Anomalous coalescence from a nonlinear Schroedinger equation with a quintic term: interpretation through Thompson's approach

Abstract

Inspired by models for A+ A→ A(0) reactions with non-Brownian diffusion, we suggest a possible analytical explanation for the phenomena of anomalous coalescence of bubbles found in one-dimension (1d) by Josserand and Rica through numerical work [Phys. Rev. Letters 78 (1997) 1215]. The explanation firstly requires an exponent γ, which is sometimes used to describe anomalous diffusion. Here it displays an explicit dependence on the dimensionality ( γ= γ( d)=4/ d for d⩽2). So we have d c =2, coinciding with the upper critical dimension of A+ A→ A(0) reactions (Mod. Phys. Lett. B 13 (1999) 829; Mod. Phys. Lett. B 15(26) (2001) 1205) with Brownian diffusion condition ( γ=2). Thus anomalous coalescence emerges, only below the critical dimension ( d<2). We show that the typical size of the structures (bubbles) grows as R( t)∼ t 1/4 in 1d. An alternative explanation could also be thought as a diffusion constant D which depends on the average concentration (〈 n〉), namely D= D 0〈 n〉 α . It is introduced into an effective action for A+ A→ A(0) reactions. Therefore we are also able to reproduce the anomalous behavior for n( t) and R( t) in 1d, being α=0 for d⩾2 (mean field behavior) and α=2(2− d)/ d 2 for d⩽2.