Injection on a line (slope -3/2)

If one looks at a snapshot of the 2D picture for ``injection on a line'', one recognizes that one can describe this as a structure of cracks which are all anchored at the injection line. There are $L$ such cracks (some of length zero), and cracks can merge with increasing distance from the injection line, but they cannot branch.

According to Ref. [14], this leads naturally to a size exponent of $-3/2$, as found in the simulations. The argument is the following: The whole area, $L2$, is covered by $\int ds\; \hspace{0.17em}\; s\; \hspace{0.17em}\; n(s)\; ,$ where $n(s)$ is the number of clusters of size $s$ on a linear scale. We assume $n(s)\; s-\tau$, however the normalization is missing. If all clusters are anchored at a line of size $L$, then a doubling of the length of the line will result in twice as many clusters. In consequence, the normalization constant is $\propto L$, and thus $n(s)\; L\; \hspace{0.17em}\; s-\tau$. Now we balance the total area, $L2$, with what we just learned about the covering clusters: $L2\int ds\; \hspace{0.17em}\; s\; \hspace{0.17em}\; L\; \hspace{0.17em}\; s-\tau =\; L\; \hspace{0.17em}\; \int ds\; \hspace{0.17em}\; s1-\tau L\; \hspace{0.17em}\; s2-\tau |$₀^S .

Assuming that , then the integral does not converge for , and we need to take into account how the cut-off $S$ scales with $L$. This depends on how the cracks move in space as a function of the distance from the injection line. If the cracks are roughly straight, then the size of the largest cluster is $L2$. If the cracks are random walks, then the size of the largest cluster is $L3/2$. In consequence:

• For ``straight'' lines: $\Rightarrow$ $2\; =\; 1\; +\; 2\; \hspace{0.17em}\; (2-\tau )\; =\; 1\; +\; 4\; -\; 2\; \tau$ $\Rightarrow$ $\tau =\; -\; 3/2\; .$
• For random walk: $2\; =\; 1\; +\; 3/2\; \hspace{0.17em}\; (2-\tau )\; =\; 1\; +\; 3\; -\; 3\; \tau /2\; \Rightarrow \; \tau =\; -\; 4/3\; .$

Since our simulations result in $\tau \approx -3/2$, we conclude that our lines between clusters are not random walks. This is intuitively reasonable: When a cluster is killed, then the growth is biased towards the center of the deleted cluster, thus resulting in random walks which are all differently biased. This bias then leads to the ``straight line'' behavior. -- This implies that the $s-3/2$ steady state scaling law hinges on two ingredients (in a 2D system):

• The injection comes from a 1D structure.
• The boundaries between clusters follow something that corresponds to straight lines. As we have seen, the biasing of a random walk is already enough to obtain this effect.