Algebraic coarsening in voter models with intermediate states

Abstract

The introduction of intermediate states in the dynamics of the voter model modifies the ordering process and restores an effective surface tension. The logarithmic coarsening of the conventional voter model in two dimensions is eliminated in favour of an algebraic decay of the density of interfaces with time, compatible with Model A dynamics at low temperatures. This phenomenon is addressed by deriving Langevin equations for the dynamics of appropriately defined continuous fields. These equations are analyzed using field theoretical arguments and by means of a recently proposed numerical technique for the integration of stochastic equations with multiplicative noise. We find good agreement with lattice simulations of the microscopic model. * Electronic address: dallasta@ictp.it † Electronic address: tobias.galla@manchester.ac.uk

Social influencing and associated random walk models: Asymptotic consensus times on the complete graph

“…Such models with intermediate states have only recently begun receiving attention 30,31 in statistical physics literature, and we believe that this study is an important contribution to this body of literature.…”

Systems with two symmetric absorbing states: Relating the microscopic dynamics with the macroscopic behavior

“…To illustrate our previous results, we now analyze a general class of 3-state models [13,14], known to exhibit curvature driven by surface tension, as recently shown in [15]. They are composed by two external absorbing states S = ±1, and an intermediate state S = 0.…”

“…Interesting dynamical behaviors arise depending on the specific updating rules, the number of states and the functional form of the transition probabilities between configurations. For instance, it has been found that the addition of memory or inertia in the spin dynamics [11,12], the introduction of intermediate states [13][14][15], or the use of non-linear transitions [16], result in a drastic change of the coarsening properties and final outcome of the system.Despite that the dynamical rules of the models are very different in nature, many of them seem to share the same macroscopic behavior, such as coarsening and criticality. However, the minimal conditions that a microscopic dynamics must hold in order to observe a particular behavior have not been clearly identified yet.…”

“…The analysis of this equation helps to understand some of the open questions about phase ordering in these systems, that is, whether the coarsening is driven by curvature like in the Ising model [17], or it is without surface tension like in the original voter model (VM) [4]. This approach also explains, from a different perspective than in [15], why adding intermediate states to the VM leads to an effective surface tension. Moreover, numerical simulations of the spin dynamics reveal the conditions on the interaction range, to observe the three possible classes of phase transitions unveiled in the Langevin equation [3].…”

“…We want to derive a Langevin equation for the field φ r (t), that is a continuous representation of the spin at site r, at time t. For this we follow a standard approach (see [15]), and consider an ensemble of Ω copies of the system, each copy representing a particular spin configuration. This is equivalent to assume Ω spin particles at each site of the lattice (our microscopic model corresponds exactly to Ω = 1, but this substitution can be made at the end of the calculation).…”