Stochastic diagrams for critical point spectra

A new technique for calculating the time-evolution, correlations and steady state spectra for nonlinear stochastic differential equations is presented. To illustrate the method, we consider examples involving cubic nonlinearities in an N-dimensional phase-space. These serve as a useful paradigm for describing critical point phase transitions in numerous equilibrium and non-equilibrium systems, ranging from chemistry, physics and biology, to engineering, sociology and economics. The technique consists in developing the stochastic variable as a power series in time, and using this to compute the short time expansion for the correlation functions. This is then extrapolated to large times, and Fourier transformed to obtain the spectrum. Stochastic diagrams are developed to facilitate computation of the coefficients of the relevant power series expansion. Two different types of long-time extrapolation technique, involving either simple exponentials or logarithmic rational approximations, are evaluated for third-order diagrams. The analytical results thus obtained are compared with numerical simulations, together with exact results available in special cases. The agreement is found to be excellent up to and including the neighborhood of the critical point. Exponential extrapolation works especially well even above the critical point at large N values, where the dynamics is one of phase-diffusion in the presence of a spontaneously broken symmetry. This method also enables the calculation of the steady state spectra of polynomial functions of the stochastic variables. In these cases, the final correlations can be non-bistable even above threshold. Here logarithmic rational extrapolation has the greater accuracy of the two extrapolation methods. Stochastic diagrams are also applicable to more general problems involving spatial variation, in addition to temporal variation.