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

Based 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.