Phase transitions induced by microscopic disorder: A study based on the order parameter expansion

Abstract

a b s t r a c tBased on the order parameter expansion, we present an approximate method which allows us to reduce large systems of coupled differential equations with diverse parameters to three equations: one for the global, mean field, variable and two which describe the fluctuations around this mean value. The method is based on a systematic perturbation expansion and can be applied around the vicinity of the homogeneous state. With this tool we analyze phase transitions induced by microscopic disorder in three prototypical models of phase transitions which have been studied previously in the presence of thermal noise. We study how macroscopic order is induced or destroyed by time-independent local disorder and analyze the limits of the approximation by comparing the results with the numerical solutions of the self-consistency equation which arises from the property of self-averaging. Finally, we carry on a finite-size analysis of the numerical results and calculate the corresponding critical exponents.

Order parameter expansion and finite-size scaling study of coherent dynamics induced by quenched noise in the active rotator model

“…As figure 5(a) shows, the maximum value χ max (N ) scales as N c with c = 0.65 ± 0.03 at the first transition and c = 0.61 ± 0.07 at the second. Interestingly enough, the values of the critical exponent at both transitions seem to be consistent with the value c = 2/3 observed in a phase transition induced by quenched noise in a Ginzburg-Landau model [10]. It turns out that the full dependence of N and σ at both transitions can be fitted using standard finite-size-scaling theory [24,25] as…”

“…Due to the complexity of the problem, the analytical treatments are usually very difficult to be carried out in full detail and most results rely on extensive numerical simulations. However, a recently introduced technique named "order parameter expansion" [5][6][7][8][9][10] offers a simple approximate way of analyzing the effect of the random quenched terms in the dynamical equations. The approximation scheme allows the reduction of the large number of coupled differential equations for the microscopic variables to just a few effective equations for the relevant macroscopic dynamical variables: the mean values and dispersions from the mean.…”

Integrable order parameter dynamics of globally coupled oscillators

“…Immediately after crossing the border between regions IV and V, in region V the saddle point (0, 0) transforms into the unstable node (0, 0) originating two saddles (± √ η, ±ξ √ η) while the stable nodes (± √ ξ, ±η √ ξ) remain. In region V both ξ and η are positive and line (15) is again an ellipse. However, contrary to region I, this ellipse now consists of four separatrices which start at the saddles (± √ η, ±ξ √ η) and end at the stable nodes (± √ ξ, ±η √ ξ).…”

“…When the coupling is absent (K = 0) system (1) describes the dynamics of noninteracting oscillators with a stable circular limit cycles |z j | = 1. In [10] system (1) was studied in the framework of two order parameter approximation (for more about the order parameter expansion method, see [11][12][13][14][15][16]):…”

The constructive role of diversity in the global response of coupled neuron systems

“…It is possible to perform an approximate analysis of the effect of the diversity in the case of diffusive (electrical) coupling. The analysis allows one to gain an insight into the amplification mechanism by showing how the effective nullclines of the global variable X (t) are modified when varying s. The theoretical analysis is based upon a modification of the so-called order parameter expansion developed by Monte & D'Ovidio (2002) and Monte et al (2005) along the lines of Komin & Toral (2010). The approximation begins by expanding the dynamical variables around their average values…”