Composite Bound States and Broken U(1) symmetry in the Chemical Master Equation derivation of the Gray-Scott Model

Abstract

We give a first principles derivation of the stochastic partial differential equations that describe the chemical reactions of the Gray-Scott model (GS): $U+2V {\stackrel λ{\rightarrow}} 3 V;$ and $V {\stackrel μ{\rightarrow}} P$, $U {\stackrel ν{\rightarrow}} Q$, with a constant feed rate for $U$. We find that the conservation of probability ensured by the chemical master equation leads to a modification of the usual differential equations for the GS model which now involves two composite fields and also intrinsic noise terms. One of the composites is $ψ_1 = ϕ_v^2$, where $ < ϕ_v >_η = v$ is the concentration of the species $V$ and the averaging is over the internal noise $η_{u,v,ψ_1}$. The second composite field is the product of three fields $ χ= λϕ_u ϕ_v^2$ and requires a noise source to ensure probability conservation. A third composite $ψ_2 = ϕ_{u} ϕ_{v}$ can be also be identified from the noise-induced reactions. The Hamiltonian that governs the time evolution of the many-body wave function, associated with the master equation, has a broken U(1) symmetry related to particle number conservation. By expanding around the (broken symmetry) zero energy solution of the Hamiltonian (by performing a Doi shift) one obtains from our path integral formulation the usual reaction diffusion equation, at the classical level. The Langevin equations that are derived from the chemical master equation have multiplicative noise sources for the density fields $ϕ_u, ϕ_v, χ$ that induce higher order processes such as $n \rightarrow n$ scattering for $n > 3$. The amplitude of the noise acting on $ ϕ_v$ is itself stochastic in nature.