Anomalous coalescence phenomena under stationary regime condition through Thompson's method

In this work, anomalous coalescence phenomena of type A+A→A(0) are considered in the case of stationary regime, namely the case of an external homogeneous source (h) of single particles A, in the limit of zero external field rate. The main purpose here is to pick up new critical exponents in the stationary regime for the phenomenon of anomalous coalescence of bubbles found in one-dimension by Josserand and Rica (JR) (Phys. Rev. Lett. 78 (1997) 1215). The critical exponents of statical concentration (δ), relaxation time (Δ′) and the exponent for concentration decay (ξ) are evaluated in the limit h→0 (zero field rate). We show that such critical exponents depend on the parameter γ (Physica A 334 (2004) 335), which characterizes the anomalous diffusion condition. In order to explain the anomalous coalescence of JR's work as already made before by the present author (Physica A 334 (2004) 335), the parameter γ must display an explicit dependence on the dimensionality [γ=γ(d)=4/dford⩽2]. We use such a result [γ(d)] in order to pick up new critical exponents for the special case of anomalous coalescence found by JR (we find δ=5, Δ′=45 and ξ=14 in 1-d). We also verify that the known scaling relations among such critical indexes (Phys. Rev. A 32(2) (1985) 1129) are obeyed, so that in the special case of γ=2 (brownian diffusion condition given to mean field regime), we naturally recover those results previously obtained by Rácz (Phys. Rev. A 32(2) (1985) 1129). We go further to give a better explanation for such new results, taking into account another exponent α (Physica A 334 (2004) 335) to explain the behavior of diffusion constant, which depends on concentration (D=D0〈n〉α), that is to say, we obtain analytically new critical indexes which controll the diffusion behavior (D) for long-time regime, in the presence of an external homogeneous source.