State-space renormalization group theory of nonequilibrium reaction networks: Exact solutions for hypercubic lattices in arbitrary dimensions

Abstract

Nonequilibrium reaction networks (NRNs) underlie most biological functions. Despite their diverse dynamic properties, NRNs share the signature characteristics of persistent probability fluxes and continuous energy dissipation even in the steady state. Dynamics of NRNs can be described at different coarse-grained levels. Our previous work showed that the apparent energy dissipation rate at a coarse-grained level follows an inverse power law dependence on the scale of coarse-graining. The scaling exponent is determined by the network structure and correlation of stationary probability fluxes. However, it remains unclear whether and how the (renormalized) flux correlation varies with coarse-graining. Following Kadanoff's real space renormalization group (RG) approach for critical phenomena, we address this question by developing a State-Space Renormalization Group (SSRG) theory for NRNs, which leads to an iterative RG equation for the flux correlation function. In square and hypercubic lattices, we solve the RG equation exactly and find two types of fixed point solutions: a family of nontrivial fixed points where the correlation exhibits power-law decay and a trivial fixed point where the correlation vanishes beyond the nearest neighbors. The power-law fixed point is stable if and only if the power exponent is less than the lattice dimension $n$. Consequently, the correlation function converges to the power-law fixed point only when the correlation in the fine-grained network decays slower than $r^{-n}$ and to the trivial fixed point otherwise. If the flux correlation in the fine-grained network contains multiple stable solutions with different exponents, the RG iteration dynamics select the fixed point solution with the smallest exponent. We also discuss a possible connection between the RG flows of flux correlation with those of the Kosterlitz-Thouless transition.