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Confined diffusion in a random Lorentz gas environment

We study the diffusive behavior of biased Brownian particles in a two dimensional confined geometry filled with the freezing obstacles. The transport properties of these particles are investigated for various values of the obstacles density $η$ and the scaling parameter $f$, which is the ratio of work done to the particles to available thermal energy. We show that, when the thermal fluctuations dominate over the external force, i.e., small $f$ regime, particles get trapped in the given environment when the system percolates at the critical obstacles density $η_c \approx 1.2$. However, as $f$ increases, we observe that particles trapping occurs prior to $η_c$. In particular, we find a relation between $η$ and $f$ which provides an estimate of the minimum $η$ up to a critical scaling parameter $f_c$ beyond which the Fick-Jacobs description is invalid. Prominent transport features like nonmonotonic behavior of the nonlinear mobility, anomalous diffusion, and greatly enhanced effective diffusion coefficient are explained for various strengths of $f$ and $η$. Also, it is interesting to observe that particles exhibit different kinds of diffusive behaviors, i.e., subdiffusion, normal diffusion, and superdiffusion. These findings, which are genuine to the confined and random Lorentz gas environment, can be useful to understand the transport of small particles or molecules in systems such as molecular sieves and porous media which have a complex heterogeneous environment of the freezing obstacles.