Irreversibility in linear systems with colored noise.

Time irreversibility is a distinctive feature of nonequilibrium dynamics and several measures of irreversibility have been introduced to assess the distance from thermal equilibrium of a stochastically driven system. While the dynamical noise is often approximated as white, in many real applications the time correlations of the random forces can actually be significantly long-lived compared to the relaxation times of the driven system. We analyze the effects of temporal correlations in the noise on commonly used measures of irreversibility and demonstrate how the theoretical framework for white-noise-driven systems naturally generalizes to the case of colored noise. Specifically, we express the autocorrelation function, the area enclosing rates, and mean phase space velocity in terms of solutions of a Lyapunov equation and in terms of their white-noise limit values.

Learning the non-equilibrium dynamics of Brownian movies.

Time-lapse microscopy imaging provides direct access to the dynamics of soft and living systems. At mesoscopic scales, such microscopy experiments reveal intrinsic thermal and non-equilibrium fluctuations. These fluctuations, together with measurement noise, pose a challenge for the dynamical analysis of these Brownian movies. Traditionally, methods to analyze such experimental data rely on tracking embedded or endogenous probes. However, it is in general unclear, especially in complex many-body systems, which degrees of freedom are the most informative about their non-equilibrium nature. Here, we introduce an alternative, tracking-free approach that overcomes these difficulties via an unsupervised analysis of the Brownian movie. We develop a dimensional reduction scheme selecting a basis of modes based on dissipation. Subsequently, we learn the non-equilibrium dynamics, thereby estimating the entropy production rate and time-resolved force maps. After benchmarking our method against a minimal model, we illustrate its broader applicability with an example inspired by active biopolymer gels.

Mesoscopic non-equilibrium measures can reveal intrinsic features of the active driving.

Biological assemblies such as chromosomes, membranes, and the cytoskeleton are driven out of equilibrium at the nanoscale by enzymatic activity and molecular motors. Similar non-equilibrium dynamics can be realized in synthetic systems, such as chemically fueled colloidal particles. Characterizing the stochastic non-equilibrium dynamics of such active soft assemblies still remains a challenge. Recently, new non-invasive approaches have been proposed to determine the non-equilibrium behavior, which are based on detecting broken detailed balance in the stochastic trajectories of several coordinates of the system. Inspired by the method of two-point microrheology, in which the equilibrium fluctuations of a pair of probe particles reveal the viscoelastic response of an equilibrium system, here, we investigate whether we can extend such an approach to non-equilibrium assemblies: can one extract information on the nature of the active driving in a system from the analysis of a two-point non-equilibrium measure? We address this question theoretically in the context of a class of elastic systems, driven out of equilibrium by a spatially heterogeneous stochastic internal driving. We consider several scenarios for the spatial features of the internal driving that may be relevant in biological and synthetic systems, and investigate how such features of the active noise may be reflected in the long-range scaling behavior of two-point non-equilibrium measures.

Scaling behavior of nonequilibrium measures in internally driven elastic assemblies.

Detecting and quantifying nonequilibrium activity is essential for studying internally driven assemblies, including synthetic active matter and complex living systems such as cells or tissue. We discuss a noninvasive approach of measuring nonequilibrium behavior based on the breaking of detailed balance. We focus on "cycling frequencies"-the average frequency with which the trajectories of pairs of degrees of freedom revolve in phase space-and explain their connection with other nonequilibrium measures, including the area enclosing rate and the entropy production rate. We test our approach on simple toy models composed of elastic networks immersed in a viscous fluid with site-dependent internal driving. We prove both numerically and analytically that the cycling frequencies obey a power law as a function of distance between the tracked degrees of freedom. Importantly, the behavior of the cycling frequencies contains information about the dimensionality of the system and the amplitude of active noise. The mapping we use in our analytical approach thus offers a convenient framework for predicting the behavior of two-point nonequilibrium measures for a given activity distribution in the network.

Nonequilibrium Scaling Behavior in Driven Soft Biological Assemblies.

Measuring and quantifying nonequilibrium dynamics in active biological systems is a major challenge because of their intrinsic stochastic nature and the limited number of variables accessible in any real experiment. We investigate what nonequilibrium information can be extracted from noninvasive measurements using a stochastic model of soft elastic networks with a heterogeneous distribution of activities, representing enzymatic force generation. In particular, we use this model to study how the nonequilibrium activity, detected by tracking two probes in the network, scales as a function of the distance between the probes. We quantify the nonequilibrium dynamics through the cycling frequencies, a simple measure of circulating currents in the phase space of the probes. We find that these cycling frequencies exhibit power-law scaling behavior with the distance between probes. In addition, we show that this scaling behavior governs the entropy production rate that can be recovered from the two traced probes. Our results provide insight into how internal enzymatic driving generates nonequilibrium dynamics on different scales in soft biological assemblies.