Convolutionless Nakajima–Zwanzig equations for stochastic analysis in nonlinear dynamical systems

Abstract

Determining the statistical properties of stochastic nonlinear systems is of major interest across many disciplines. Currently, there are no general efficient methods to deal with this challenging problem that involves high dimensionality, low regularity and random frequencies. We propose a framework for stochastic analysis in nonlinear dynamical systems based on goal-oriented probability density function (PDF) methods. The key idea stems from techniques of irreversible statistical mechanics, and it relies on deriving evolution equations for the PDF of quantities of interest, e.g. functionals of the solution to systems of stochastic ordinary and partial differential equations. Such quantities could be low-dimensional objects in infinite dimensional phase spaces. We develop the goal-oriented PDF method in the context of the time-convolutionless Nakajima-Zwanzig-Mori formalism. We address the question of approximation of reduced-order density equations by multi-level coarse graining, perturbation series and operator cumulant resummation. Numerical examples are presented for stochastic resonance and stochastic advection-reaction problems.

Statistical analysis and simulation of random shocks in stochastic Burgers equation

“…This can be done in a formally exact way by using the MZ formalism, in particular, the convolutionless form [29,31,32]. This yields…”

“…Substituting (2.20) into equation (2.14) gives the reduced order kinetic equation 21) where [29,43,44] relative to the joint PDF of the random vector ξ . For instance, the first two cumulants are .22) and…”

“…We now introduce approximations. To this end, we expand the right-hand side of equation (2.15) in formal power series in the coupling parameter σ (see [29] for details). This yields…”

“…Here, we follow a different approach based on modelling, via deterministic equations, the probability density function (pdf) of the solution to the stochastic Burgers equation. The key idea stems from techniques of irreversible statistical mechanics, in particular, the Mori-Zwanzig (MZ) formalism [29][30][31][32][33], and it relies on deriving reduced-order kinetic equations for the stochastic velocity field in the limits of small viscosity and small perturbations. We combine this approach with the adaptive discontinuous Galerkin (DG) …”

Parametric reduced models for the nonlinear Schrödinger equation

“…An elegant framework for deriving the effective model is the Mori-Zwanzig projection [2][3][4], which has recently become an extremely important tool to simplify complex dynamical systems [5][6][7][8][9][10][11][12][13][14][15]. In particular, this derivation led to a set of generalized Langevin equation (GLEs), a typical result of the Mori-Zwanzig procedure.…”