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

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 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. There is a family of nontrivial fixed points where the correlation exhibits power-law decay, characterized by a power exponent that can take any value within a continuous range. There is also 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. The analytical results are supported by numerical simulations. We also discuss a possible connection between the RG flows of flux correlation with those of the Kosterlitz-Thouless transition.

Figure 1

An illustration of the coarse-graining of the reaction network and the definition of correlation functions. Starting with the network on the left, all the states in the same shaded area are combined together to form a coarse-grained state, which is indicated by the same color on the right. The longitudinal (transversal) correlation function is defined as the correlation between the net fluxes on the red (blue) links, which are parallel (perpendicular) to the direction in which their separation is reduced by half.

Figure 2

Fixed points and linear stability analysis in the square lattice. (a, b) Profiles of the correlation $C_a^{★}\left(x\right)$ and its cumulative sum $S_a^{★}\left(x\right)$ at the power-law fixed point for different exponents $a$. (c) The spectral radius of the Jacobian matrix truncated at dimensions $(2^m-1)$ for different $a$. The black dashed line indicates ${\left|\lambda \right|}_{\max }=1$, which is the boundary separating stable and unstable fixed points. (d) The spectral radius extrapolated to infinite $m$ for different exponents $a$, compared with theory (black dashed line for $a<2$ and red dashed line for $a>2$).

Figure 3

The evolution of nearest-neighbor correlation $C_t\left(1\right)$ under RG iterations, starting from power-law initialization (a) or exponential initialization (b). The legends of solid lines specify initial conditions $C_0\left(x\right)$, and broken lines indicate the values at fixed points. For power-law initialization (a), the exponents are $a_1=1.4, a_2=1.8, a_3=2.2$. The mixing ratio is $b=0.3$. For exponential initialization (b), the decay rates are ${\gamma }_1={10}^{-5}, {\gamma }_2={10}^{-4}, {\gamma }_3={10}^{-3}$.

Figure 4

RG flows for the square lattice model. (a–c) RG flows in the $(p_t,q_t)$ plane for systems with a combination of two nontrivial solutions [Eq. (23)] with three different combinations of exponents: (a), $a=1.7, b=1.8$; (b), $a=1.7, b=2.2$; (c), $a=2.2, b=2.3$. Empty and filled red circles represent unstable and stable fixed points, respectively. Inset of C shows the trajectories starting with $p_t+q_t>1$, which diverge to a large number before converging back to the origin which is the only stable fixed point. (d) RG flows starting with a linear combination of power-law functions. The initial conditions are linear combination of power laws (i.e., $C_0\left(x\right)=b_1{x}^{-a}+b_2{x}^{-(a+\mathrm{\Delta }a)}$). $\mathrm{\Delta }a=0.5$. $b_1$ and $b_2$ are chosen to visualize RG flows in a sufficiently large dynamic range. $L=3$ is used to calculate $a_{\mathrm{eff}}$. Solid and broken red lines indicate families of stable and unstable fixed points, and the transition point $a_c=2.0$ is marked with a cross. The solid blue lines show the RG flows from 22 direct RG iterations, and dashed blue lines are extrapolation using the last 6 iterations assuming exponential convergence to a fixed point.

Figure 5

RG iterations in the cubic lattice network with isotropic initialization. (a) Dynamics of $C_t^{*}=C_t(0,1)+C_t(1,0)+C_t(1,1)$, which determines the dissipation scaling exponent ${\lambda }_{3d}$. (b) Dynamics of a nonneighboring correlation $C_t(1,2)$, which captures the behavior of correlation at generic separations. The legends of solid lines specify initial conditions with $A_{1,2,3}$ being the proper normalization constants. The dashed lines indicate values at the fixed point. The exponents are $a_1=2.4, a_2=2.7, a_3=3.3$. The mixing ratio is $b=0.5$. Note that the curves that come together in panel (a) also converge in panel (b).

Figure 6

RG iterations in the cubic lattice network with initial conditions that break different symmetries. (a–b) start with $f\left(\theta \right)=f_0{(1+b|cos\theta |}^c)$, which breaks isotropy. (c–d) start with $f\left(\theta \right)=f_0[1+bsin\left(2\theta \right)+ccos\left(2\theta \right)]$, which breaks axial reflection symmetry. (a) the dynamics of $C_t^{*}=C_t(0,1)+C_t(1,0)+C_t(1,1)$, which determines the dissipation scaling exponent ${\lambda }_{3d}$. (b) the dynamics of a nonneighboring correlation $C_t(1,2)$, which captures the behavior of correlation at generic separations. (c) Dynamics of the generalized $C_t^{*}$ [defined in Eq. (46)]. (d) Dynamics of a nonneighboring correlation $C_t(-1,2)$. In all cases, the exponent is $a=2.5$, and the values of $b$ and $c$ are given in the legends. $f_0$ ensures proper normalization. Panels (a) and (b) share the same legends, and so do panels (c) and (d). The dashed lines indicate values at the fixed point. Note that the curves converge in panels (a) and (c) but not in panel (b) or panel (d).