A comprehensive review of spatiotemporal pattern formation in systems driven away from equilibrium is presented, with emphasis on comparisons between theory and quantitative experiments. Examples include patterns in hydrodynamic systems such as thermal convection in pure fluids and binary mixtures, Taylor-Couette flow, parametric-wave instabilities, as well as patterns in solidification fronts, nonlinear optics, oscillatory chemical reactions and excitable biological media. The theoretical starting point is usually a set of deterministic equations of motion, typically in the form of nonlinear partial differential equations. These are sometimes supplemented by stochastic terms representing thermal or instrumental noise, but for macroscopic systems and carefully designed experiments the stochastic forces are often negligible. An aim of theory is to describe solutions of the deterministic equations that are likely to be reached starting from typical initial conditions and to persist at long times. A unified description is developed, based on the linear instabilities of a homogeneous state, which leads naturally to a classification of patterns in terms of the characteristic wave vector q0 and frequency ω0 of the instability. Type Is systems (ω0=0, q0≠0) are stationary in time and periodic in space; type IIIo systems (ω0≠0, q0=0) are periodic in time and uniform in space; and type Io systems (ω0≠0, q0≠0) are periodic in both space and time. Near a continuous (or supercritical) instability, the dynamics may be accurately described via "amplitude equations," whose form is universal for each type of instability. The specifics of each system enter only through the nonuniversal coefficients. Far from the instability threshold a different universal description known as the "phase equation" may be derived, but it is restricted to slow distortions of an ideal pattern. For many systems appropriate starting equations are either not known or too complicated to analyze conveniently. It is thus useful to introduce phenomenological order-parameter models, which lead to the correct amplitude equations near threshold, and which may be solved analytically or numerically in the nonlinear regime away from the instability. The above theoretical methods are useful in analyzing "real pattern effects" such as the influence of external boundaries, or the formation and dynamics of defects in ideal structures. An important element in nonequilibrium systems is the appearance of deterministic chaos. A greal deal is known about systems with a small number of degrees of freedom displaying "temporal chaos," where the structure of the phase space can be analyzed in detail. For spatially extended systems with many degrees of freedom, on the other hand, one is dealing with spatiotemporal chaos and appropriate methods of analysis need to be developed. In addition to the general features of nonequilibrium pattern formation discussed above, detailed reviews of theoretical and experimental work on many specific systems are presented. These include Rayleigh-Benard convection in a pure fluid, convection in binary-fluid mixtures, electrohydrodynamic convection in nematic liquid crystals, Taylor-Couette flow between rotating cylinders, parametric surface waves, patterns in certain open flow systems, oscillatory chemical reactions, static and dynamic patterns in biological media, crystallization fronts, and patterns in nonlinear optics. A concluding section summarizes what has and has not been accomplished, and attempts to assess the prospects for the future.

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Pattern formation outside of equilibrium

Bose-Einstein condensation (BEC) has been observed in a dilute gas of sodium atoms. A Bose-Einstein condensate consists of a macroscopic population of the ground state of the system, and is a coherent state of matter. In an ideal gas, this phase transition is purely quantum-statistical. The study of BEC in weakly interacting systems which can be controlled and observed with precision holds the promise of revealing new macroscopic quantum phenomena that can be understood from first principles.

Bose-Einstein condensation in a gas of sodium atoms

The boundary layer equations for plane, incompressible, and steady flow are 

$$\matrix{ {u{{\partial u} \over {\partial x}} + v{{\partial u} \over {\partial y}} = - {1 \over \varrho }{{\partial p} \over {\partial x}} + v{{{\partial ^2}u} \over {\partial {y^2}}},} \cr {0 = {{\partial p} \over {\partial y}},} \cr {{{\partial u} \over {\partial x}} + {{\partial v} \over {\partial y}} = 0.} \cr }$$

Boundary Layer Theory

This Colloquium gives an overview of recent theoretical and experimental progress in the area of nonequilibrium dynamics of isolated quantum systems There is particularly a focus on quantum quenches: the temporal evolution following a sudden or slow change of the coupling constants of the system Hamiltonian Several aspects of the slow dynamics in driven systems are discussed and the universality of such dynamics in gapless systems with specific focus on dynamics near continuous quantum phase transitions is emphasized Recent progress on understanding thermalization in closed systems through the eigenstate thermalization hypothesis is also reviewed and relaxation in integrable systems is discussed Finally key experiments probing quantum dynamics in cold atom systems are overviewed and put into the context of our current theoretical understanding

Colloquium: Nonequilibrium dynamics of closed interacting quantum systems

We continue to investigate the possibility that interstellar turbulence is caused by nonlinear interactions
among shear Alfven waves. Here, as in Paper I, we restrict attention to the symmetric case where the oppositely directed waves carry equal energy fluxes. This precludes application to the solar wind in which the outward flux significantly exceeds the ingoing one. All our detailed calculations are carried out for an incompressible
magnetized fluid. In incompressible magnetohydrodynamics (MHD), nonlinear interactions only occur between oppositely direct waves. Paper I contains a detailed derivation of the inertial range spectrum for the weak turbulence of shear Alfven waves. As energy cascades to high perpendicular wavenumbers, interactions become so strong that the assumption of weakness is no longer valid. Here, we present a theory for the strong turbulence of shear Alfven waves. It has the following main characteristics. (1) The inertial-range energy spectrum exhibits a critical balance beween linear wave periods and nonlinear turnover timescales. (2) The "eddies" are elongated in the direction of the field on small spatial scales; the parallel and perpendicular components of the wave vector, k_z and k_⊥, are related by k_z ≈ k^(2/3) _⊥L^(-1/3), where L is the outer scale of the turbulence. (3) The "one-dimensional" energy spectrum is proportional to k^(-5/3) _⊥-an anisotropic Kolmogorov energy spectrum. Shear Alfvenic turbulence mixes specific entropy as a passive contaminant. This gives rise to an electron density power spectrum whose form mimics the energy spectrum of the turbulence. Radio, wave scattering by these electron density fluctuations produces anisotropic scatter-broadened images. Damping by ion-neutral collisions restricts Alfvenic turbulence to highly ionized regions of the interstellar medium. We expect negligible generation of compressive MHD waves by shear Alfven waves belonging to the critically balanced cascade. Viscous and collisionless damping are also unimportant in the interstellar medium (ISM). Our calculations support the general picture of interstellar turbulence advanced by Higdon.

Toward a theory of interstellar turbulence. 2. Strong Alfvenic turbulence

Kolmogorov Spectra of Turbulence I

Kolmogorov Spectra of Turbulence I: Wave Turbulence

Kolmogorov spectra of turbulence I: Wave turbulence

The saga of YIG: Spectra, thermodynamics, interaction and relaxation of magnons in a complex magnet

. . . . . . . . . . . . . . 6 Advanced S-Theory: Supplementary Sections 121 6.1 Ground State Evolution of System . . . . . . . . . . . . . . . . . with Increasing Pumping Amplitude 123 6.1.1 Ground State of Parametric Waves . . . . . . for Complex Pair Interaction Amplitudes 124 . . . . . . 6.1.2 The Second and Intermediate Thresholds 125 6.1.3 Nonlinear Behavior . . . of Non-Analytic Pair Interaction Amplitudes 128 6.2 Influence of Nonlinear Damping . . . . . . . . . . . . . . . . . . . . . . . . . . on Parametric Excitation 132 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Simple Theory 132 6.2.2 Influence of Non-Analyticity . . . . . . . . . . . . . . . . . . . . . . on Nonlinear Damping 135 6.3 Parametric Excitation Under the Feedback Effect on Pumping . . . . . . . . . . . . . . 139 6.3.1 Harniltonian of the Problem . . . . . . . . . . . . . . . . . . 139 . . . 6.3.2 General Analysis of the Equations of Motion 141 6.3.3 First-Order Processes . . . . . . . . . . . . . . . . . . . . . . . . 143 6.3.4 SecondOrder Processes . . . . . . . . . . . . . . . . . . . . . . 146 6.4 Nonlinear Theory of Parametric wave Excitation at Finite Temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 6.4.1 Different Time Correlators and Frequency Spectrum . . . . . . . . . . . . . . . . . . . . 147 . . . . . 6.4.2 Basic Equations of Temperature S-Theory 146 6.4.3 Separation of Waves into Parametric and Thermal 150 6.4.4 Two-Dimensional Reduction of Basic Equations . 151 6.4.5 Distribution of Parametric Waves in k . . . . . . . . . 152 6.4.6 Spectrum of Pararnctr-ic Waves . . . . . . . . . . . . . . . 153 6.4.7 Heating Below Threshold . . . . . . . . . . . . . . . . . . . . 153 6.4.8 Influence of Thermal Bath on Total Characteristics 153 6.5 Introduction to Spatially Inhomogeneous S-Theory . . . . . 155 6.5.1 Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 6.5.2 Parametric Threshold in Inhomogeneous Media . 157 . . . 6.5.3 Stationary State in Non-Homogeneous Media 160 6.6 Nonlinear Behavior of Parametric Waves from Various Branches. Asymmetrical S-Theory . . . . . . . . . . . 165 6.6.1 Derivation of Basic Equations . . . . . . . . . . . . . . . . 165 6.6.2 Stationary States in Isotropic Case . . . . . . . . . . . . 167 6.7 Parametric Excitation of Waves by Noise Pumping . . . . 172 6.7.1 Equations of S-Theory Under Noise Pumping . . 173 6.7.2 Distribution of Parametric Waves ! 6 Advanced S-Theory: i Supplementary Sections / I

Victor S. L'vov

Papers

Kolmogorov Spectra of Turbulence I

Kolmogorov Spectra of Turbulence I: Wave Turbulence

Kolmogorov spectra of turbulence I: Wave turbulence

The saga of YIG: Spectra, thermodynamics, interaction and relaxation of magnons in a complex magnet

Wave Turbulence Under Parametric Excitation