Stochastic Dynamics of ϕ4 Kinks: Numerics and Analysis

The stationary density of the overdamped ϕ4 SPDE corresponds to a mean number of kinks and antikinks that is maintained by a balance between nucleation of new kink-antikink pairs and annihilation whenever a kink and antikink collide. We consider numerical methods for solution of the SPDE, and a definition of the location of the centre of a kink using a smoothing function that is the derivative of the function describing the shape and energy of an isolated kink. This definition allows calculation of the diffusivity of a kink as a function of noise intensity or “temperature” and defines the parameter characterising the “small-noise” régime. In the reaction-diffusion description of the dynamics (where kink-antikink pairs are nucleated with rate Γ, diffuse with diffusivity, D and annihilate on collision) the number of kinks per unit length in the steady-state has a simple exact expression.