Intended for graduates and researchers in physics, chemistry, biology, and applied mathematics, this book provides an up-to-date introduction to current research in fluctuations in spatially extended systems. It covers the theory of stochastic partial differential equations and gives an overview of the effects of external noise on dynamical systems with spatial degrees of freedom. Starting with a general introduction to noise-induced phenomena in dynamical systems, the text moves on to an extensive discussion of analytical and numerical tools needed to gain information from stochastic partial differential equations. It then turns to particular problems described by stochastic PDEs, covering a wide part of the rich phenomenology of spatially extended systems, such as nonequilibrium phase transitions, domain growth, pattern formation, and front propagation. The only prerequisite is a minimal background knowledge of the Langevin and Fokker-Planck equations.

Inhalt

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-Bénard 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.