An introduction to the Ginzburg-Landau theory of phase transitions and nonequilibrium patterns

Abstract This paper presents an introduction to phase transitions and critical phenomena on the one hand, and nonequilibrium patterns on the other, using the Ginzburg–Landau theory as a unified language. In the first part, mean-field theory is presented, for both statics and dynamics, and its validity tested self-consistently. As is well known, the mean-field approximation breaks down below four spatial dimensions, where it can be replaced by a scaling phenomenology. The Ginzburg–Landau formalism can then be used to justify the phenomenological theory using the renormalization group, which elucidates the physical and mathematical mechanism for universality. In the second part of the paper it is shown how near pattern forming linear instabilities of dynamical systems, a formally similar Ginzburg–Landau theory can be derived for nonequilibrium macroscopic phenomena. The real and complex Ginzburg–Landau equations thus obtained yield nontrivial solutions of the original dynamical system, valid near the linear instability. Examples of such solutions are plane waves, defects such as dislocations or spirals, and states of temporal or spatiotemporal (extensive) chaos.

• Publication year

  2014

• Venue

  Physics Reports

• Publication date

  2014-10-27

• Fields of study

  Physics

• Identifiers
  DOI 10.1016/j.physrep.2015.01.001arXiv 1410.7285
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar

• No claims are published for this paper.

• No concepts are published for this paper.

• CONTRIBUTION TO THE THEORY OF LIGHT SCATTERING NEAR THE SECOND-ORDER PHASE-TRANSITION POINTS

  2015cited by this paper
• Pattern Formation and Dynamics in Nonequilibrium Systems

  2004influential reference
• Conceptual foundations of quantum field theory: Renormalization group theory: its basis and formulation in statistical physics

  1998influential reference
• Pattern formation outside of equilibrium

  1993influential reference
• Introduction to the renormalization group and to critical phenomena

  1977cited by this paper
• Lectures On Phase Transitions And The Renormalization Group

  1972cited by this paper
• statistical-physics-part-1

  1958cited by this paper

• Phase-field simulation study on the multifaceted migration of microcavities on silicon under multi-field coupling effects

  2026cites this paper
• Parallel phase-field-LBM framework for Accelerated Sea ice crystallization simulation

  2026cites this paper
• Non-equilibrium processing science of additive friction stir deposition

  2026cites this paper
• On the convergence of PINNs for inverse source problem in the complex Ginzburg–Landau equation

  2025cites this paper
• Optically-Induced Faraday-Goldstone Waves

  2025cites this paper
• Weakly Nonlinear Instability of a Convective Flow in a Plane Vertical Channel

  2025cites this paper
• Ion migration at the ice–water interface during the freezing process of salt solution – Numerical investigation

  2025cites this paper
• Wavelet-based density sketching with functional hierarchical tensor

  2025cites this paper
• Extension of global renormalization group theory to include symmetric and asymmetric fluctuations for critical phase behavior

  2025cites this paper
• Computational modeling of sea ice freezing dynamics across scales

  2025cites this paper
• Theory of Order-Disorder Phase Transitions Induced by Fluctuations Based on Network Models

  2025cites this paper
• Extending the Flory-Huggins Theory for Crystalline Multicomponent Mixtures.

  2025cites this paper
• Autowave Physics of Inhomogeneous Plastic Flow

  2025cites this paper
• Exact fluctuation relation for open systems beyond the Zwanzig FEP equation

  2025cites this paper
• A Review of Indirect Freezing Desalination: Key Parameters, Experimental Research, and Numerical Modeling

  2025cites this paper
• Topological constraints on self-organisation in locally interacting systems

  2025cites this paper
• Critical current density in advanced superconductors

  2025cites this paper
• Induced phase transitions and spontaneous symmetry breaking based on the renormalized Ginzburg-Landau theory

  2025influential citation
• Invariant modules and noninvariant exact solutions for Infeld–Rowlands equation

  2025cites this paper
• Pattern formation in a coupled driven diffusive system.

  2025cites this paper
• Comparison of the Charge Density Wave Transition in Monolayer 1T-TaS2 and 1T-TaSe2 from First Principle

  2025cites this paper
• Dynamic phase transitions in mean-field Ginzburg–Landau models: Conjugate fields and Fourier-mode scaling

  2025cites this paper
• Equivalence of stationary dynamical solutions in a directed chain and a Delay Differential Equation of neuroscientific relevance

  2025cites this paper
• Lüders and Portevin–Le Chatelier Bands at the Stage of Elastoplastic Transition: Nucleation and Propagation

  2025cites this paper
• Physics-Inspired All-Pair Interaction Learning for 3D Dynamics Modeling

  2025cites this paper
• Dynamics and frictional dissipation for treading slowly in a puddle.

  2025cites this paper
• Identification and properties of topological states in the bulk of quasicrystals

  2025cites this paper
• Topological Defect Formation beyond the Kibble-Zurek Mechanism in Crossover Transitions with Approximate Symmetries.

  2025cites this paper
• Defining and finding lifelike entities with a lazy filter

  2025cites this paper
• Characterizing phase transitions and criticality in non-Hermitian extensions of the XY model

  2025cites this paper
• Phase transitions in tensorial group field theory: Landau-Ginzburg analysis of the causally complete Lorentzian Barrett-Crane model

  2024cites this paper
• Terahertz spectroscopy of collective charge density wave dynamics at the atomic scale

  2024cites this paper
• Propagation properties of bright solitons generated by the complex Ginzburg-Landau equation with high-order dispersion and nonlinear gradient terms

  2024cites this paper
• Exploring Superconductivity: The Interplay of Electronic Orders in Topological Quantum Materials

  2024cites this paper
• Stacking-Engineered Ferroelectricity and Multiferroic Order in van der Waals Magnets.

  2024cites this paper
• Asymmetric Symmetry Breaking: Unequal Probabilities of Vacuum Selection

  2024cites this paper
• Solving high-dimensional Kolmogorov backward equations with functional hierarchical tensor operators

  2024cites this paper
• Twisted Vortices in Two‐Component Ginzburg–Landau Theory

  2024influential citation
• Scale invariance beyond criticality within the mean-field analysis of tensorial field theories

  2024cites this paper
• TRAVELING FRONTS OF A REAL SUPERCRITICAL QUINTIC GINZBURG-LANDAU EQUATION COUPLED BY A SLOW DIFFUSION MODE

  2024cites this paper
• Ensemble-Based Annealed Importance Sampling

  2024cites this paper
• Temperature-dependent evolution of micromagnetic structure of B20-type Fe1-xCoxGe studied by Mössbauer spectroscopy

  2024cites this paper
• Global null controllability  of stochastic semilinear complex Ginzburg-Landau equations

  2024cites this paper
• Machine Learning of Slow Collective Variables and Enhanced Sampling via Spatial Techniques

  2024cites this paper
• Spontaneous symmetry breaking in a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>S</mml:mi><mml:mi>O</mml:mi><mml:mo>(</mml:mo><mml:mn>3</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math> non-Abelian lattice gauge theory in <mml:math xmlns:mml="http://www.w3.org/1998/Ma

  2024cites this paper
• Inter-event time power laws in heterogeneous systems

  2024cites this paper
• Effective theory of collective deep learning

  2024cites this paper
• Research on controllable processing technology of microsphere cavity inside silicon substrates utilizing thermoelectric coupling effect

  2024cites this paper
• Localized Plasticity Pattern at the Prefracture Stage: Origin and Development

  2024cites this paper
• Pattern formation from gauge/gravity duality

  2024cites this paper
• Simulation of Stochastic Non-Equilibrium Thermal Effects of Particle Inclusions within Fluid Interfaces and Membranes

  2024cites this paper
• An Overview of Sustainable Desalination with Freezing Crystallization: Current Development, Future Challenges, and Prospects

  2024cites this paper
• Coexistence of ferroelectricity and antiferroelectricity in 2D van der Waals multiferroic

  2024cites this paper
• Atomic wires on substrates: Physics between one and two dimensions

  2024cites this paper
• Localisation of stress-affected chemical reactions in solids described by coupled mechanics-diffusion-reaction models

  2024cites this paper
• Multiphase critical point type phase boundary induced superior magnetocaloric properties in ferromagnetic Tb1-xGdxCo2 alloys

  2024cites this paper
• Theoretical and numerical study of the Landau-Khalatnikov model describing a formation of 2D domain patterns in ferroelectrics

  2024cites this paper
• Interplay of Electronic Orders in Topological Quantum Materials

  2024cites this paper
• Frequency stabilization of adiabatic temperature change in Fe50Rh50 alloy in a cyclic magnetic field of 1.2 T

  2024cites this paper
• Conducting viscous bodies with phase transitions: Deriving Gurtin’s postulate from the second law structure invariance

  2024cites this paper
• Solving high-dimensional Fokker-Planck equation with functional hierarchical tensor

  2023cites this paper
• Novel transition dynamics of topological solitons

  2023cites this paper
• Exact solutions to a family of nonlinear Schrödinger equations

  2023cites this paper
• Exact solutions to a family of complex Ginzburg-Landau equations with cubic-quintic nonlinearity

  2023cites this paper
• Time-reversal and parity-time symmetry breaking in non-Hermitian field theories.

  2023cites this paper
• Quantized vortex dynamics of the complex Ginzburg-Landau equation on the torus

  2023cites this paper
• Multi-Physical Field Simulations of Directional Seawater Freeze-Crystallisation Under a Magnetic Field

  2023cites this paper
• Topologically protected metastable states in classical dynamics

  2023cites this paper
• Symbiotic dynamics in living liquid crystals.

  2023cites this paper
• Two-Dimensional Discommensurations: An Extension to McMillan’s Ginzburg–Landau Theory

  2023cites this paper
• An effective theory of collective deep learning

  2023cites this paper
• Manifold learning in atomistic simulations: a conceptual review

  2023cites this paper
• Investigation and application of microscopic mechanism of multiple bubble movement on directional frozen crystallization of seawater

  2023cites this paper
• Kibble–Zurek scaling of nonequilibrium phase transition in barium titanate

  2023cites this paper
• Landau theory-based relaxational modeling of first-order magnetic transition dynamics in magnetocaloric materials

  2023cites this paper
• Topological or not? A unified pattern description in the one-dimensional anisotropic quantum XY model with a transverse field

  2023cites this paper
• Collective dynamics and the Anderson-Higgs mechanism in a bona fide holographic superconductor

  2023influential citation
• Entanglement entropy as an order parameter for strongly coupled nodal line semimetals

  2023cites this paper
• Pore-scale numerical study: brine water crystallization with ice crystal particle motion using the LBM-PFM-IBM

  2023cites this paper
• Phase Diagram and Specific Heat of a Nonequilibrium Curie–Weiss Model

  2023cites this paper
• Gelation and Orientation Dynamics Induced by Contact of Protein Solution with Transglutaminase Solution

  2023cites this paper
• Relationship between Rate-Limiting Process and Scaling Law in Gel Growth Induced by Liquid-Liquid Contact

  2023cites this paper
• Nonadiabatically Driven Subcritical Crack Nucleation in Solids

  2023cites this paper
• Synergistic chemomechanical dynamics of feedback-controlled microreactors.

  2023cites this paper
• Mapping of multiphase field model parameters to physical factors in order to simulate desired phase transformations

  2023cites this paper
• Entropy production in the nonreciprocal Cahn-Hilliard model.

  2023cites this paper
• Tensorizing flows: a tool for variational inference

  2023cites this paper
• Relative Entropy and Mutual Information in Gaussian Statistical Field Theory

  2023cites this paper
• Physics-based analytical model for trap assisted biosensing in dual cavity negative capacitance junctionless accumulation mode FET

  2023cites this paper
• Phase transition modelling of relapse in major depressive disorder: Developing and reflecting on an interdisciplinary conceptual translation

  2023cites this paper
• Inhibition effect of adsorption on brine pockets formation during seawater freeze desalination

  2022cites this paper
• Dynamical phase transitions in the collisionless pre-thermal states of isolated quantum systems: theory and experiments

  2022cites this paper
• Formation of an Irregular Pattern of Localized Plasticity in the Linear-To-Parabolic Hardening Transition

  2022cites this paper
• Numerical and experimental investigation of phase change characteristics and desalination effect of seawater freezing desalination in narrow channels

  2022cites this paper
• Phase transitions in TGFT: a Landau-Ginzburg analysis of Lorentzian quantum geometric models

  2022cites this paper
• PTFE Crystal Growth in Composites: A Phase-Field Model Simulation Study

  2022cites this paper
• Breather modes of fully nonlinear mass-in-mass chains.

  2022influential citation
• Mechanism of the Resistivity Switching Induced by the Joule Heating in Crystalline NbO2

  2022cites this paper
• Sea water frozen crystalisation impacted by flow and heterogeneous nucleation: PFM-LBM coupled modeling, simulation and experiments

  2022cites this paper
• LBM–PFM simulation of directional frozen crystallisation of seawater in the presence of a single bubble

  2022cites this paper