Thermodynamic bounds for diffusion in nonequilibrium systems with multiple timescales.

We derive a thermodynamic uncertainty relation bounding the mean squared displacement of a Gaussian process with memory, driven out of equilibrium by unbalanced thermal baths and/or by external forces. Our bound is tighter with respect to previous results and also holds at finite time. We apply our findings to experimental and numerical data for a vibrofluidized granular medium, characterized by regimes of anomalous diffusion. In some cases our relation can distinguish between equilibrium and nonequilibrium behavior, a nontrivial inference task, particularly for Gaussian processes.

Power spectrum and critical exponents in the 2D stochastic Wilson-Cowan model.

The power spectrum of brain activity is composed by peaks at characteristic frequencies superimposed to a background that decays as a power law of the frequency, [Formula: see text], with an exponent [Formula: see text] close to 1 (pink noise). This exponent is predicted to be connected with the exponent [Formula: see text] related to the scaling of the average size with the duration of avalanches of activity. "Mean field" models of neural dynamics predict exponents [Formula: see text] and [Formula: see text] equal or near 2 at criticality (brown noise), including the simple branching model and the fully-connected stochastic Wilson-Cowan model. We here show that a 2D version of the stochastic Wilson-Cowan model, where neuron connections decay exponentially with the distance, is characterized by exponents [Formula: see text] and [Formula: see text] markedly different from those of mean field, respectively around 1 and 1.3. The exponents [Formula: see text] and [Formula: see text] of avalanche size and duration distributions, equal to 1.5 and 2 in mean field, decrease respectively to [Formula: see text] and [Formula: see text]. This seems to suggest the possibility of a different universality class for the model in finite dimension.

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

We review generalized fluctuation-dissipation relations, which are valid under general conditions even in "nonstandard systems," e.g., out of equilibrium and/or without a Hamiltonian structure. The response functions can be expressed in terms of suitable correlation functions computed in the unperturbed dynamics. In these relations, typically, one has nontrivial contributions due to the form of the stationary probability distribution; such terms take into account the interaction among the relevant degrees of freedom in the system. We illustrate the general formalism with some examples in nonstandard cases, including driven granular media, systems with a multiscale structure, active matter, and systems showing anomalous diffusion.

Induced and endogenous acoustic oscillations in granular faults.

The frictional properties of disordered systems are affected by external perturbations. These perturbations usually weaken the system by reducing the macroscopic friction coefficient. This friction reduction is of particular interest in the case of disordered systems composed of granular particles confined between two plates, as this is a simple model of seismic fault. Indeed, in the geophysical context frictional weakening could explain the unexpected weakness of some faults, as well as earthquake remote triggering. In this manuscript, we review recent results concerning the response of confined granular systems to external perturbations, considering the different mechanisms by which the perturbation could weaken a system, the relevance of the frictional reduction to earthquakes, as well as discussing the intriguing scenario whereby the weakening is not monotonic in the perturbation frequency, so that a re-entrant transition is observed, as the system first enters a fluidized state and then returns to a frictional state.This article is part of the theme issue 'Statistical physics of fracture and earthquakes'.

Tracer diffusion in crowded narrow channels.

We summarise different results on the diffusion of a tracer particle in lattice gases of hard-core particles with stochastic dynamics, which are confined to narrow channels-single-files, comb-like structures and quasi-one-dimensional channels with the width equal to several particle diameters. We show that in such geometries a surprisingly rich, sometimes even counter-intuitive, behaviour emerges, which is absent in unbounded systems. This is well-documented for the anomalous diffusion in single-files. Less known is the anomalous dynamics of a tracer particle in crowded branching single-files-comb-like structures, where several kinds of anomalous regimes take place. In narrow channels, which are broader than single-files, one encounters a wealth of anomalous behaviours in the case where the tracer particle is subject to a regular external bias: here, one observes an anomaly in the temporal evolution of the tracer particle velocity, super-diffusive at transient stages, and ultimately a giant diffusive broadening of fluctuations in the position of the tracer particle, as well as spectacular multi-tracer effects of self-clogging of narrow channels. Interactions between a biased tracer particle and a confined crowded environment also produce peculiar patterns in the out-of-equilibrium distribution of the environment particles, very different from the ones appearing in unbounded systems. For moderately dense systems, a surprising effect of a negative differential mobility takes place, such that the velocity of a biased tracer particle can be a non-monotonic function of the force. In some parameter ranges, both the velocity and the diffusion coefficient of a biased tracer particle can be non-monotonic functions of the density. We also survey different results obtained for a tracer particle diffusion in unbounded systems, which will permit a reader to have an exhaustively broad picture of the tracer diffusion in crowded environments.

Anomalous mobility of a driven active particle in a steady laminar flow.

We study, via extensive numerical simulations, the force-velocity curve of an active particle advected by a steady laminar flow, in the nonlinear response regime. Our model for an active particle relies on a colored noise term that mimics its persistent motion over a time scale [Formula: see text]. We find that the active particle dynamics shows non-trivial effects, such as negative differential and absolute mobility (NDM and ANM, respectively). We explore the space of the model parameters and compare the observed behaviors with those obtained for a passive particle ([Formula: see text]) advected by the same laminar flow. Our results show that the phenomena of NDM and ANM are quite robust with respect to the details of the considered noise: in particular for finite [Formula: see text] a more complex force-velocity relation can be observed.

Controlled Viscosity in Dense Granular Materials.

We experimentally investigate the fluidization of a granular material subject to mechanical vibrations by monitoring the angular velocity of a vane suspended in the medium and driven by an external motor. On increasing the frequency, we observe a reentrant transition, as a jammed system first enters a fluidized state, where the vane rotates with high constant velocity, and then returns to a frictional state, where the vane velocity is much lower. While the fluidization frequency is material independent, the viscosity recovery frequency shows a clear dependence on the material that we rationalize by relating this frequency to the balance between dissipative and inertial forces in the system. Molecular dynamics simulations well reproduce the experimental data, confirming the suggested theoretical picture.

Anomalous force-velocity relation of driven inertial tracers in steady laminar flows.

We study the nonlinear response to an external force of an inertial tracer advected by a two-dimensional incompressible laminar flow and subject to thermal noise. In addition to the driving external field F, the main parameters in the system are the noise amplitude [Formula: see text] and the characteristic Stokes time [Formula: see text] of the tracer. The relation velocity vs. force shows interesting effects, such as negative differential mobility (NDM), namely a non-monotonic behavior of the tracer velocity as a function of the applied force, and absolute negative mobility (ANM), i.e. a net motion against the bias. By extensive numerical simulations, we investigate the phase chart in the parameter space of the model, [Formula: see text], identifying the regions where NDM, ANM and more common monotonic behaviors of the force-velocity curve are observed.

Heat fluctuations of Brownian oscillators in nonstationary processes: Fluctuation theorem and condensation transition.

We study analytically the probability distribution of the heat released by an ensemble of harmonic oscillators to the thermal bath, in the nonequilibrium relaxation process following a temperature quench. We focus on the asymmetry properties of the heat distribution in the nonstationary dynamics, in order to study the forms taken by the fluctuation theorem as the number of degrees of freedom is varied. After analyzing in great detail the cases of one and two oscillators, we consider the limit of a large number of oscillators, where the behavior of fluctuations is enriched by a condensation transition with a nontrivial phase diagram, characterized by reentrant behavior. Numerical simulations confirm our analytical findings. We also discuss and highlight how concepts borrowed from the study of fluctuations in equilibrium under symmetry-breaking conditions [Gaspard, J. Stat. Mech. (2012) P0802110.1088/1742-5468/2012/08/P08021] turn out to be quite useful in understanding the deviations from the standard fluctuation theorem.

Self-propulsion against a moving membrane: Enhanced accumulation and drag force.

Self-propulsion (SP) is a main feature of active particles (AP), such as bacteria or biological micromotors, distinguishing them from passive colloids. A renowned consequence of SP is accumulation at static interfaces, even in the absence of hydrodynamic interactions. Here we address the role of SP in the interaction between AP and a moving semipermeable membrane. In particular, we implement a model of noninteracting AP in a channel crossed by a partially penetrable wall, moving at a constant velocity c. With respect to both the cases of passive colloids with c>0 and AP with c=0, the AP with finite c show enhancement of accumulation in front of the obstacle and experience a largely increased drag force. This effect is understood in terms of an effective potential localised at the interface between particles and membrane, of height proportional to cτ/ξ, where τ is the AP's reorientation time and ξ the width characterizing the surface's smoothness (ξ→0 for hard core obstacles). An approximate analytical scheme is able to reproduce the observed density profiles and the measured drag force, in very good agreement with numerical simulations. The effects discussed here can be exploited for automatic selection and filtering of AP with desired parameters.

Nonlinear Response of Inertial Tracers in Steady Laminar Flows: Differential and Absolute Negative Mobility.

We study the mobility and the diffusion coefficient of an inertial tracer advected by a two-dimensional incompressible laminar flow, in the presence of thermal noise and under the action of an external force. We show, with extensive numerical simulations, that the force-velocity relation for the tracer, in the nonlinear regime, displays complex and rich behaviors, including negative differential and absolute mobility. These effects rely upon a subtle coupling between inertia and applied force that induces the tracer to persist in particular regions of phase space with a velocity opposite to the force. The relevance of this coupling is revisited in the framework of nonequilibrium response theory, applying a generalized Einstein relation to our system. The possibility of experimental observation of these results is also discussed.

Nonlinear response and emerging nonequilibrium microstructures for biased diffusion in confined crowded environments.

We study analytically the dynamics and the microstructural changes of a host medium caused by a driven tracer particle moving in a confined, quiescent molecular crowding environment. Imitating typical settings of active microrheology experiments, we consider here a minimal model comprising a geometrically confined lattice system (a two-dimensional striplike or a three-dimensional capillary-like system) populated by two types of hard-core particles with stochastic dynamics (a tracer particle driven by a constant external force and bath particles moving completely at random). Resorting to a decoupling scheme, which permits us to go beyond the linear-response approximation (Stokes regime) for arbitrary densities of the lattice gas particles, we determine the force-velocity relation for the tracer particle and the stationary density profiles of the host medium particles around it. These results are validated a posteriori by extensive numerical simulations for a wide range of parameters. Our theoretical analysis reveals two striking features: (a) We show that, under certain conditions, the terminal velocity of the driven tracer particle is a nonmonotonic function of the force, so in some parameter range the differential mobility becomes negative, and (b) the biased particle drives the whole system into a nonequilibrium steady state with a stationary particle density profile past the tracer, which decays exponentially, in sharp contrast with the behavior observed for unbounded lattices, where an algebraic decay is known to take place.

Diffusion and Subdiffusion of Interacting Particles on Comblike Structures.

We study the dynamics of a tracer particle (TP) on a comb lattice populated by randomly moving hard-core particles in the dense limit. We first consider the case where the TP is constrained to move on the backbone of the comb only. In the limit of high density of the particles, we present exact analytical results for the cumulants of the TP position, showing a subdiffusive behavior ∼t^{3/4}. At longer times, a second regime is observed where standard diffusion is recovered, with a surprising nonanalytical dependence of the diffusion coefficient on the particle density. When the TP is allowed to visit the teeth of the comb, based on a mean-field-like continuous time random walk description, we unveil a rich and complex scenario with several successive subdiffusive regimes, resulting from the coupling between the geometrical constraints of the comb lattice and particle interactions. In this case, remarkably, the presence of hard-core interactions asymptotically speeds up the TP motion along the backbone of the structure.

Fluctuations in partitioning systems with few degrees of freedom.

We study the behavior of a moving wall in contact with a particle gas and subjected to an external force. We compare the fluctuations of the system observed in the microcanonical and canonical ensembles, by varying the number of particles. Static and dynamic correlations signal significant differences between the two ensembles. Furthermore, velocity-velocity correlations of the moving wall present a complex two-time relaxation that cannot be reproduced by a standard Langevin-like description. Quite remarkably, increasing the number of gas particles in an elongated geometry, we find a typical time scale, related to the interaction between the partitioning wall and the particles, which grows macroscopically.

Nonequilibrium fluctuation-dissipation theorem and heat production.

We use a relationship between response and correlation function in nonequilibrium systems to establish a connection between the heat production and the deviations from the equilibrium fluctuation-dissipation theorem. This scheme extends the Harada-Sasa formulation [Phys. Rev. Lett. 95, 130602 (2005)], obtained for Langevin equations in steady states, as it also holds for transient regimes and for discrete jump processes involving small entropic changes. Moreover, a general formulation includes two times and the new concepts of two-time work, kinetic energy, and of a two-time heat exchange that can be related to a nonequilibrium "effective temperature." Numerical simulations of a chain of anharmonic oscillators and of a model for a molecular motor driven by adenosine triphosphate hydrolysis illustrate these points.

Microscopic theory for negative differential mobility in crowded environments.

We study the behavior of the stationary velocity of a driven particle in an environment of mobile hard-core obstacles. Based on a lattice gas model, we demonstrate analytically that the drift velocity can exhibit a nonmonotonic dependence on the applied force, and show quantitatively that such negative differential mobility (NDM), observed in various physical contexts, is controlled by both the density and diffusion time scale of the obstacles. Our study unifies recent numerical and analytical results obtained in specific regimes, and makes it possible to determine analytically the region of the full parameter space where NDM occurs. These results suggest that NDM could be a generic feature of biased (or active) transport in crowded environments.

Time asymmetry of the Kramers equation with nonlinear friction: fluctuation-dissipation relation and ratchet effect.

We show by numerical simulations that the presence of nonlinear velocity-dependent friction forces can induce a finite net drift in the stochastic motion of a particle in contact with an equilibrium thermal bath and in an asymmetric periodic spatial potential. In particular, we study the Kramers equation for a particle subjected to Coulomb friction, namely a constant force acting in the direction opposite to the particle's velocity. We characterize the nonequilibrium irreversible dynamics by studying the generalized fluctuation-dissipation relation for this ratchet model driven by Coulomb friction.

Ratchet effect driven by Coulomb friction: the asymmetric Rayleigh piston.

The effect of Coulomb friction is studied in the framework of collisional ratchets. It turns out that the average drift of these devices can be expressed as the combination of a term related to the lack of equipartition between the probe and the surrounding bath, and a term featuring the average frictional force. We illustrate this general result in the asymmetric Rayleigh piston, showing how Coulomb friction can induce a ratchet effect in a Brownian particle in contact with an equilibrium bath. An explicit analytical expression for the average velocity of the piston is obtained in the rare collision limit. Numerical simulations support the analytical findings.

Nonequilibrium fluctuations in a driven stochastic Lorentz gas.

We study the stationary state of a one-dimensional kinetic model where a probe particle is driven by an external field E and collides, elastically or inelastically, with a bath of particles at temperature T. We focus on the stationary distribution of the velocity of the particle, and of two estimates of the total entropy production Δs(tot). One is the entropy production of the medium Δs(m), which is equal to the energy exchanged with the scatterers, divided by a parameter θ, coinciding with the particle temperature at E=0. The other is the work W done by the external field, again rescaled by θ. At small E, a good collapse of the two distributions is found: in this case, the two quantities also verify the fluctuation relation (FR), indicating that both are good approximations of Δs(tot). Differently, for large values of E, the fluctuations of W violate the FR, while Δs(m) still verifies it.