Critical behavior of a Ginzburg-Landau model with additive quenched noise

Abstract

We address a mean-field zero-temperature Ginzburg–Landau, or 4, model subjected to quenched additive noise, which has been used recently as a framework for analyzing collective effects induced by diversity. We first make use of a self-consistent theory to calculate the phase diagram of the system, predicting the onset of an order–disorder critical transition at a critical value σc of the quenched noise intensity σ, with critical exponents that follow the Landau theory of thermal phase transitions. We subsequently perform a numerical integration of the system's dynamical variables in order to compare the analytical results (valid in the thermodynamic limit and associated with the ground state of the global Lyapunov potential) with the stationary state of the (finite-size) system. In the region of the parameter space where metastability is absent (and therefore the stationary state coincides with the ground state of the Lyapunov potential), a finite-size scaling analysis of the order parameter fluctuations suggests that the magnetic susceptibility diverges quadratically in the vicinity of the transition, which constitutes a violation of the fluctuation–dissipation relation. We derive an effective Hamiltonian and accordingly argue that its functional form does not allow one to straightforwardly relate the order parameter fluctuations to the linear response of the system, at odds with equilibrium theory. In the region of the parameter space (a > 1, a being a parameter of the Lyapunov potential) where the system is susceptible to having a large number of metastable states (and therefore the stationary state does not necessarily correspond to the ground state of the global Lyapunov potential), we numerically find a phase diagram that strongly depends on the initial conditions of the dynamical variables. Specifically, for symmetrically distributed initial conditions, the system shows a disorder–order transition for σc' < σc, yielding a reentrant transition in the full picture. The location of σc' increases with the parameter a and eventually coalesces with σc, yielding in this case the disappearance of both transitions. On the other hand, for positive-definite initial conditions the order–disorder transition is eventually smoothed for large values of a, and accordingly no critical behavior is found. At this point we conclude that structural diversity can induce both the creation and annihilation of order in a nontrivial way.