Theory of Diffusion Controlled Growth

We present a new theoretical framework for diffusion limited aggregation and associated dielectric breakdown models in two dimensions. Key steps are understanding how these models interrelate when the ultraviolet cutoff strategy is changed, the analogy with turbulence and the use of logarithmic field variables. Within the simplest, Gaussian, truncation of mode-mode coupling, all properties can be calculated. The agreement with prior knowledge from simulations is encouraging, and a new superuniversality of the tip scaling exponent is both predicted and confirmed.

Figure 1

The size fluctuations (“output noise”) $A_{\mathrm{o}\mathrm{u}\mathrm{t}}=(\delta N/N{)}^2$, measured at fixed radius $R$ for DLA clusters grown with off-lattice noise reduction [11] at various (“input”) noise levels $A_{\mathrm{i}\mathrm{n}}$. For low $A_{\mathrm{i}\mathrm{n}}$, $A_{\mathrm{o}\mathrm{u}\mathrm{t}}$ self-organizes from below in the manner of a turbulent system.

Figure 2

Cumulative contribution to the mean growth velocity plotted against wave vector as $k^{{\eta }_1}$ with logarithmic scales. The data are (bottom to top) for ${\eta }_1=0.5,0.6,0.7,0.8,0.9,1.0$ and all exhibit a common power law slope $1/\alpha -1\approx 0.35\pm 0.04$ per the guidelines shown.

Figure 3

Multifractal spectra from the Gaussian theory ($\alpha =2/3$), compared to measured values for DLA [20]. Agreement is excellent for the active region $\tau \ge 0$, $\alpha \le 1$, and there are no adjustable parameters.