A biased intruder in a dense quiescent medium: looking beyond the force–velocity relation

Abstract

We study the dynamics of a biased intruder (BI) pulled by a constant force F through a dense molecular crowding environment modelled as a lattice gas of unbiased, randomly moving hard-core particles. Going beyond the usual analysis of the force–velocity relation (FVR), we focus on the behaviour of the higher moments of the BI vector displacement Rn at time n (the FVR is just the first moment) in the leading order in the density ρ0 of vacancies (O(ρ0)). We prove that in infinite 2D systems the probability distribution P(Rn) converges to a Gaussian as n → ∞, despite the fact that the BI drives the system into a non-equilibrium steady state with a non-homogeneous spatial distribution of the lattice gas particles. We show that in infinite 2D systems the variance of the distribution P(Rn) along the direction of the bias grows (weakly) super-diffusively: . In the direction perpendicular to the bias, the variance . The coefficients ν1 and ν2, which we determine exactly for arbitrary bias in O(ρ0), mirror the interplay between the bias, vacancy-controlled transport and the back-flow effects of the medium on the BI. We observe that ν1 ∼ |F|2 for small bias, which signifies that the super-diffusive behaviour emerges beyond the linear-response approximation. We present analytical arguments showing that such an anomalous, field-induced broadening of fluctuations is dramatically enhanced in confined, quasi-1D geometries—infinite 2D stripes and 3D capillaries. We argue that in such systems, exhibits a strongly super-diffusive behaviour, . Monte Carlo simulations confirm our analytical results.