Why patchy diffusion-limited aggregation belongs to the directed-percolation universality class.

Abstract

We present a possible link between nonequilibrium phase transition observed in patchy diffusion-limited aggregation (DLA) [M. J. Kartha and A. Sayeed, Phys. Lett. A 380, 2791 (2016)10.1016/j.physleta.2016.06.036] and directed bond percolation (DP). A system of directed percolation with patchy particles (patchy DP) in which the bond connectivity is established depending on patch size p is analyzed. It is observed that patchy DP starting from a single seed shows a nonequilibrium phase transition. Below a critical value of the patch size p_{c}, the system reaches an absorbing state above which is a fluctuating active state as observed in the DP system. The value of this nonuniversal parameter p_{c} is observed to be slightly higher than the value observed in patchy DLA. Close to the critical value, the order parameter P(∞)∼(p-p_{c})^{β} where β=0.272±0.010, which is consistent with the directed-percolation universality class. Therefore the intrinsic nature of patchy DP is responsible for the phase transition in patchy DLA. This study reveals that the estimated critical value of patch size p_{c}=0.80625±0.00020 in patchy DP is different from the critical bond probability p_{c}=0.6447 in the DP system. This elucidates that the bond probability in DP is not equivalent to the patch probability of a particular site. Our work also gives an insight into the problem related with formation of an extended network of pentagon subunits in connection with the virus capsid.