Anomalous Mobility Enhancement in Restricted Geometries

We investigate the effect of geometrical confinement on driven transport in anomalous media within the framework of the quenched trap model. The particle performs a biased random walk on a lattice, where quenched disorder introduces power-law distributed trapping times. We consider channels that are infinite along one direction and have a finite cross-section in the remaining $d-1$ directions. In the glassy regime, where $α=T/T_g<1$, we derive an analytical expression for the mean longitudinal displacement in the long-time and weak-force limits. We find the scaling $\langle x_\parallel(t)\rangle \propto F_\parallel^αt^αΩ^{-(1-α)}$, where $F_\parallel>0$ is the effective longitudinal force and $Ω$ is the number of sites in a transverse cross-section. Thus, the anomalous mobility is enhanced for narrower channels. Our analytical result applies to general Bravais lattices in $d$ dimensions with periodic transverse boundary conditions. Numerical simulations confirm the analytical predictions for different lattice geometries and show quantitative agreement even for channels with reflecting boundary conditions.