Similarity transformation in one–dimensional reaction–diffusion systems; voting model as an example

The exact solution for a system with two–particle coagulation and decoagulation has been studied. The spectrum of the Hamiltonian of the system is found. It is shown that the steady state is two–fold degenerate. The average number density in each cite 〈ni(t)〉 and the equal time two–point functions 〈ni(t), nj(t)〉 are calculated. Any equal time correlation functions at large times, 〈ni(∞), nj(∞), · · ·〉, is also calculated. The relaxation behaviour of the system toward its final state is investigated and it is shown that generally it is exponential, as it is expected. For the special symmetric case, the relaxation behaviour of the system is a power law. For the asymmetric case, it is shown that the profile of deviation from the final values is an exponential function of the position.