Time-Dependent Fluctuations and Superdiffusivity in the Driven Lattice Lorentz Gas

Abstract

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...

Time-dependent dynamics of the three-dimensional driven lattice Lorentz gas

“…We are interested in the following scenario: The system is initially in equilibrium and at time t = 0 a force on the tracer is switched on, driving the system into a new stationary state. This choice is in accordance with previous work on the driven lattice Lorentz gas in two dimensions [19,20], but also different choices are conceivable [24,26]. Let us emphasize that using normalized rates, while keeping the relative weights of the transition probabilites, changes only the time scale in a forcedependent manner such that all observables can be trivially rescaled.…”

“…There has been recent progress on the driven lattice Lorentz system in two dimensions, where the equilibrium dynamics in first order of the density [16][17][18] has been generalized to an exact analytic solution for the case of a force pulling on the tracer particle [19][20][21]. Furthermore, the complementary case of a biased intruder in a dense environment of mobile particles on a lattice has also been considered [22][23][24].…”

Complex dynamics induced by strong confinement – From tracer diffusion in strongly heterogeneous media to glassy relaxation of dense fluids in narrow slits

“…The superdiffusively growing fluctuations at intermediate times can be rationalized in terms of an asymptotic model in the limit of large forces [94]. In this case, the transition rates along the field dominate the motion of the tracer and transitions perpendicular and against the field can be ignored.…”

“…Although the calculations are the same in principle, the formulas become much more involved due to the inclusion of the force and one has to rely on computer algebra to obtain closed expressions [93,94].…”

“…In the driven lattice Lorentz gas the system is initially in equilibrium and a uniform step force is switched on at time zero pulling the tracer along a lattice direction [93,94]. The dimensionless force F = force · a/k B T measures the strength of the applied force on the tracer and it introduces a bias in the nearest-neighbor transition probabilities W (d ∈ N ).…”