the "reactive" fluxes contribute to the ("housekeeping") entropy production, in the steady state. The time-reversal even and odd parts of the nonequilibrium Hamiltonian contribute in qualitatively different but physically instructive ways to the entropy. We find instances where fluctuations due to noise are solely responsible for the dissipation. Finally, this structure gives rise to a new, physically pertinent instance of frenesy.

Spectral fingerprints of nonequilibrium dynamics: The case of a Brownian gyrator

The same system can exhibit a completely different dynamical behavior when it evolves in equilibrium conditions or when it is driven out-of-equilibrium by, e.g., connecting some of its components to heat baths kept at different temperatures. Here we concentrate on an analytically solvable and experimentally relevant model of such a system-the so-called Brownian gyrator-a two-dimensional nanomachine that performs a systematic, on average, rotation around the origin under nonequilibrium conditions, while no net rotation takes place under equilibrium ones. On this example, we discuss a question whether it is possible to distinguish between two types of a behavior judging not upon the statistical properties of the trajectories of components but rather upon their respective spectral densities. The latter are widely used to characterize diverse dynamical systems and are routinely calculated from the data using standard built-in packages. From such a perspective, we inquire whether the power spectral densities possess some "fingerprint" properties specific to the behavior in nonequilibrium. We show that indeed one can conclusively distinguish between equilibrium and nonequilibrium dynamics by analyzing the cross-correlations between the spectral densities of both components in the short frequency limit, or from the spectral densities of both components evaluated at zero frequency. Our analytical predictions, corroborated by experimental and numerical results, open a new direction for the analysis of a nonequilibrium dynamics.

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

We introduce a general formulation of the fluctuation-dissipation relations (FDRs) holding also in far-from-equilibrium stochastic dynamics. A great advantage of this version of the FDR is that it does not require explicit knowledge of the stationary probability density function. Our formula applies to Markov stochastic systems with generic noise distributions: When the noise is additive and Gaussian, the relation reduces to those known in the literature; for multiplicative and non-Gaussian distributions (e.g., Cauchy noise) it provides exact results in agreement with numerical simulations. Our formula allows us to reproduce, in a suitable small-noise limit, the response functions of deterministic, strongly nonlinear dynamical models, even in the presence of chaotic behavior: This could have important practical applications in several contexts, including geophysics and climate. As a case of study, we consider the Lorenz '63 model, which is paradigmatic for the chaotic properties of deterministic dynamical systems.

Response and flux of information in extended nonequilibrium dynamics

It is well known that entropy production is a proxy to the detection of nonequilibrium, i.e., of the absence of detailed balance; however, due to the global character of this quantity, its knowledge does not allow to identify spatial currents or fluxes of information among specific elements of the system under study. In this respect, much more insight can be gained by studying transfer entropy and response, which allow quantifying the relative influence of parts of the system and the asymmetry of the fluxes. In order to understand the relation between the above-mentioned quantities, we investigate spatially asymmetric extended systems. First, we consider a simplified linear stochastic model, which can be studied analytically; then, we include nonlinear terms in the dynamics. Extensive numerical investigation shows the relation between entropy production and the above-introduced degrees of asymmetry. Finally, we apply our approach to the highly nontrivial dynamics generated by the Lorenz 96 model for Earth oceanic circulation.

Spectroscopy of phase transitions for multiagent systems

In this paper, we study phase transitions for weakly interacting multiagent systems. By investigating the linear response of a system composed of a finite number of agents, we are able to probe the emergence in the thermodynamic limit of a singular behavior of the susceptibility. We find clear evidence of the loss of analyticity due to a pole crossing the real axis of frequencies. Such behavior has a degree of universality, as it does not depend on either the applied forcing or on the considered observable. We present results relevant for both equilibrium and nonequilibrium phase transitions by studying the Desai-Zwanzig and Bonilla-Casado-Morillo models.

Fluctuation–Dissipation Relations in Active Matter Systems

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.

Response theory and phase transitions for the thermodynamic limit of interacting identical systems

We study the response to perturbations in the thermodynamic limit of a network of coupled identical agents undergoing a stochastic evolution which, in general, describes non-equilibrium conditions. All systems are nudged towards the common centre of mass. We derive Kramers–Kronig relations and sum rules for the linear susceptibilities obtained through mean field Fokker–Planck equations and then propose corrections relevant for the macroscopic case, which incorporates in a self-consistent way the effect of the mutual interaction between the systems. Such an interaction creates a memory effect. We are able to derive conditions determining the occurrence of phase transitions specifically due to system-to-system interactions. Such phase transitions exist in the thermodynamic limit and are associated with the divergence of the linear response but are not accompanied by the divergence in the integrated autocorrelation time for a suitably defined observable. We clarify that such endogenous phase transitions are fundamentally different from other pathologies in the linear response that can be framed in the context of critical transitions. Finally, we show how our results can elucidate the properties of the Desai–Zwanzig model and of the Bonilla–Casado–Morillo model, which feature paradigmatic equilibrium and non-equilibrium phase transitions, respectively.

Effective temperatures in inhomogeneous passive and active bidimensional Brownian particle systems

We study the stationary dynamics of an active interacting Brownian particle system. We measure the violations of the fluctuation dissipation theorem, and the corresponding effective temperature, in a locally resolved way. Quite naturally, in the homogeneous phases the diffusive properties and effective temperature are also homogeneous. Instead, in the inhomogeneous phases (close to equilibrium and within the MIPS sector) the particles can be separated in two groups with different diffusion properties and effective temperatures. Notably, at fixed activity strength the effective temperatures in the two phases remain distinct and approximately constant within the MIPS region, with values corresponding to the ones of the whole system at the boundaries of this sector of the phase diagram. We complement the study of the globally averaged properties with the theoretical and numerical characterization of the fluctuation distributions of the single-particle diffusion, linear response, and effective temperature in the homogeneous and inhomogeneous phases. We also distinguish the behavior of the (time-delayed) effective temperature from the (instantaneous) kinetic temperature, showing that the former is independent of the friction coefficient.