Renormalisation for Reaction-Diffusion Systems with Non-Local Interactions

Abstract

Models of reaction diffusion processes usually employ discrete lattice models with particles interacting at the same site, resulting in localized reactions in the continuum limit. Here, various non-local interactions are considered, and two features reported. Firstly, it is shown that sufficiently non-local interactions will regulate ultra-violet divergences that perturbative methods with local interactions produce. However, in asymptotic regimes, infra-red divergences persist and ultra-violet divergences can reappear. Renormalisation methods are shown to report the same universal behaviour as local interactions at critical points. Secondly, the renormalisation group can be interpreted as a space-time-field rescaling that preserves action structure. This can be used to extract solutions to Callan-Symanzik equations directly without having to solve (or construct) the equation. These observations are exemplified for two paradigm models; annihilation $A_p+A_q\rightarrow φ$, and this process paired with branching, birth and death.

Figures (6)

• Figure 1: A) Single loop Feynman diagram for the annihilation model in Lee1994 with local interaction rate $R$. B) Corresponding diagram for non-local interaction function $R(k)$ (in momentum representation). Dotted lines indicate truncated propagators. In both cases external momenta are zero and the loop momentum is $k$.
• Figure 2: Feynman perturbative diagrams for action post Hubbard-Stratonovich transforms. A) Three classes of propagator. B) Three vertices in perturbative expansion for annihilation terms, with coefficients. C) Annihilation node pairs connected by an annihilation propagator $R$ with momentum $k$ in each direction (each with coefficient $-\frac{1}{2}$) correspond to a single, directionless propagator (with coefficient $-\frac{R(k)+R(-k)}{2}$). D) Three vertices in perturbative expansion for branching terms (all have coefficient $1$). E) Possible node pairs for branching propagator $Q$. F) Terminal nodes. The $n_0$ (filled) node is fixed at time $t=0$ and carries no momentum. The lower birth node carries weight $B(k)$ with momentum $k$ to be integrated over.
• Figure 3: A) Dressing of propagator $G_{\bar{\psi}\psi}$ via a Dyson series involving branching propagator $Q$ edges. Feynman graph components are B) final, internal and initiating ladder structures involving annihilation propagator edges $R$ and ladder terminations via a branching propagator edge $Q$, along with C) initiating nodes. Edge directions can be inferred and so are dropped.
• Figure 4: Feynman diagrams. A) Diagrams corresponding to the vertex function $\Gamma^{(1,2)}$. Dotted lines indicate truncated propagators. B) Dyson recursion diagrams for tree approximation to mean density.
• Figure 5: A) Feynman diagrams for the response functional up to second order. Time runs from right to left. The texture filled edge represents the response functional $G_{cl}(k;t_1,t_2)$ from time $t_1$ to $t_2$ for momentum $k$. The spiral edges indicate tree structures which all extend from time $0$ and carry zero momentum. Black line edges are propagators and all carry momentum $k$. Double edges are interaction propagators, with momenta stated (either zero or $k$). B) The first two diagrams for the mean density using response functionals.