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

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 R n at time n (the FVR is just the first moment) in the leading order in the density ρ 0 of vacancies (O(ρ 0 )). We prove that in infinite 2D systems the probability distribution P ( R n ) 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 ( R n ) along the direction of the bias grows (weakly) super-diffusively: . In the direction perpendicular to the bias, the variance . The coefficients ν 1 and ν 2 , which we determine exactly for arbitrary bias in O(ρ 0 ), mirror the interplay between the bias, vacancy-controlled transport and the back-flow effects of the medium on the BI. We observe that ν 1 ∼ | 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.