The generalization of dynamical scaling to local scale invariance is reviewed. Starting from a recapitulation of the phenomenology of ageing phenomena, the generalization of dynamical scaling to local scale transformation for any given dynamical exponent $z$ is described and the two distinct types of local scale invariance are presented. The special case $z=2$ and the associated Ward identity of Schrödinger invariance is treated. Local scale invariance predicts the form of the two-point functions. Existing confirmations of these predictions for (I) the Lifshitz points in spin systems with competing interactions such as the ANNNI model and (II) non-equilibrium ageing phenomena as occur in the kinetic Ising model with Glauber dynamics are described.The generalization of dynamical scaling to local scale invariance is reviewed. Starting from a recapitulation of the phenomenology of ageing phenomena, the generalization of dynamical scaling to local scale transformation for any given dynamical exponent $z$ is described and the two distinct types of local scale invariance are presented. The special case $z=2$ and the associated Ward identity of Schrödinger invariance is treated. Local scale invariance predicts the form of the two-point functions. Existing confirmations of these predictions for (I) the Lifshitz points in spin systems with competing interactions such as the ANNNI model and (II) non-equilibrium ageing phenomena as occur in the kinetic Ising model with Glauber dynamics are described.

arXiv:cond-mat/0307649Jul 25, 20030 citations

Vieillissement et comportement d’échelle dynamique hors équilibre

Florian Baumann

Ageing phenomena and dynamical scaling behaviour have been observed in many out-of-equilibrium systems, but a general framework for the description of such systems is still missing. A first step in this direction is the theory of local scale-invariance (LSI), which attempts to identify generalised forms of spatio-temporal dynamical scaling. For systems with a dynamical exponent z = 2, it has already been known how to treat stochastic partial differential equations and the consequences have been verified in many explicit models. In this thesis a reformulation and extension of LSI for systems with z ? 2 is presented. We infer for the first time generalised Bargmann superselection and discuss extended dynamical symmetries of Langevin equations with z ? 2. We can establish a formalism for the calculation of non-equilibrium correlation -and response functions and the results are confirmed in several new model calculations. Secondly, the ageing behaviour in reaction-diffusion systems is investigated. Although the main features of ageing as seen in magnets are still valid, important differences in exponent relations are found. Explicitly, the contact process is studied through field-theoretical methods and two bosonic models are solved exactly. For the latter, we show how to extend LSI with z = 2 to nonlinear models. Thirdly, the ageing behaviour in semi-infinite magnetic systems close to the surface is considered. The results show that the general scaling picture known from infinite systems remains valid, but some ageing exponents and scaling functions differ from the bulk quantities.Ageing phenomena and dynamical scaling behaviour have been observed in many out-of-equilibrium systems, but a general framework for the description of such systems is still missing. A first step in this direction is the theory of local scale-invariance (LSI), which attempts to identify generalised forms of spatio-temporal dynamical scaling. For systems with a dynamical exponent z = 2, it has already been known how to treat stochastic partial differential equations and the consequences have been verified in many explicit models. In this thesis a reformulation and extension of LSI for systems with z ? 2 is presented. We infer for the first time generalised Bargmann superselection and discuss extended dynamical symmetries of Langevin equations with z ? 2. We can establish a formalism for the calculation of non-equilibrium correlation -and response functions and the results are confirmed in several new model calculations. Secondly, the ageing behaviour in reaction-diffusion systems is investigated. Although the main features of ageing as seen in magnets are still valid, important differences in exponent relations are found. Explicitly, the contact process is studied through field-theoretical methods and two bosonic models are solved exactly. For the latter, we show how to extend LSI with z = 2 to nonlinear models. Thirdly, the ageing behaviour in semi-infinite magnetic systems close to the surface is considered. The results show that the general scaling picture known from infinite systems remains valid, but some ageing exponents and scaling functions differ from the bulk quantities.

Nov 7, 20071 citation

Scaling and phase transitions in random systems

Marek Cieplak, Jayanth R. Banavar

Physica A Statistical Mechanics and its ApplicationsMar 1, 19932 citations

Scale Invariance: From Phase Transitions to Turbulence

Annick Lesne, Michel Laguës

Nov 4, 201146 citations

z

is described and the two distinct types of local scale invariance are presented. The special case

z=2

and the associated Ward identity of Schrödinger invariance is treated. Local scale invariance predicts the form of the two-point functions. Existing confirmations of these predictions for (I) the Lifshitz points in spin systems with competing interactions such as the ANNNI model and (II) non-equilibrium ageing phenomena as occur in the kinetic Ising model with Glauber dynamics are described.