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

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)] 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 $P\left(\infty \right)\sim {(p-p_c)}^{\beta }$ where $\beta =0.272\pm 0.010$, 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 $p_c=0.80625\pm 0.00020$ 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.

Figure 1

(a) Schematic shows a typical patchy disk assigned to each of the diagonal sites of a square lattice. (b) A condition for bond connectivity between two disks is illustrated.

Figure 2

Typical patchy-DP cluster starting from the single seed for $p=0.6$. The red (gray) arrows indicate the percolation path. The arrow to the right indicates the time advance.

Figure 3

Typical snapshots of the patchy-DP systems for random initial configuration starting from the top corner seed and corresponding clusters on the right side for patch sizes $p=0.6,0.8$, and 0.9, respectively.

Figure 4

Average number of particles in the cluster as a function of time for different values of patch size $p$. Arrow indicates the increasing value of $p$.

Figure 5

Survival probability $P\left(t\right)$ as a function of time. The arrow indicates increasing value of patch size. Here the estimated critical value of the patch size is $p_c=0.80625\pm 0.00020$.

Figure 6

(a) Comparison of survival probability against patch size for patchy DLA [1] and patchy DP. (b) The plot of $lnP\left(\infty \right)$ against $ln\left(p-p_c\right)$ for a patchy-DP system.

Figure 7

The packing of identical pentagons in a plane for two types of trimer connections between the pentagon is depicted in (a, b) [33].

Figure 8

Schematic of patchy pentamer (virus subunit) inscribed in a patchy disk. The interacting portion $\left(\frac{4}{5}\right)$ favors bonding to neighbors and the deactivated side is clearly depicted.

Figure 9

The packing of equal patchy pentagons (virus subunit) in a plane (a, b) shows the trimer connection between the pentagons.