Universality classes in nonequilibrium lattice systems

This article reviews our present knowledge of universality classes in nonequilibrium systems defined on regular lattices. The first section presents the most important critical exponents and relations, as well as the field-theoretical formalism used in the text. The second section briefly addresses the question of scaling behavior at first-order phase transitions. In Sec. III the author looks at dynamical extensions of basic static classes, showing the effects of mixing dynamics and of percolation. The main body of the review begins in Sec. IV, where genuine, dynamical universality classes specific to nonequilibrium systems are introduced. Section V considers such nonequilibrium classes in coupled, multicomponent systems. Most of the known nonequilibrium transition classes are explored in low dimensions between active and absorbing states of reaction-diffusion-type systems. However, by mapping they can be related to the universal behavior of interface growth models, which are treated in Sec. VI. The review ends with a summary of the classes of absorbing-state and mean-field systems and discusses some possible directions for future research.

Figure 1

(Color in online edition) Phase diagram of two-dimensional, $Z_2$-symmetric nonequilibrium spin models. Dashed lines correspond to the noisy voter model (V), the Ising model (I), and the majority vote model (M). The low-temperature phase is located in the upper right corner, above the transition line (solid line). From 114.

Figure 2

Schematic plot for $q_c\left(d\right)$ (solid line). Open symbols correspond to continuous phase transition, filled symbol to a known first-order transition. Below the dashed line the transition is continuous, too. From 427.

Figure 3

Directed site percolation in $d=1+1$ dimensions.

Figure 4

(Color in online edition) Local slopes of the density decay in a Bosonic BARW1 model (Sec. 4A3). Different curves correspond to $\lambda =0.12883$,0.12882,0.12881,0.1288,0.12879 (from bottom to top). From 347.

Figure 5

Schematic of the update rule for the Domany-Kinzel model.

Figure 6

Phase diagram of the one-dimensional Domany-Kinzel stochastic cellular automation. From 181.

Figure 7

Schematic mean-field phase diagram for boundary directed percolation. The transitions are labeled by $\mathrm{O}=\text{ordinary}$, $\mathrm{E}=\text{extraordinary}$, $\mathrm{S}=\text{surface}$, and $\mathrm{Sp}=\text{special}$. From 139.

Figure 8

The space-time evolution of the critical, $(1+1)$-dimensional directed site percolation process confined by a parabola (a) $\sigma <1∕Z$, (b) $\sigma =1∕Z$, (c) $\sigma >1∕Z$. From 242.

Figure 9

Estimates for the exponent $\beta$ and the derived exponents ${\nu }_{\perp }$ and ${\nu }_{\Vert }$ in comparison with the field-theoretic results (solid lines) and the DP exponents (dotted lines). The quantities ${\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_2$ represent deviations from the scaling relations (116, 117), respectively. The vertical, dashed lines distinguish between regions of constant and continuously changing exponents. From 191.

Figure 10

Isotropic site percolation in $d=2$ dimensions.

Figure 11

(Color in online edition) Parabola boundary confinement cluster simulations for compact directed percolation. Middle curves: number of active sites ($C=2$,1.5,1.2,1, top to bottom); lower curves: survival probability ($C=2$,1.5,1.2,1, top to bottom); upper curves: $R^2\left(t\right)$ ($C=2$,1.5,1.2,1, top to bottom). From 346.

Figure 12

The anomalous annihilation process: the graph from 191 shows direct estimates and extrapolations for the decay exponent $\alpha$, as a function of $\sigma$. The solid line represents the exact result (neglecting log corrections at $\sigma =1$).

Figure 13

(Color in online edition) Local slopes [Eq. (93)] of the density decay in a bosonic BARW2 model. Different curves correspond to $\sigma =0.466$,0.468,0.469,0.47 (from bottom to top). From 347.

Figure 14

(Color in online edition) Phase diagram of the two-parameter model. The transversal dotted line indicates the critical point that was investigated in more detail. From 346.

Figure 15

(Color in online edition) ${\beta }_{\mathrm{eff}}=d\ln {\rho }_{\infty }∕d\ln ϵ$ of kinks (circles) near the critical point $(ϵ=\mid \tilde{\delta }-{\tilde{\delta }}^{\prime }\mid )$ and linear extrapolation (line) to the asymptotic value $[\beta =0.95\left(2\right)]$. Simulations were performed on a one-dimensional NEKIM ring of size $L=24000$. From 346.

Figure 16

Phase diagram of the monomer-monomer model. Here $p$ denotes the adsorption probability of $A$-$s$, $r$ the reduced adsorption probability of $A$-$s$ as the consequence of nearest-neighbor repulsion. From 48.

Figure 17

Domain-wall dynamics in the interacting monomer-monomer model: $A$ particle reaction, branching; $B$ particle reaction, annihilation.

Figure 18

(Color in online edition) Kink density $\ln \left[{\rho }_k\left(t\right)\right]$ vs $\ln \left(t\right)$ in NEKIM simulations for $\sigma =0,0.1,0.2,\dots ,1$ initial conditions (from bottom to top curves). From 314.

Figure 19

Schematic mean-field boundary phase diagram for BARW. See text for an explanation of the labeling. From 139.

Figure 20

Schematic surface phase diagrams for BARW in $d=1$ for (a) ${\sigma }_m<{\sigma }_{m,\text{critical}}$, and (b) ${\sigma }_m={\sigma }_{m,\text{critical}}$. See text for an explanation of the labeling. From 139.

Figure 21

(Color in online edition) Initial concentration dependence of the exponent $\eta$ for the pair-contact model. Linear regression gives a slope 0.320(7) between $\eta -{\eta }_{DP}$ and ${\rho }_1\left(0\right)-{\rho }_1^{\mathrm{nat}}$. From 342.

Figure 22

(Color in online edition) Schematic phase diagram of the one-dimensional pair-contact process with particle diffusion model: circles, simulation and DMRG results; vertical solid line at $p=\frac{1}{3}$, mean-field; dashed line, pair approximation.

Figure 23

(Color in online edition) Effective $\beta$ exponents for different diffusion rates: ●, $D=0.05$; ∎, $D=0.1$; ◆, $D=0.2$; ▴, $D=0.5$; ▾, $D=0.7$.

Figure 24

(Color in online edition) Flow of the interspecies couplings in the two-component, DP model under renormalization: D, decoupled; S, symmetric; U, unidirectional fixed points. From 215.

Figure 25

(Color in online edition) Space-time evolution of the asymmetric production and $m$-particle annihilating random walk (2-PARWas) model at the critical point. Black pixels correspond to $A$ particles, others to $B$’s. From 335.

Figure 26

Mapping between spins, kinks and surfaces.

Figure 27

(Color in online edition) Absorption of dimers with probability $p$ and desorbtion at the edges of terraces with probability $1-p$. Evaporation from the middle of plateaus is not allowed. From 195.

Figure 28

Extended particle interpretation. Dimers are adsorbed $(2A\to \varnothing )$ and desorbed $(A\to 3A)$ at the bottom layer. Similar processes take place at higher levels. From 195.

Figure 29

(Color in online edition) Phase diagram of one-dimensional dimer models.