Dynamics of a Toom interface in three dimensions

Abstract

We introduce a novel three-dimensional Toom model on a bcc lattice, and study its physical properties. In the low-noise limit, the model leads to an effective solid-on-solid type model, which exhibits a stationary interface via depositions and evaporations with an avalanche process. We find that the model is described by the Edwards-Wilkinson equation for the unbiased case and the anisotropic Kardar-Parisi-Zhang equation in the weak -coupling limit for the biased case. Thus the square of the surface width diverges logarithmically with space and time for both unbiased and biased cases.

Anisotropic Surface Growth Model in Disordered Media

“…Nevertheless, since the morphologies of Fig. 1a and 1b are similar to each other and that of λ > 0 and λ ⊥ < 0 can be proven using the tilt argument for the stochastic model [17], we believe that the stochastic model belongs to the AQKPZ universality. The PD transition turns out to be continuous as depicted in Fig.…”

“…Such anisotropic avalanche process is to be distinguished from the isotropic avalanche process on tilted substrates, in which the roughness exponent along the tilt direction (r ⊥ -direction) is 1/3 [8]. Using the tilt argument, it can be shown that our model includes alternative signs of the nonlinear terms, that is, λ > 0 and λ ⊥ < 0 [17]. A typical surface morphology is shown in Fig.…”

Strong-coupling phases of the anisotropic Kardar-Parisi-Zhang equation

“…That is, if both couplings were chosen with an equal sign, the isotropic KPZ behavior was recovered at long distances whereas in the case of an opposite sign or with one vanishing coupling the weak-coupling Edwards-Wilkinson (EW) behavior with logarithmic roughness was obtained. The results of Wolf were confirmed by numerical simulations [34][35][36] and the logarithmic correlations by exact arguments [37]. In contrast, studies on variants of the KPZ equation [38][39][40][41][42] as well as on related models [43,44] indicated that anisotropy may become important in some cases.…”

Characterisation of Surface Activity of Carbon Black and its Relation to Polymer‐Filler Interaction

“…The nontrivial effect of the anisotropy can be observed in the case of stepped surfaces, [75,76] or in the threedimensional discrete Toom model, [68,[77][78][79] which is described by the AKPZ equation. The original twodimensional Toom model has attracted much attention since its non-ergodicity in the presence of small perturbations has been proved, [20] leading to the possibility that the model is 'generic' for a variety of physical systems, including carbon black growth.…”

“…Therein, the dynamics of spin flips may be regarded as a 'deposition-evaporation' process of particles which occurs in an avalanche fashion. [78] This physics has been generalised in Reference [78] where the spin dynamics in three dimensions are mapped into particle dynamics via the deposition and evaporation process with an avalanche on a checkerboard lattice. For the biased case the interface is described by the AKPZ equation.…”