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Journal of Physics A Mathematical and General

Nonequilibrium critical relaxation with coupling to a conserved density

Klaus Oerding, H. K. Janssen

Published Jul 21, 1993• 49 citations
Publisher

Abstract

The authors study nonequilibrium critical relaxation properties of model C (purely dissipative relaxation of an order parameter coupled to a conserved density) starting from a macroscopically prepared initial state with short-range correlations. Using a field-theoretic renormalization group approach they show that all the stages of growth of the correlation length display universal behaviour governed by a new critical exponent theta . This exponent is calculated to second order in in =4-d where d is the spatial dimension of the system.

DOI
10.1088/0305-4470/26/14/006

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The authors study nonequilibrium critical relaxation properties of model C (purely dissipative relaxation of an order parameter coupled to a conserved density) starting from a macroscopically prepared initial state with short-range correlations. Using a field-theoretic renormalization group approach they show that all the stages of growth of the correlation length display universal behaviour governed by a new critical exponent theta . This exponent is calculated to second order in in =4-d where d is the spatial dimension of the system.
We study the purely relaxational tricritical dynamics of a non-conserved order parameter (model A) in the upper critical dimension d c =3 and in 3- epsilon dimensions. We are especially interested in the relaxation, starting from a macroscopically prepared initial state with a small correlation length. Using the methods of renormalized field theory we obtain the scaling behaviour of the correlation and response functions and study the nonlinear relaxation of the order parameter M(t). In three dimensions M(t) displays a crossover from the purely logarithmic short-time behaviour M(t) approximately (In(t/t 0 )) -a to a t - 1 4 / power law with logarithmic corrections. For dimensions d<3 we obtain the exponents which govern the tricritical relaxation at lowest non-trivial order in epsilon =3-d. The dynamic scaling exponent z is calculated at second order in epsilon .
The non-equilibrium critical relaxation of systems in which the order parameter couples reversibly to conserved densities is investigated. The initial state which may be macroscopically prepared by a quench from a temperature T >> T c to the critical temperature T c is characterized by short-range correlations. In the case where the order parameter itself is conserved the relaxation shows universal scaling. If, on the other hand, the order parameter is not conserved the scaling is governed by a new exponent which depends on the width of the initial distribution of the conserved fields. However, the response function of the conserved fields still shows universal scaling.
We investigate the critical relaxation of dilute Ising systems starting from a macroscopically prepared initial state with short-range correlations. Using the methods of renormalized field theory we calculate the exponent theta ' which describes the initial increase of the magnetization to second order in square root epsilon , where epsilon =4-d. Since computer simulations of the dilute Ising model have shown that a large part of the critical region is governed by crossover phenomena, we also discuss the influence of the slow crossover on the relaxation.
The influence of nonequilibrium initial values of the order parameter on its evolution at a critical point is described using a renormalization group approach of the field theory. The dynamic critical exponent θ of the short time evolution of a system with an n -component order parameter is calculated within a dynamical dissipative model using the method of Σ-expansion in a three-loop approximation. Numerical values of θ for three-dimensional systems are determined using the Padé-Borel method for the summation of asymptotic series.
We study nonequilibrium critical relaxation properties of systems with quenched extended defects, correlated in ${\ensuremath{\varepsilon}}_{d}$ dimensions and randomly distributed in the remaining $d\ensuremath{-}{\ensuremath{\varepsilon}}_{d}$ dimensions. Using a field-theoretic renormalization-group approach, we find the scaling behavior of the nonequilibrium response and correlation functions and calculate the initial slip exponents $\ensuremath{\theta}$ and ${\ensuremath{\theta}}^{\ensuremath{'}},$ which describe the growth of correlations during the initial stage of the critical relaxation, in the two-loop approximation.
We present a method to calculate short-time nonequilibrium universal exponents within the functional-renormalization-group scheme. As an example, we consider the classical critical dynamics of the relaxational model A after a quench of the temperature of the system and calculate the initial-slip exponent which characterizes the nonequilibrium universal short-time behavior of both the order parameter and correlation functions. The value of this exponent is found to be consistent with the result of a perturbative dimensional expansion and of Monte Carlo simulations in three spatial dimensions.