Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- Part I Near-equilibrium critical dynamics
- Part II Scale invariance in non-equilibrium systems
- 8 Non-equilibrium critical dynamics
- 9 Reaction-diffusion systems
- 10 Active to absorbing state transitions
- 11 Driven diffusive systems and growing interfaces
- Index
- References
11 - Driven diffusive systems and growing interfaces
from Part II - Scale invariance in non-equilibrium systems
Published online by Cambridge University Press: 05 June 2014
Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- Part I Near-equilibrium critical dynamics
- Part II Scale invariance in non-equilibrium systems
- 8 Non-equilibrium critical dynamics
- 9 Reaction-diffusion systems
- 10 Active to absorbing state transitions
- 11 Driven diffusive systems and growing interfaces
- Index
- References
Summary
The emergence of generic scale invariance, i.e., algebraic behavior without tuning to special critical points, appears to be remarkably common in systems that are settled in a non-equilibrium steady state. Prototypical examples are simple non-linear Langevin equations that describe driven diffusive systems and driven interfaces or growth models far from thermal equilibrium, whose distinct phases are characterized by non-trivial RG fixed points and hence universal scaling exponents. We start with driven lattice gases with particle exclusion that are described by generalizations of the one-dimensional noisy Burgers equation for fluid hydrodynamics. Symmetries and conservation laws completely determine the ensuing stationary power laws, as well as the intermediate aging scaling regime and even the large-deviation function for the particle current fluctuations. Next we address the non-equilibrium critical point for driven Ising lattice gases, whose critical exponents can again be computed exactly. We then turn our attention to the prominent Kardar–Parisi–Zhang equation, originally formulated to describe growing crystalline surfaces and the dynamics of driven interfaces, but also closely related to the noisy Burgers equation and even to the equilibrium statistical mechanics of directed lines in disordered environments. After introducing the scaling theory for interface fluctuations, we proceed to a renormalization group analysis at fixed dimension d. For d > 2, a non-trivial unstable RG fixed point separates a phase with Gaussian or Edwards–Wilkinson scaling exponents from a strong-coupling rough phase that is inaccessible by perturbative methods.
Information
- Type
- Chapter
- Information
- Critical DynamicsA Field Theory Approach to Equilibrium and Non-Equilibrium Scaling Behavior, pp. 443 - 498Publisher: Cambridge University PressPrint publication year: 2014
References
and , 1995, Fractal Concepts in Surface Growth, Cambridge: Cambridge University Press.CrossRefGoogle Scholar
, , , , and , 2005, Current fluctuations in stochastic lattice gases, Phys. Rev. Lett. 94, 030601-1–1.CrossRefGoogle ScholarPubMed
, , , and , 2010, Nonperturbative renormalization group for the Kardar–Parisi–Zhang equation, Phys. Rev. Lett. 104, 150601-1–4.CrossRefGoogle Scholar
, , , and , 2011, Nonperturbative renormalization group for the Kardar–Parisi–Zhang equation: general framework and first applications, Phys. Rev. E 84, 061128-1–18.CrossRefGoogle Scholar
, , , and , 2004a, Finite-size scaling in the driven lattice gas, J. Stat. Phys. 115, 281–322.CrossRefGoogle Scholar
, , , and , 2004b, Comment on “Dynamic behavior of anisotropic nonequilibrium driving lattice gases”, Phys. Rev. Lett. 92, 029601–1.CrossRefGoogle Scholar
, , , and , 1986, Finite-size scaling and correlation lengths for disordered systems, Phys. Rev. Lett. 57, 2999–3002.CrossRefGoogle ScholarPubMed
, 2011, Monte Carlo Analysis of Non-equilibrium Steady States and Relaxation Kinetics in Driven Lattice Gases, Ph.D. dissertation, VirginiaPolytechnic Institute and State University.Google Scholar
and , 2011, Slow relaxation and aging kinetics for the driven lattice gas, Phys. Rev. E 83, 051107-1–11.CrossRefGoogle ScholarPubMed
and , 2012, Nonequilibrium relaxation and critical aging for driven Ising lattice gases, Phys. Rev. Lett. 108, 110602-1–5.CrossRefGoogle ScholarPubMed
, 1998, An exactly soluble non-equilibrium system: the asymmetric simple exclusion process, Phys. Rep. 301, 65–83.CrossRefGoogle Scholar
and , 1999, Universal large deviation function of the Kardar–Parisi–Zhang equation in one dimension, J. Stat. Phys. 94, 1–30.CrossRefGoogle Scholar
and , 1998, Exact large deviation function in the asymmetric exclusion process, Phys. Rev. Lett. 80, 209–213.CrossRefGoogle Scholar
and , 1992, Exact dynamical exponent at the Kardar–Parisi–Zhang roughening transition, Phys. Rev. Lett. 69, 1979–1981.CrossRefGoogle ScholarPubMed
and , 1982, The surface statistics of a granular aggregate, Proc. R. Soc. London A 381, 17–31.CrossRefGoogle Scholar
and , 1991, Directed paths in a random potential, Phys. Rev. B 43, 10 728-10 742.CrossRefGoogle Scholar
, , and , 1995, Continuum limit, Galilean invariance, and solitons in the quantum equivalent of the noisy Burgers equation, Phys. Rev. Lett. 75, 1883–1886.CrossRefGoogle ScholarPubMed
, , and , 1977, Large-distance and long-time properties of a randomly stirred fluid, Phys. Rev. A 16, 732–749.CrossRefGoogle Scholar
and , 1994, Two-loop renormalization-group analysis of the Burgers–Kardar–Parisi–Zhang equation, Phys. Rev. E 50, 1024–1045.CrossRefGoogle ScholarPubMed
, , and , 1996, Mode-coupling and renormalization group results for the noisy Burgers equation, Phys. Rev. E 53, 4424–4438.CrossRefGoogle ScholarPubMed
and , 1995, Kinetic roughening phenomena, stochastic growth, directed polymers and all that, Phys. Rep. 254, 215–414.CrossRefGoogle Scholar
and , 2010, Nonequilibrium Phase Transitions,Vol.2: Ageing and Dynamical Scaling Far from Equilibrium, Dordrecht: Springer.CrossRefGoogle Scholar
and , 1994, Anomalous fluctuations of directed polymers in random media, Phys. Rev. B 49, 3136–3154.CrossRefGoogle ScholarPubMed
, 1997, On critical exponents and the renormalization of the coupling constant in growth models with surface diffusion, Phys. Rev. Lett. 78, 1082–1085.
and , 1986a, Field theory of long time behaviour in driven diffusive systems, Z. Phys. B Cond. Matt. 63, 517–520.CrossRefGoogle Scholar
and , 1986b, Field theory of critical behaviour in driven diffusive systems, Z. Phys. B Cond. Matt. 64, 503–514.
, , and , 1999, Exact results for the Kardar–Parisi–Zhang equation with spatially correlated noise, Eur. Phys. J. B 9, 491–511.CrossRefGoogle Scholar
, , and , 1986, Dynamic scaling of growing interfaces, Phys. Rev. Lett. 56, 889–892.CrossRefGoogle ScholarPubMed
and , 1987, Scaling of directed polymers in random media, Phys. Rev. Lett. 58, 2087–2090.CrossRefGoogle ScholarPubMed
, , and , 1983, Phase transitions in stationary nonequilibrium states of model lattice systems, Phys. Rev. B 28, 1655–1658.CrossRefGoogle Scholar
, , and , 1984, Nonequilibrium steady states of stochastic lattice gas models of fast ionic conductors, J. Stat. Phys. 34, 497–537.CrossRefGoogle Scholar
, , and , 2012, Nonperturbative renormalization group for the Kardar–Parisi–Zhang equation: scaling functions and amplitude ratios in 1 + 1, 2 + 1, and 3 + 1 dimensions, Phys. Rev. E 86, 051124-1–19.CrossRefGoogle Scholar
, 1997, Short-time scaling behavior of growing interfaces, Phys. Rev. E 55, 668–679; err. Phys. Rev. E 56, 1285.CrossRefGoogle Scholar
, 1997, Origins of scale invariance in growth processes, Adv. Phys. 46, 139–282.CrossRefGoogle Scholar
and , 1992, Kinetic roughening of growing surfaces, in: Solids Far from Equilibrium, ed. , Cambridge: Cambridge University Press, 479–582.Google Scholar
, 1995, On the renormalization of the Kardar–Parisi–Zhang equation, Nucl. Phys. B 448, 559–574.CrossRefGoogle Scholar
, 1998, On growth, disorder, and field theory, J. Phys. Cond. Matt. 10, 9905–9950.CrossRefGoogle Scholar
, , and , 2007, Current distribution in systems with anomalous diffusion: renormalization group approach, J. Phys. A: Math. Theor. 40, 1447–1465.CrossRefGoogle Scholar
, 1991, Finite-size scaling of driven diffusive systems: theory and Monte Carlo studies, Phys. Rev. Lett. 66, 453–456.CrossRefGoogle ScholarPubMed
and , 1986, Field theory of critical behavior in a driven diffusive system, J. Stat. Phys. 44, 567–588.CrossRefGoogle Scholar
and , 1999, Nonequilibrium Phase Transitions in Lattice Models, Cambridge: Cambridge University Press.CrossRefGoogle Scholar
, , , , , and , 2001, Kinetic roughening in slow combustion of paper, Phys. Rev. E 64, 036101-1–12.CrossRefGoogle ScholarPubMed
and , 2000, Vortex-glass phases in type-II superconductors, Adv. Phys. 49, 607–704.CrossRefGoogle Scholar
, , , and , 1999, Validity of the Kardar–Parisi–Zhang equation in the asymptotic limit of metal electrodeposition, Phys. Rev. B 59, 4638–4641.CrossRefGoogle Scholar
and , 1995, Statistical mechanics of driven diffusive systems, in: Phase Transitions and Critical Phenomena, Vol. 17, eds. and , London: Academic Press.Google Scholar
, 2001, Exactly solvable models for many-body systems far from equilibrium, in: Phase Transitions and Critical Phenomena, Vol. 19, eds. and , London: Academic Press.Google Scholar
, , and , 1989, Dynamics of driven interfaces with a conservation law, Phys. Rev. A 40, 6763–6766.CrossRefGoogle ScholarPubMed
and , 2010, Universal fluctuations of growing interfaces: evidence in turbulent liquid crystals, Phys. Rev. Lett. 104, 230601-1–4.CrossRefGoogle ScholarPubMed
and , 2012, Evidence for geometry-dependent universal fluctuations of the Kardar–Parisi–Zhang interfaces in liquid-crystal turbulence, J. Stat. Phys. 147, 853–890.CrossRefGoogle Scholar
, 1996, Anisotropic finite-size scaling analysis of a two-dimensional driven diffusive system, J. Stat. Phys. 82, 1409–1427.CrossRefGoogle Scholar
, 1998, On the perturbation expansion of the KPZ equation, J. Stat. Phys. 93, 143–154.CrossRefGoogle Scholar
and , 1993, Self-consistent approach to the Kardar-Parisi-Zhang equation, Phys.Rev. E 47, R1455–R1458.CrossRefGoogle ScholarPubMed
, , and , 2011, Non-equilibrium statistical mechanics: from a paradigmatic model to biological transport, Rep. Prog. Phys. 74, 116601-1-41.CrossRefGoogle Scholar
and , 2001a, Upper critical dimension, dynamic exponent, and scaling functions in the mode-coupling theory for the Kardar-Parisi-Zhang equation, Phys. Rev. Lett. 86, 3946–3949.CrossRefGoogle ScholarPubMed
and , 2001b, Stretched exponential relaxation in the mode-coupling theory for the Kardar-Parisi-Zhang equation, Phys. Rev. E 63, 057103-1-1.CrossRefGoogle ScholarPubMed
and , 2001c, Numerical solution of the mode-coupling equations for the Kardar-Parisi-Zhang equation in one dimension, Phys. Rev. E 65, 017105-1-3.CrossRefGoogle ScholarPubMed
and , 1988, Polymers on disordered trees, spin glasses, and traveling waves, J. Stat. Phys. 51, 817–840.CrossRefGoogle Scholar
, 2001, Scaling function for the noisy Burgers equation in the soliton approximation, Europhys. Lett. 56, 492–498.CrossRefGoogle Scholar
, 2005, Localized growth modes, dynamic textures, and upper critical dimension for the Kardar-Parisi-Zhang equation in the weak-noise limit, Phys. Rev. Lett. 94, 195702-1-4.CrossRefGoogle ScholarPubMed
, 2006, Kardar-Parisi-Zhang equation in the weak noise limit: pattern formation and upper critical dimension, Phys.Rev.E 73, 031104-1-26.CrossRefGoogle ScholarPubMed
, , and , 1999, Scaling regimes and critical dimensions in the Kardar-Parisi-Zhang problem, Europhys. Lett. 47, 14–20.CrossRefGoogle Scholar
and , 1994, Kardar-Parisi-Zhang model and anomalous elasticity of two- and three-dimensional smectic-A liquid crystals, Phys. Rev. E 49, 2567–2578.CrossRefGoogle ScholarPubMed
, , , , and , 2012, Short-time growth of a Kardar-Parisi-Zhang interface with flat initial conditions, Phys. Rev. E 86, 041151-1-8.CrossRefGoogle ScholarPubMed
, 2012, (2 + 1)-dimensional directed polymer in a random medium: scaling phenomena and universal distributions, Phys. Rev. Lett. 109, 170602-1-5.CrossRefGoogle Scholar
and , 1991, Exact scaling function of interface growth dynamics, Phys. Rev. A 44, R7873–R7876.CrossRefGoogle ScholarPubMed
and , 2004, Numerical evidence for stretched exponential relaxations in the Kardar-Parisi-Zhang equation, Phys.Rev.E 69, 052603-1-4.CrossRefGoogle ScholarPubMed
and , 2011, Extremely large-scale simulation of a Kardar-Parisi-Zhang model using graphics cards, Phys. Rev. E 84, 061150-1-7.CrossRefGoogle ScholarPubMed
, 1995, Bethe ansatz solution for crossover scaling functions of the asymmetric XXZ chain and the Kardar-Parisi-Zhang-type growth model, Phys. Rev. E 52, 3512–3524.CrossRefGoogle ScholarPubMed
and , 2012, Fluctuations of current in nonstationary diffusive lattice gases, Phys. Rev. E 86, 031106-1-11.CrossRefGoogle ScholarPubMed
and , 1997, Upper critical dimension of the Kardar-Parisi-Zhang equation, Phys. Rev. Lett. 78, 903–906.CrossRefGoogle Scholar
, , , and , 2002, Width distributions and the upper critical dimension of Kardar-Parisi-Zhang interfaces, Phys.Rev.E 65, 026136-1-4.CrossRefGoogle ScholarPubMed
, , , and , 1989, Burgers equation with correlated noise: renormalization-group analysis and applications to directed polymers and interface growth, Phys. Rev. A 39, 3053–3075.CrossRefGoogle ScholarPubMed
, , , , , and , 1995, Glassy solutions of the Kardar-Parisi-Zhang equation, Phys. Rev. Lett. 74, 4257–4260.CrossRefGoogle ScholarPubMed
and , 1992, Kinetic surface roughening. I. The Kardar-Parisi-Zhang equation in the weak-coupling regime, Phys. Rev. A 45, 7156–7161.CrossRefGoogle ScholarPubMed
, , and , 2013, Dimensional fragility of the Kardar-Parisi-Zhang universality class, J. Stat. Mech., P11001, 1–11.Google Scholar
, , and , 2009, Mapping of (2 + 1)-dimensional Kardar-Parisi-Zhang growth onto a driven lattice gas model of dimers, Phys. Rev. E 79, 021125-1-5.CrossRefGoogle Scholar
and , 2004, Exact scaling functions for one-dimensional stationary KPZ growth, J. Stat. Phys. 115, 255–279.CrossRefGoogle Scholar
, 2012, Extremes of N vicious walkers for large N : application to the directed polymer and KPZ interfaces, J. Stat. Phys. 149, 385–410.CrossRefGoogle Scholar
and , 1998, Driven diffusive systems: an introduction and recent developments, Phys. Rep. 301, 45–64.CrossRefGoogle Scholar
and , 1992, Nonlinear deposition: a new approach, Europhys. Lett. 20, 301–306.CrossRefGoogle Scholar
and , 2008, The ideas behind self-consistent expansion, J. Stat. Mech., P04023-1-12.CrossRefGoogle Scholar
, , and , 2001, Critical behaviour of driven bilayer systems: a field-theoretic renormalisation group study, J. Phys. A: Math. Gen. 34, L583–L589.CrossRefGoogle Scholar
and , 2002, Universality classes in the anisotropic Kardar-Parisi-Zhang model, Europhys. Lett. 59, 655–661.CrossRefGoogle Scholar
Accessibility standard: Unknown
Why this information is here
This section outlines the accessibility features of this content - including support for screen readers, full keyboard navigation and high-contrast display options. This may not be relevant for you.Accessibility Information
Accessibility compliance for the PDF of this chapter is currently unknown
and may be updated in the future.