Similarity transformation in one-dimensional reaction-diffusion systems: the voting model as an example

Abstract

The exact solution for a system with two-particle annihilation 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.

Exactly solvable reaction diffusion models on a Bethe Lattice through the empty-interval method

“…Again the root z 2 (which is either greater than one or negative) give rise to unphysical solutions. The reason is that although the dynamical solution [29] need not be nonnegative and decreasing in n by itself, the sum of such a solution and the stationary solution can be a complete physical solution to [11] and [12] and hence should satisfy these conditions. For sufficiently large n, a contribution coming from z 2 would be dominant and either blows up or becomes negative.…”

“…This set of equations, together with the boundary condition [11], can be solved iteratively. The equation for E 1 (t) becomes…”

“…Approximation methods are generally different in different dimensions, as for example the mean filed techniques, working good for high dimensions, generally do not give correct results for low-dimensional systems. A large fraction of analytical studies belong to low-dimensional (especially one-dimensional) systems, as solving low-dimensional systems should in principle be easier [1][2][3][4][5][6][7][8][9][10][11].…”

Phase transition in an asymmetric generalization of the zero-temperature Glauber model

“…Specially, at L → ∞, the relaxation time is the same as the relaxation time for A = B = 0, that is the same as (23). If, for example, |A| > 1, then the total phase change of (1 + AZ) is no longer zero.…”

“…The above system on an infinite lattice has been investigated in [23], where its n-point functions, its equilibrium states, and its relaxation towards these states are studied. It can be easily shown that the time evolution equation for the average densities of the system with the above interactions are the same as that of a system with the following interactions, where diffusion is also present:…”