In many applications, the primary objective of numerical simulation of timeevolving systems is the prediction of coarse-grained, or macroscopic, quantities. The purpose of this review is twofold: first, to describe a number of simple model systems where the coarse-grained or macroscopic behaviour of a system can be explicitly determined from the full, or microscopic, description; and second, to overview some of the emerging algorithmic approaches that have been introduced to extract effective, lower-dimensional, macroscopic dynamics. The model problems we describe may be either stochastic or deterministic in both their microscopic and macroscopic behaviour, leading to four possibilities in the transition from microscopic to macroscopic descriptions. Model problems are given which illustrate all four situations, and mathematical tools for their study are introduced. These model problems are useful in the evaluation of algorithms. We use specific instances of the model problems to illustrate these algorithms. As the subject of algorithm development and analysis is, in many cases, in its infancy, the primary purpose here is to attempt to unify some of the emerging ideas so that individuals new to the field have a structured access to the literature. Furthermore, by discussing the algorithms in the context of the model problems, a platform for understanding existing algorithms and developing new ones is built.

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Extracting macroscopic dynamics: model problems and algorithms

In a previous paper, a method was presented to integrate numerically nonlinear stochastic differential equations (SDEs) with additive, Gaussian, white noise. The method, a generalization of the Range Kutta algorithm, extrapolates from one point to the next applying functional evaluations at stochastically determined points. This paper extends (and at one point corrects) algorithms for the simple class of equations considered in the previous paper. In addition, the method is expanded to treat vector SDEs, equations with time-dependent functions, and SDEs higher than first order. The parameters for several explicit integration schemes are displayed.

Numerical integration of stochastic differential equations — ii

We study a variant of the Kac–Zwanzig model of a particle in a heat bath. The heat bath consists of n particles which interact with a distinguished particle via springs and have random initial data. As n → ∞ the trajectories of the distinguished particle weakly converge to the solution of a stochastic integro-differential equation—a generalized Langevin equation (GLE) with power-law memory kernel and driven by 1/f

 α
 -noise. The limiting process exhibits fractional sub-diffusive behaviour. We further consider the approximation of non-Markovian processes by higher-dimensional Markovian processes via the introduction of auxiliary variables and use this method to approximate the limiting GLE. In contrast, we show the inadequacy of a so-called fractional Fokker–Planck equation in the present context. All results are supported by direct numerical experiments.

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Fractional kinetics in Kac-Zwanzig heat bath models

Recent Developments in Classical Density Functional Theory

The climate is a forced and dissipative nonlinear system featuring nontrivial dynamics on a vast range of spatial and temporal scales. The understanding of the climate's structural and multiscale properties is crucial for the provision of a unifying picture of its dynamics and for the implementation of accurate and efficient numerical models. We present some recent developments at the intersection between climate science, mathematics, and physics, which may prove fruitful in the direction of constructing a more comprehensive account of climate dynamics. We describe the Nambu formulation of fluid dynamics and the potential of such a theory for constructing sophisticated numerical models of geophysical fluids. Then, we focus on the statistical mechanics of quasi-equilibrium flows in a rotating environment, which seems crucial for constructing a robust theory of geophysical turbulence. We then discuss ideas and methods suited for approaching directly the nonequilibrium nature of the climate system. First, we describe some recent findings on the thermodynamics of climate, characterize its energy and entropy budgets, and discuss related methods for intercomparing climate models and for studying tipping points. These ideas can also create a common ground between geophysics and astrophysics by suggesting general tools for studying exoplanetary atmospheres. We conclude by focusing on nonequilibrium statistical mechanics, which allows for a unified framing of problems as different as the climate response to forcings, the effect of altering the boundary conditions or the coupling between geophysical flows, and the derivation of parametrizations for numerical models.

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Mathematical and physical ideas for climate science

A New Expansion of the Master Equation

A new expansion scheme to evaluate the eigenvalues of the generalized evolution operator (Frobenius-Perron operator) $H_{q}$ relevant to the fluctuation spectrum and poles of the order-$q$ power spectrum is proposed. The ``partition function'' is computed in terms of unstable periodic orbits and then used in a finite pole approximation of the continued fraction expansion for the evolution operator. A solvable example is presented and the approximate and exact results are compared; good agreement is found.

Continued-fraction expansion of eigenvalues of generalized evolution operators in terms of periodic orbits

A new expansion scheme to evaluate the eigenvalues of the generalized evolution operator (Frobenius-Perron operator)Hq relevant to the fluctuation spectrum and poles of the order-q power spectrum is proposed. The “partition function” is computed in terms of unstable periodic orbits and then used in a finite pole approximation of the continued fraction expansion for the evolution operator. A solvable example is presented and the approximate and exact results are compared; good agreement is found.

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Hideto Shigematsu

Papers

A New Expansion of the Master Equation

Continued-fraction expansion of eigenvalues of generalized evolution operators in terms of periodic orbits

Continued-fraction expansion of eigenvalues of generalized evolution operators in terms of periodic orbits

Non-Markov Brownian Motions of Macrovariables