Journal of Statistical Physics, Vol. 65, Nos. 5/6, 1991 Scaling Properties of Diffusion-Limited Reactions on Fractal and Euclidean Geometries Katja Lindenberg, 1 Wen-Shyan Sheu, 1,2 and Raoul Kopelman 3 We review our scaling results for the diffusion-limited reactions A + A ~ 0 and A + B ~ 0 on Euclidean and fractal geometries. These scaling results embody the anomalies that are observed in these reactions in low dimensions; we collect these observations under a single phenomenological umbrella. Although we are not able to fix all the exponents in our scaling expressions from first principles, we establish bounds that bracket the observed numerical results. KEY WORDS: Diffusion-limited reactions; fractals; scaling; aggregates. 1. INTRODUCTION The kinetic laws that describe diffusion-controlled annihilation reactions of the form A+A ~0 and A+ B ~0 in low dimensions differ from the "classical" mean field forms and are frequently called "anomalous." The so-called anomalous behavior arises from the fact that diffusion in low dimensions is not an effective mixing mechanism; consequently, the spatial distributions of reactants differ from the thoroughly mixed ones that are implicit in the classical rate laws. It is useful to begin by describing what we mean by "thoroughly mixed." In the single-species case A + A--, 0, thorough mixing implies a distribution of nearest-neighbor distances of the Hertz form. (1~ This dis- tribution includes small nearest-neighbor distances with a high probability; as nearby reactants annihilate one another, an effective mixing mechanism t Department of Chemistry, 0340, and Institute for Nonlinear Science, 0402, University of California at San Diego, La Jolla, California 92093. z Current address: Department of Chemistry, University of Texas, Austin, Texas 78712. 3 Departments of Chemistry and Physics, University of Michigan, Ann Arbor, Michigan 48109. 1269 822/65/5-6-28 0022-4715/91/1200-126950650/0 9 1991 Plenum Publishing Corporation