Entrainment and Unit Velocity: Surprises in an Accelerated Exclusion Process

We introduce a class of distance-dependent interactions in an accelerated exclusion process inspired by the observation of transcribing RNA polymerase speeding up when “pushed” by a trailing one. On a ring, the accelerated exclusion process steady state displays a discontinuous transition, from being homogeneous (with augmented currents) to phase segregated. In the latter state, the holes appear loosely bound and move together, much like a train. Surprisingly, the current-density relation is simply $J=1-\rho$, signifying that the “hole train” travels with unit velocity.

Figure 1

AEP with ${\ell }_{\max }=3$. (a) Backward hops and overlaps are forbidden. (b) A particle hopping to the back of a 3-cluster triggers an additional hop. No other hops are triggered, e.g., (c): particle hits a cluster with $\ell >{\ell }_{\max }$.

Figure 2

(a) $J_{\mathrm{AEP}}(\rho )$ for various ${\ell }_{\max }$’s. Detailed views of ${\ell }_{\max }=20$, 100, 500 are plotted with symbols in (b) and dashed lines for $J_{\mathrm{AC}}$ and $J_{\mathrm{UV}}$; in (c), $J(\rho )+\rho$ is plotted to accentuate the transition into UV.

Figure 3

Cluster-size distribution for $H=5$ in $L=1000$. Broad distributions for the ordinary TASEP and AC are essentially identical. In phase-segregated UV, dominant maxima prevail at both extremes. For quantitative comparisons, the TASEP distribution (black connected line) is shown in all panels.