Many exciting frontiers of science and engineering require understanding the spatiotemporal properties of sustained nonequilibrium systems such as fluids, plasmas, reacting and diffusing chemicals, crystals solidifying from a melt, heart muscle, and networks of excitable neurons in brains. This introductory textbook for graduate students in biology, chemistry, engineering, mathematics, and physics provides a systematic account of the basic science common to these diverse areas. This book provides a careful pedagogical motivation of key concepts, discusses why diverse nonequilibrium systems often show similar patterns and dynamics, and gives a balanced discussion of the role of experiments, simulation, and analytics. It contains numerous worked examples and over 150 exercises. This book will also interest scientists who want to learn about the experiments, simulations, and theory that explain how complex patterns form in sustained nonequilibrium systems.

A three-dimensional MHD equilibrium code is described that does not assume the existence of good flux surfaces. Given an initial guess for the magnetic field, the code proceeds by calculating the pressure-driven current and then by updating the field using Ampere's law. The numerical algorithm to solve the magnetic differential equation for the pressure-driven current is described, and demonstrated for model fields having islands and stochastic regions. The numerical algorithm which solves Ampere's law In three dimensions is also described. Finally, the convergence of the code is illustrated for a particular steilarator equilibrium with no large islands."Presented as an invited talk at the 8th Europhysics Conference on Computational Physics, Elbsee, Fed. Rep. of Germany, May 13-16, 1986. To appear in a special issue of Computer Physics Communications. In this paper we describe a three-dimensional MHD equilibrium code that does not assume the existence of good flux surfaces. The code uses an algorithm for solving the equilibrium equations that is different from the energy-minimizing algorithms which have been developed extensively and successfully for 3D equilibrium codes that assume good surfaces [1][2][3]. We describe the algorithm, our reasons for choosing It, and the numerical methods we have developed to implement it. We also describe the numerical issues raised by the presence of Islands and stochastic regions, and how we have dealt with those issues. In particular, we describe how the code solves magnetic differential equations, using as examples model fields with islands and stochastic regions. Finally, we describe the convergence of the code for a particular stellarator equilibrium with no large inlands.We specialize In this paper to the case of zero net current within each flux surface, as is appropriate for applications to stellarators. Our code has been written to deal with the more general case of arbitrary net currents, except for some small modifications that need to be made to deal with the currents in the islands and stochastic regions.The applications we have in mind for the code are the study of the breaking of flux surfaces in stellarators, and the study of The magnetic field becomes increasingly distorted, it is not known under what conditions the field produced by the pressure-driven currents will cause the formation of large islands and stochastic regions. In particular, it is important to know whether there is an equilibrium beta limit due to such an effect, and what that beta limit is. The convention In evaluating stellarator designs has been to estimate the equilibrium beta limit to be the value of beta at which the magnetic axis shifts halfway to the wall. An analytical calculation of equilibrium island formation in one proposed helical axis stellarator found a much lower beta limit [4].Nominally axisymmetric devices, such as tokamaks, are in practice generally nonaxisymmetrlc because of the presence of tearing instabilities, magnetic ripple from discrete coils, and field error...

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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 Runge‐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.

The free energy is calculated for the various phases possible in a superconductor containing a periodic array of magnetic ions with ferromagnetic interactions. Suggestions are made for experimental observation of coexisting superconductivity and long-range magnetic order.

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A recently developed space-time adaptive mesh refinement algorithm (AMRA) for simulating isotropic one- and two-dimensional excitable media is generalized to simulate three-dimensional anisotropic media. The accuracy and efficiency of the algorithm is investigated for anisotropic and inhomogeneous 2D and 3D domains using the Luo-Rudy 1 (LR1) and FitzHugh-Nagumo models. For a propagating wave in a 3D slab of tissue with LR1 membrane kinetics and rotational anisotropy comparable to that found in the human heart, factors of 50 and 30 are found, respectively, for the speedup and for the savings in memory compared to an algorithm using a uniform space-time mesh at the finest resolution of the AMRA method. For anisotropic 2D and 3D media, we find no reduction in accuracy compared to a uniform space-time mesh. These results suggest that the AMRA will be able to simulate the 3D electrical dynamics of canine ventricles quantitatively for 1 s using 32 1-GHz Alpha processors in approximately 9 h.

Analytical and numerical methods are used to study the linear stability of spatia11y periodic solutions for various two-dimensional equations which model thermal convection in fluids. This analysis suggests new model equations that will be useful for investigating questions such as wavenumber selection, pattern formation, and the onset of turbulence in large-aspect-ratio Rayleigh-Benard systems. In particular, we construct a nonrelaxational model that has stability boundaries similar to those calculated for intermediate Prandtl-number fluids.