# Diffusion-limited reaction in one dimension: Paired
and unpaired nucleation.

*Journal of Chemical Physics*
**115** 73 (2001)

#### Salman Habib

Theoretical Division T-8, Los Alamos National Laboratory

New Mexico 87545, USA.
#### Katja Lindenberg

Department of Chemistry and Biochemistry 0340,
University of California San Diego

La Jolla, California 92093-0340.
#### Grant Lythe

Theoretical Division T-8, Los Alamos National Laboratory

Los Alamos, New Mexico 87545

and

GISC, Departamento de Matemáticas,
Universidad Carlos III de Madrid

28911 Leganés, Spain.
#### Carmen Molina-París

Theoretical Division T-8, Los Alamos National Laboratory

Los Alamos, New Mexico 87545

and

Centro de Astrobiología CSIC-INTA

Carretera de Ajalvir,
28850 Torrejón, Madrid, Spain.

## Abstract

We study the dynamics of diffusing particles in one space
dimension with annihilation on collision and nucleation (creation of
particles) with constant probability per unit time and length. The
cases of nucleation of single particles and nucleation in pairs are
considered. A new method of analysis permits exact calculation of the
steady state density and its time evolution in terms of the three
parameters describing the microscopic dynamics: the nucleation rate,
the initial separation of nucleated pairs and the diffusivity of a
particle. For paired nucleation at sufficiently small initial
separation the nucleation rate is proportional to the square of the
steady state density. For unpaired nucleation, and for paired
nucleation at sufficiently large initial separation, the nucleation
rate is proportional to the cube of the steady state density.