Transitions in Stochastic Non-equilibrium Systems: Efficient Reduction and Analysis

Abstract

A central challenge in physics is to describe non-equilibrium systems driven by randomness, such as a randomly growing interface, or fluids subject to random fluctuations that account e.g. for local stresses and heat fluxes not related to the velocity and temperature gradients. For deterministic systems with infinitely many degrees of freedom, normal form and center manifold theory have shown a prodigious efficiency to often completely characterize how the onset of linear instability translates into the emergence of nonlinear patterns. However, in presence of random fluctuations, this reduction procedure is seriously challenged due to large excursions caused by the noise, and the approach needs to be revisited. We present an alternative framework to cope with these difficulties exploiting the approximation theory of stochastic invariant manifolds and energy estimates measuring the defect of parameterization of the high-modes. To operate for fluid problems, these error estimates are derived under assumptions regarding dissipation effects brought by the high-modes that suitably counterbalance the loss of regularity due to the nonlinear terms. The approach enables us to predict, from the reduced equations, the occurrence in large probability of a stochastic analogue to the pitchfork bifurcation, as long as the noise's intensity and the eigenvalue's magnitude of the mildly unstable mode scale accordingly. Our parameterization formulas involve non-Markovian coefficients, which depend explicitly on the history of the noise path that drives the SPDE dynamics, and their memory content is self-consistently determined by the intensity of the random force and its interaction through the SPDE's nonlinear terms. Applications to a stochastic Rayleigh-Benard problem are detailed, for which conditions for a stochastic pitchfork bifurcation (in large probability) to occur, are clarified.