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.
In these lectures we introduce the functional renormalization group out of equilibrium. While in thermal equilibrium typically a Euclidean formulation is adequate, nonequilibrium properties require real-time descriptions. For quantum systems specified by a given density matrix at initial time, a generating functional for real-time correlation functions can be written down using the Schwinger-Keldysh closed time path. This can be used to construct a nonequilibrium functional renormalization group along similar lines as for Euclidean field theories in thermal equilibrium. Important differences i…
We present a numerical study on the ordering dynamics of a one-dimensional nonequilibrium Ising spin system with chirality. This system is characterized by a direction-dependent spin update rule. Pairs of $+-$ spins can flip to $++$ or $--$ with probability $(1-u)$ or to $-+$ with probability $u$ while $-+$ pairs are frozen. The system was found to evolve into the ferromagnetic ordered state at any $u<1$ exhibiting the power-law scaling of the characteristic length scale $ξ\sim t^{1/z}$ and the domain wall density $ρ\sim t^{-δ}$. The scaling exponents $z$ and $δ$ were found to vary continuousl…
In this work we formulate the nonequilibrium dynamical renormalization group (ndRG). The ndRG represents a general renormalization-group scheme for the analytical description of the real-time dynamics of complex quantum many-body systems. In particular, the ndRG incorporates time as an additional scale which turns out to be important for the description of the long-time dynamics. It can be applied to both translational invariant and disordered systems. As a concrete application we study the real-time dynamics after a quench between two quantum critical points of different universality classes.…
Using a tensor renormalization group method with oblique projectors for an anisotropic tensor network, we confirm that the rescaled spectrum of transfer matrices at nonequilibrium critical points in the (1+1)-dimensional directed percolation, a canonical model of nonequilibrium critical phenomena, is scale-invariant and its structure is universal.
We study renormalization group multicritical fixed points in the $ε$-expansion of scalar field theories characterized by the symmetry of the (hyper)cubic point group $H_N$. After reviewing the algebra of $H_N$-invariant polynomials and arguing that there can be an entire family of multicritical (hyper)cubic solutions with $ϕ^{2n}$ interactions in $d=\frac{2n}{n-1}-ε$ dimensions, we use the general multicomponent beta functionals formalism to study the special cases $d = 3-ε$ and $d =\frac{8}{3}-ε$, deriving explicitly the beta functions describing the flow of three- and four-critical (hyper)cu…
The energy dissipation rate in a nonequilibirum reaction system can be determined by the reaction rates in the underlying reaction network. By developing a coarse-graining process in state space and a corresponding renormalization procedure for reaction rates, we find that energy dissipation rate has an inverse power-law dependence on the number of microscopic states in a coarse-grained state. The dissipation scaling law requires self-similarity of the underlying network, and the scaling exponent depends on the network structure and the flux correlation. Implications of this inverse dissipatio…
Scaling and hyperscaling laws provide exact relations among critical exponents describing the behavior of a system at criticality. For nonequilibrium growth models with a conserved drift there exist few of them. One such relation is $α+z=4$, found to be inexact in a renormalization group calculation for several classical models in this field. Herein we focus on the two-dimensional case and show that it is possible to construct conserved surface growth equations for which the relation $α+z=4$ is exact in the renormalization group sense. We explain the presence of this scaling law in terms of th…
We study the nonequilibrium critical behavior of the pair contact process with diffusion (PCPD) by means of nonperturbative functional renormalization group techniques. We show that usual perturbation theory fails because the effective potential develops a nonanalyticity at a finite length scale: Perturbatively forbidden terms are dynamically generated and the flow can be continued once they are taken into account. Our results suggest that the critical behavior of PCPD can be either in the directed percolation or in a new (conjugated) universality class.
The totally asymmetric simple exclusion process along with particle adsorption and evaporation kinetics is a model of boundary-induced nonequilibrium phase transition. In the continuum limit, the average particle density across the system is described by a singular differential equation involving multiple scales which lead to the formation of boundary layers (BL) or shocks. A renormalization group analysis is developed here by using the location and the width of the BL as the renormalization parameters. It not only allows us to cure the large distance divergences in the perturbative solution f…
A generically observed mechanism that drives the self-organization of living systems is interaction via chemical signals among the individual elements -- which may represent cells, bacteria, or even enzymes. Here we propose a novel mechanism for such interactions, in the context of chemotaxis, which originates from the polarity of the particles and which generalizes the well-known Keller--Segel interaction term. We study the resulting large-scale dynamical properties of a system of such chemotactic particles using the exact stochastic formulation of Dean and Kawasaki along with dynamical renor…
The damage spreading transition: a hierarchy of renormalization group fixed points
Laboratoire de Physique de l’École Normale Supérieure, CNRS, ENS & Université PSL, Sorbonne Université, Université Paris Cité（巴黎高等师范学院物理实验室，法国国家科学研究中心，巴黎高等师范学院及巴黎文理研究大学，索邦大学，巴黎西岱大学）；International Centre for Theoretical Sciences, Tata Institute of Fundamental Research（理论科学国际中心，塔塔基础研究所）
Scaling Behaviors in Active Model B+ via the Functional Renormalization Group
Eötvös University（厄特沃什大学）；RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS)（理化学研究所跨学科理论与数学科学中心(iTHEMS)）；HUN-REN-ELTE Theoretical Physics Research Group（匈牙利研究与创新网络-厄特沃什大学理论物理研究组）；Department of Physics, Keio University（庆应义塾大学物理学系）
