Small-scale behaviour in deterministic reaction models

TL;DR

This paper analytically confirms the small-distance behavior of gap distributions in a one-dimensional particle system with attractive forces, identifying the limits of mean-field applicability and highlighting the open question of the exponent's significance.

Contribution

It provides an analytical proof of the gap distribution exponent in a reaction model, clarifying the mean-field limit and correlation effects.

Findings

In the limit z→0, the distribution exponent β equals α for α>0.

Mean-field theory holds for small gaps beyond a certain length scale.

Correlations are negligible in the small-gap limit for similar models.

Abstract

In a recent paper published in this journal [J. Phys. A: Math. Theor. 42 (2009) 495004] we studied a one-dimensional particles system where nearest particles attract with a force inversely proportional to a power \alpha of their distance and coalesce upon encounter. Numerics yielded a distribution function h(z) for the gap between neighbouring particles, with h(z)=z^{\beta(\alpha)} for small z and \beta(\alpha)>\alpha. We can now prove analytically that in the strict limit of z\to 0, \beta=\alpha for \alpha>0, corresponding to the mean-field result, and we compute the length scale where mean-field breaks down. More generally, in that same limit correlations are negligible for any similar reaction model where attractive forces diverge with vanishing distance. The actual meaning of the measured exponent \beta(\alpha) remains an open question.