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Vieillissement et comportement d’échelle dynamique hors équilibre

Florian Baumann

Published Nov 7, 2007• 1 citation

Abstract

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.

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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.

Recent studies on the phenomenology of ageing in certain many-particle systems which are at a critical point of their non-equilibrium steady states are reviewed. Examples include the contact process, the parity-conserving branching-annihilating random walk, two exactly solvable particle reaction models and kinetic growth models. While the generic scaling descriptions known from magnetic systems can be taken over, some of the scaling relations between the ageing exponents are no longer valid. In particular, there is no obvious generalization of the universal limit fluctuation–dissipation ratio. The form of the scaling function of the two-time response function is compared with the prediction of the theory of local scale-invariance.

A generalised form of time-translation-invariance permits to re-derive the known generic phenomenology of ageing, which arises in classical many-body systems after a quench from an initially disordered system to a temperature T ≤ T c , at or below the critical temperature T c . Generalised time-translation-invariance is obtained, out of equilibrium, from a change of representation of the Lie algebra generators of the dynamical symmetries of scale-invariance and time-translation-invariance. Observable consequences include the algebraic form of the scaling functions for large arguments of the two-time auto-correlators and auto-responses, the equality of the auto-correlation and the auto-response exponents λ C = λ R , the cross-over scaling form for an initially magnetised critical system and the explanation of a novel finite-size scaling if the auto-correlator or auto-response converge for large arguments y = t / s ≫ 1 to a plateau. For global two-time correlators, the time-dependence involving the initial critical slip exponent Θ is confirmed and is generalised to all temperatures below criticality and to the global two-time response function, and their finite-size scaling is derived as well. This also includes the time-dependence of the squared global order-parameter. The celebrate Janssen-Schaub-Schmittmann scaling relation with the auto-correlation exponent is thereby extended to all temperatures below the critical temperature. A simple criterion on the relevance of non-linear terms in the stochastic equation of motion is derived, taking the dimensionality of couplings into account. Its applicability in a wide class of models is confirmed, for temperatures T ≤ T c . Relevance to experiments is also discussed.

The explicit calculation of the scaling form of the two-time autocorrelation function in phase-ordering kinetics and in those cases of non-equilibrium critical dynamics where the dynamical exponent z = 2 through the extension of dynamical scaling to local scale invariance is reviewed. Conceptually, this mainly requires an extension from the usually considered d -dimensional ageing or Schrödinger algebras to a new kind of representation of the conformal algebra in d + 2 dimensions. Explicit tests in several exactly solved models of simple magnets and through simulations in the 2D Ising and q -states Potts models ( q = 2, 3, 8) quenched to T < T c are presented and the extension to systems with non-equilibrium steady states is discussed through two exactly solvable models as well. In the context of surface growth models, possible generalizations for a dynamical exponent z = 4 and beyond are discussed.

We investigate some aspects of the ageing behaviour observed in the contact process after a quench from its active phase to the critical point. In particular we discuss the scaling properties of the two-time response function and we calculate it and its universal ratio to the two-time correlation function up to first order in the field-theoretical -expansion ( = 4− d ). The scaling form of the response function does not fit the prediction of the theory of local scale invariance. Our findings are in good qualitative agreement with recent numerical results.