Various challenges are faced when animalcules such as bacteria, protozoa, algae, or sperms move autonomously in aqueous media at low Reynolds number. These active agents are subject to strong stochastic fluctuations, that compete with the directed motion. So far most studies consider the lowest order moments of the displacements only, while more general spatio-temporal information on the stochastic motion is provided in scattering experiments. Here we derive analytically exact expressions for the directly measurable intermediate scattering function for a mesoscopic model of a single, anisotropic active Brownian particle in three dimensions. The mean-square displacement and the non-Gaussian parameter of the stochastic process are obtained as derivatives of the intermediate scattering function. These display different temporal regimes dominated by effective diffusion and directed motion due to the interplay of translational and rotational diffusion which is rationalized within the theory. The most prominent feature of the intermediate scattering function is an oscillatory behavior at intermediate wavenumbers reflecting the persistent swimming motion, whereas at small length scales bare translational and at large length scales an enhanced effective diffusion emerges. We anticipate that our characterization of the motion of active agents will serve as a reference for more realistic models and experimental observations.

We determine the nonlinear time-dependent response of a tracer on a lattice with randomly distributed hard obstacles as a force is switched on. The calculation is exact to first order in the obstacle density and holds for arbitrarily large forces. Whereas, on the impurity-free lattice, the nonlinear drift velocity in the stationary state is analytic in the driving force, interactions with impurities introduce logarithmic contributions beyond the linear regime. The long-time decay of the velocity toward the steady state is exponentially fast for any finite value of the force, in striking contrast to the power-law relaxation predicted within linear response. We discuss the range of validity of our analytic results by comparison to stochastic simulations.

We consider a tracer particle on a lattice in the presence of immobile obstacles. Starting from equilibrium, a force pulling on the particle is switched on, driving the system to a new stationary state. We solve for the complete transient dynamics of the fluctuations of the tracer position along the direction of the force. The analytic result, exact in first order of the obstacle density and for arbitrarily strong driving, is compared to stochastic simulations. Upon strong driving, the fluctuations grow superdiffusively for intermediate times; however, they always become diffusive in the stationary state. The diffusion constant is nonanalytic for small driving and is enhanced by orders of magnitude by increasing the force.The material properties of complex fluids such as colloidal dispersions [1,2], solutions of biopolymers [3,4], or biomaterials [5,6] can be probed by pulling a mesoscopic tracer particle through the medium. In linear response by the fluctuation-dissipation theorem it is sufficient to monitor the force-free thermally agitated motion of the tracer which is the principle of passive microrheology [7][8][9]. Then by a generalized Stokes-Einstein relation the dynamic mobility is connected to the linear macroscopic frequency-dependent viscosity or elastic modulus. In contrast, in active microrheology the particle is manipulated by optical or magnetic tweezers and pulled through the environment in principle by arbitrarily strong forces [10][11][12]. Here, the system is intrinsically strongly out of equilibrium and a plethora of new phenomena have been found experimentally and in simulations, such as force thinning [13][14][15], (transient) superdiffusive behavior, and enhanced diffusivites [16,17].To make theoretical progress in the nonlinear regime, generic models have been investigated that focus on the mutual exclusion originating from the strong repulsive interaction between the tracer and its environment as the most important ingredient. The underlying dynamics of the tracer is usually modeled as a random walk on a lattice or Brownian motion in continuum, while the surroundings range from dilute and immobile obstacles to dynamic and crowded environments. For lattice systems, progress and even exact results have been achieved [18][19][20][21][22][23][24][25] Here we rely on a lattice model for a driven tracer in a crowded environment to investigate the growth of the fluctuations as time progresses. The crowding is incorporated in the model by introducing hard and immobile obstacles randomly distributed over the lattice. A force is switched on at a certain instant of time such that the tracer performs a biased obstructed diffusion through the system. To first order in the obstacle density the moment-generating function for the displacements can be determined in principle exactly; so far only the timedependent velocity response has been elaborated [19]. In equilibrium, the fluctuations are also known for low obstacle densities via the mean-square displacement [46][47][48]. Within this model, we consi...

We study dynamically crowded solutions of stiff fibers deep in the semidilute regime, where the motion of a single constituent becomes increasingly confined to a narrow tube. The spatiotemporal dynamics for wave numbers resolving the motion in the confining tube becomes accessible in Brownian dynamics simulations upon employing a geometry-adapted neighbor list. We demonstrate that in such crowded environments the intermediate scattering function, characterizing the motion in space and time, can be predicted quantitatively by simulating a single freely diffusing phantom needle only, yet with very unusual diffusion coefficients.

We study the dynamics of solutions of infinitely thin needles up to densities deep in the semidilute regime by Brownian dynamics simulations. For high densities, these solutions become strongly entangled and the motion of a needle is essentially restricted to a one-dimensional sliding in a confining tube composed of neighboring needles. From the density-dependent behavior of the orientational and translational diffusion, we extract the long-time transport coefficients and the geometry of the confining tube. The sliding motion within the tube becomes visible in the non-Gaussian parameter of the translational motion as an extended plateau at intermediate times and in the intermediate scattering function as an algebraic decay. This transient dynamic arrest is also corroborated by the local exponent of the mean-square displacements perpendicular to the needle axis. Moreover, the probability distribution of the displacements perpendicular to the needle becomes strongly nonGaussian, rather it displays an exponential distribution for large displacements. On the other hand, based on the analysis of higher-order correlations of the orientation we find that the rotational motion becomes diffusive again for strong confinement. At coarse-grained time and length scales, the spatiotemporal dynamics of the needle for the high entanglement is captured by a single freely diffusing phantom needle with long-time transport coefficients obtained from the needle in solution. The time-dependent dynamics of the phantom needle is also assessed analytically in terms of spheroidal wave functions. The dynamic behavior of the needle in solution is found to be identical to needle Lorentz systems, where a tracer needle explores a quenched disordered array of other needles.

Abstract. We provide an overview of recent advances of the complex dynamics of particles in strong confinements. The first paradigm is the Lorentz model where tracers explore a quenched disordered host structure. Such systems naturally occur as limiting cases of binary glassforming systems if the dynamics of one component is much faster than the other. For a certain critical density of the host structure the tracers undergo a localization transition which constitutes a critical phenomenon. A series of predictions in the vicinity of the transition have been elaborated and tested versus computer simulations. Analytical progress is achieved for small obstacle densities. The second paradigm is a dense strongly interacting liquid confined to a narrow slab. Then the glass transition depends nonmonotonically on the separation of the plates due to an interplay of local packing and layering. Very small slab widths allow to address certain features of the statics and dynamics analytically.

We present results for the fluctuations of the displacement of a tracer particle on a planar lattice pulled by a step force in the presence of impenetrable, immobile obstacles. The fluctuations perpendicular to the applied force are evaluated exactly in first order of the obstacle density for arbitrarily strong pulling and all times. The complex time-dependent behavior is analyzed in terms of the diffusion coefficient, local exponent, and the non-Skellam parameter, which quantifies deviations from the dynamics on the lattice in the absence of obstacles. The non-Skellam parameter along the force is analyzed in terms of an asymptotic model and reveals a power-law growth for intermediate times.

In a microrheological set-up a single probe particle immersed in a complex fluid is exposed to a strong external force driving the system out of equilibrium. Here, we elaborate analytically the time-dependent response of a probe particle in a dilute suspension of Brownian particles to a large step-force, exact in first order of the density of the bath particles. The time-dependent drift velocity approaches its stationary state value exponentially fast for arbitrarily small driving in striking contrast to the power-law prediction of linear response encoded in the long-time tails of the velocity autocorrelation function. We show that the stationary-state behavior depends nonanalytically on the driving force and connect this behavior to the persistent correlations in the equilibrium state. We argue that this relation holds generically. Furthermore, we elaborate that the fluctuations in the direction of the force display transient superdiffusive behavior.