Field theory of self-organized fractal etching

Abstract

We propose a phenomenological field theoretical approach to the chemical etching of a disordered-solid. The theory is based on a recently proposed dynamical etching model. Through the introduction of a set of Langevin equations for the model evolution, we are able to map the problem into a field theory related to isotropic percolation. To the best of the authors knowledge, it constitutes the first application of field theory to a problem of chemical dynamics. By using this mapping, many of the etching process critical properties are seen to be describable in terms of the percolation renormalization group fixed point. The emerging field theory has the peculiarity of being ``{\it self-organized}'', in the sense that without any parameter fine-tuning, the system develops fractal properties up to certain scale controlled solely by the volume, $V$, of the etching solution. In the limit $V \to \infty$ the upper cut-off goes to infinity and the system becomes scale invariant. We present also a finite size scaling analysis and discuss the relation of this particular etching mechanism with Gradient Percolation. Finally, the possibility of considering this mechanism as a new generic path to self-organized criticality is analyzed, with the characteristics of being closely related to a real physical system and therefore more directly accessible to experiments.