Emergence of fluctuating traveling front solutions in macroscopic theory of noisy invasion fronts

TL;DR

This paper investigates the fluctuating behavior of invasion fronts in noisy systems, confirming that slow front probabilities are described by traveling wave solutions of a Hamiltonian field theory through numerical solutions.

Contribution

It verifies the assumption that slow front fluctuations correspond to traveling front solutions in a macroscopic Hamiltonian theory using numerical Hamilton equations.

Findings

Traveling front solutions accurately describe slow front fluctuations.

Numerical solutions support the macroscopic theory's assumptions.

The approach applies to models in the directed percolation class.

Abstract

The position of an invasion front, propagating into an unstable state, fluctuates because of the shot noise coming from the discreteness of reacting particles and stochastic character of the reactions and diffusion. A recent macroscopic theory [Meerson and Sasorov, Phys. Rev. E 84, 030101(R) (2011)] yields the probability of observing, during a long time, an unusually slow front. The theory is formulated as an effective classical Hamiltonian field theory which operates with the density field and the conjugate "momentum" field. Further, the theory assumes that the most probable density field history of an unusually slow front represents, up to small corrections, a traveling front solution of the Hamilton equations. Here we verify this assumption by solving the Hamilton equations numerically for models belonging to the directed percolation universality class.