Noise in Spatially Extended Systems

1 Introduction.- 1.1 Fluctuations in a Macroscopic World.- 1.2 Transitions in Zero-Dimensional Systems.- 1.3 Phase Transitions in d-Dimensional Systems.- 1.4 Pattern Formation.- 1.5 Other Effects of Noise in Extended Media.- 2 Fundamentals and Tools.- 2.1 Introduction to Stochastic Partial Differential Equations.- 2.2 Analytical Techniques.- 2.3 Numerical Techniques.- 3 Noise-Induced Phase Transitions.- 3.1 Additive Noise.- 3.2 Additive and Multiplicative Noise.- 3.3 Multiplicative Noise.- 4 Dynamics of Phase Transitions with Fluctuations.- 4.1 Internal Multiplicative Noise.- 4.2 Noise-Induced Phase Separation.- 5 Pattern Formation Under Multiplicative Noise.- 5.1 Multiplicative Noise in the Swift-Hohenberg Model.- 5.2 Pure Noise-Induced Patterns.- 6 Front Dynamics and External Fluctuations.- 6.1 External Fluctuations in Deterministic Fronts.- 6.2 Noise-Induced Fronts.- 6.3 Reactive Fronts under Turbulent Advection.- 7 Conclusions.- 7.1 What Has Been Done.- 7.2 What Needs to Be Done.- A Continuum and Discrete Space Descriptions.- A.1 Coarse Graining.- A.2 Continuum Limit and Functional Analysis.- B Fourier Transforms.- B.1 Continuum Fourier Transforms.- B.2 Discrete Fourier Transforms.- B.3 Discrete Fourier Transform of a Real Uncorrelated Field.- C Fokker—Planck Equation for an Additive Colored Noise.- D Colored Noise in a Linear Model.- E Fokker—Planck Equation for a Multiplicative Noise.- References.