Jamming and asymptotic behavior in competitive random parking of bidisperse cars

Abstract

We propose a generalized car parking problem where either a car of size $σ$ or of size $mσ$ ($m>1$) is sequentially parked on a line with probability $q$ and $(1-q)$, respectively. The free parameter $q$ interpolates between the classical car parking problem at either extreme ($q=0$ and $q=1$) and the competitive random sequential adsorption of a binary mixture in between. We find that the coverage in the jamming limit for a mixture always exceeds the value obtained for the uni-sized case. The introduction of a bidisperse mixture results in the slow approach ($\sim t^{-1}$) to the jamming limit by the smaller species while the larger species reach their asymptotic values exponentially fast $\sim t^{-1}e^{-(m-1)qt}$.