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 nonlineari ties 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 engin eering, sociology and economics. The technique consists in developing the s tochastic variable as a power series in time, and using this to compute the short time expansion for the correlation functions. This is then extrapola ted to large times, and Fourier transformed to obtain the spectrum. Stochas tic diagrams are developed to facilitate computation of the coefficients of the relevant power series expansion. Two different types of long-time extr apolation technique, involving either simple exponentials or logarithmic ra tional approximations, are evaluated for third-order diagrams. The analytic al 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. Expo nential extrapolation works especially well even above the critical point a t large N values, where the dynamics is one of phase-diffusion in the prese nce of a spontaneously broken symmetry. This method also enables the calcul ation 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 a ccuracy of the two extrapolation methods. Stochastic diagrams are also appl icable to more general problems involving spatial variation, in addition to temporal variation.