Noise in spatially extended systems

1 Introduction.- 1.1 Fluctuations in a Macroscopic World.- 1.1.1 Describing Stochastic Dynamics.- 1.1.2 Stochastic Partial Differential Equations.- 1.1.3 Experimental Setups.- 1.1.4 Numerical Experiments.- 1.1.5 Noise-Induced Phenomena.- 1.2 Transitions in Zero-Dimensional Systems.- 1.2.1 Internal Noise.- 1.2.2 External Noise.- 1.3 Phase Transitions in d-Dimensional Systems.- 1.3.1 Equilibrium Phase Transitions.- 1.3.2 Nonequilibrium Phase Transitions.- 1.3.3 Dynamics of Phase Transitions.- 1.4 Pattern Formation.- 1.4.1 Order-Parameter Equations.- 1.4.2 Pattern-Forming Instabilities.- 1.4.3 Amplitude Equations and Beyond.- 1.4.4 Real Patterns.- 1.5 Other Effects of Noise in Extended Media.- 1.5.1 Noise-Sustained Convective Structures.- 1.5.2 Spatial Stochastic Resonance.- 2 Fundamentals and Tools.- 2.1 Introduction to Stochastic Partial Differential Equations.- 2.1.1 Generalities and Modeling.- 2.1.2 Stochastic Calculus in SPDEs.- 2.1.3 Fokker-Planck Equation for Spatially Extended Systems.- 2.1.4 Statistical Moments and Correlations.- 2.2 Analytical Techniques.- 2.2.1 Mean-Field Analysis and Beyond.- 2.2.2 Small Noise Expansions.- 2.2.3 Linear Stability Analysis.- 2.2.4 Dynamic Renormalization Group Analysis.- 2.3 Numerical Techniques.- 2.3.1 Algorithms for Solving SPDEs.- 2.3.2 Generation of Correlated Noises.- 3 Noise-Induced Phase Transitions.- 3.1 Additive Noise.- 3.1.1 Ginzburg-Landau Model with Colored Noise.- 3.1.2 Noise-Induced Shift of the Transition Point.- 3.1.3 Fokker-Planck Analysis.- 3.1.4 Dynamic Renormalization Group Analysis.- 3.2 Additive and Multiplicative Noise.- 3.2.1 Ginzburg-Landau Model with Multiplicative Noise.- 3.2.2 Pure Noise-Induced Phase Transitions.- 3.2.3 Noise-Induced First-Order Phase Transitions.- 3.3 Multiplicative Noise.- 3.3.1 Multiplicative Noise Universality Class.- 3.3.2 Disordering Transitions Induced by Pure Multiplicative Noise.- 3.3.3 Numerical Simulation Results.- 4 Dynamics of Phase Transitions with Fluctuations.- 4.1 Internal Multiplicative Noise.- 4.1.1 Mesoscopic Derivation.- 4.1.2 Application to Phase Separation Dynamics.- 4.1.3 Extension to Nonconserved Order Parameter.- 4.2 Noise-Induced Phase Separation.- 4.2.1 External Fluctuations in Phase Separation.- 4.2.2 Stability Analysis.- 4.2.3 Phase Diagram.- 5 Pattern Formation Under Multiplicative Noise.- 5.1 Multiplicative Noise in the Swift-Hohenberg Model.- 5.1.1 A Model Equation for Rayleigh-Benard Convection.- 5.1.2 Fluctuations in the Control Parameter.- 5.1.3 Effect of a Spatially Correlated Noise.- 5.2 Pure Noise-Induced Patterns.- 6 Front Dynamics and External Fluctuations.- 6.1 External Fluctuations in Deterministic Fronts.- 6.1.1 Front Propagation in Fluctuating Media.- 6.1.2 Theoretical Approach and Predictions.- 6.1.3 Noise Effects on the Front Selection Problem.- 6.1.4 Profile Shape and Velocity Shift.- 6.1.5 Front Diffusive Spreading.- 6.2 Noise-Induced Fronts.- 6.2.1 Modeling and Analytical Results.- 6.2.2 Numerical Results.- 6.3 Reactive Fronts under Turbulent Advection.- 6.3.1 Modeling.- 6.3.2 A Gaussian Turbulence?.- 6.3.3 Theoretical Analysis and Numerical Results.- 6.3.4 The Role of Different Spectra.- 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.