Olivier Bénichou, Carlos Mejía-Monasterio, Gleb Oshanin
We present exact results on the dynamics of a biased, by an external force ${\bf F}$, intruder (BI) in a two-dimensional lattice gas of unbiased, randomly moving hard-core particles. Going beyond the usual analysis of the force-velocity relation, we study the probability distribution $P({\bf R}_n)$ of the BI displacement ${\bf R}_n$ at time {\it n}. We show that despite the fact that the BI drives the gas to a non-equilibrium steady-state, $P({\bf R}_n)$ converges to a Gaussian distribution as $n \to \infty$. We find that the variance $\sigma_x^2$ of $P({\bf R}_n)$ along ${\bf F}$ exhibits a weakly superdiffusive growth $\sigma_x^2 \sim \nu_1 \, n \, \ln(n)$, and a usual diffusive growth, $\sigma_y^2 \sim \nu_2 \, n$, in the perpendicular direction. We determine $\nu_1$ and $\nu_2$ exactly for arbitrary bias, in the lowest order in the density of vacancies, and show that $\nu_1 \sim |{\bf F}|^2$ for small bias, which signifies that superdiffusive behaviour emerges beyond the linear-response approximation. Monte Carlo simulations confirm our analytical results, and reveal a striking field-induced superdiffusive behavior $\sigma_x^2 \sim n^{3/2}$ for infinitely long 2D stripes and 3D capillaries.
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http://arxiv.org/abs/1211.0674
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