Time-dependent dynamics in the confined lattice Lorentz gas

We study a lattice model describing the non-equilibrium dynamics emerging from the pulling of a tracer particle through a disordered medium occupied by randomly placed obstacles. The model is considered in a restricted geometry pertinent for the investigation of confinement-induced effects. We analytically derive exact results for the characteristic function of the moments valid to first order in the obstacle density. By calculating the velocity autocorrelation function and its long-time tail we find that already in equilibrium the system exhibits a dimensional crossover. This picture is further confirmed by the approach of the drift velocity to its terminal value attained in the non-equilibrium stationary state. At large times the diffusion coefficient is affected by both the driving and confinement in a way that we quantify analytically. The force-induced diffusion coefficient depends sensitively on the presence of confinement. The latter is able to modify qualitatively the non-analytic behavior in the force observed for the unbounded model. We then examine the fluctuations of the tracer particle along the driving force. We show that in the intermediate regime superdiffusive anomalous behavior persists even in the presence of confinement. Stochastic simulations are employed in order to test the validity of the analytic results, exact to first order in the obstacle density and valid for arbitrary force and confinement.

💡 Research Summary

In this work the authors investigate the non‑equilibrium dynamics of a tracer particle that is pulled by a constant external force through a two‑dimensional lattice populated by randomly placed immobile obstacles. The geometry is “quasi‑confined”: periodic boundary conditions are applied in the longitudinal (x) direction while the transverse direction (y) is limited to a finite width L, effectively forming a cylindrical strip of infinite length. The tracer performs nearest‑neighbour jumps with exponentially distributed waiting times (mean τ). The bias introduced by the force F (>0) modifies the hopping probabilities according to local detailed balance, yielding forward and backward rates along the force direction W(±eₓ)=e^{±F/2}/(2cosh(F/2)) and isotropic transverse rates W(±e_y)=1/(2cosh(F/2)).

The analytical treatment proceeds in two stages. First, the “bare” system without obstacles is solved exactly. By introducing a Hilbert‑space representation of the master equation, the unperturbed Hamiltonian Ĥ₀ is diagonalized in the plane‑wave basis |k⟩, where the eigenvalue ε(k) separates into longitudinal and transverse contributions. The force appears as an imaginary shift of the longitudinal momentum, kₓ → kₓ + iF/2, which leads to a compact expression for the characteristic function F₀(k,t)=exp