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arXiv:cond-mat/0512140 · Journal of Physics Conference Series

Ageing without detailed balance: local scale invariance applied to two exactly solvable models

Florian Baumann

Published May 1, 2006• 1 citation
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Abstract

I consider ageing behaviour in two exactly solvable reaction-diffusion systems. Ageing exponents and scaling functions are determined. I discuss in particular a case in which the equality of two critical exponents, known from systems with detailed balance, does not hold any more. Secondly it is shown that the form of the scaling functions can be understood by symmetry considerations.

DOI
10.1088/1742-6596/40/1/010

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I consider ageing behaviour in two exactly solvable reaction-diffusion systems. Ageing exponents and scaling functions are determined. I discuss in particular a case in which the equality of two critical exponents, known from systems with detailed balance, does not hold any more. Secondly it is shown that the form of the scaling functions can be understood by symmetry considerations.
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known from magnetic systems can be taken over, some of the scaling relations between the ageing exponents are no longer valid.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.
Ageing in systems without detailed balance is studied in the exactly solvable bosonic contact process and the critical bosonic pair-contact process. The two-time correlation function and the two-time response function are explicitly found. In the ageing regime, the dynamical scaling of these is analyzed and exact results for the ageing exponents and the scaling functions are derived. For the critical bosonic pair-contact process the autocorrelation and autoresponse exponents agree but the ageing exponents aaa and bbb are shown to be distinct.
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.
The long-time dynamics of the 1D contact process suddenly brought out of an uncorrelated initial state is studied through a light-cone transfer-matrix renormalisation group approach. At criticality, the system undergoes ageing which is characterised through the dynamical scaling of the two-times autocorrelation and autoresponse functions. The observed non-equality of the ageing exponents a and b excludes the possibility of a finite fluctuation-dissipation ratio in the ageing regime. The scaling form of the critical autoresponse function is in agreement with the prediction of local scale-invariance.
Local scale-invariance for ageing systems without detailed balance is tested through studying the dynamical symmetries of the critical bosonic contact process and the critical bosonic pair-contact process.Their field-theoretical actions can be split into a Schrödinger-invariant term and a pure noise term. It is shown that the two-time response and correlation functions are reducible to certain multipoint response functions which depend only on the Schrödinger-invariant part of the action. For the bosonic contact process, the representation of the Schrödinger group can be derived from the free diffusion equation, whereas for the bosonic pair-contact process, a new representation of the Schrödinger group related to a non-linear Schrödinger equation with dimensionful couplings is constructed. The resulting predictions of local scale-invariance for the two-time responses and correlators are completely consistent with the exactly-known results in both models.
The scaling functions of single-time and two-time correlators in systems undergoing non-equilibrium critical dynamics with dynamical exponent z=2{z}=2z=2 are predicted from a new time-dependent non-equilibrium representation of the Schrödinger algebra. These explicit predictions are tested and confirmed in the ageing of several exactly solvable models.