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.
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.
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 .
We now begin our exposition of universal dynamic scaling behavior that emerges under non-equilibrium conditions. We are mainly concerned with the relaxational models A and B here, but also touch on other dynamic universality classes. First, we consider non-equilibrium critical relaxation from a disordered initial state, and compute the associated universal scaling exponents within the dynamic renormalization group framework. Related phenomena in the early-time regime, prior to reaching temporally translationally invariant asymptotics, are ‘aging’ and a non-equilibrium fluctuation-dissipation ratio, both of which have enjoyed prominence in the literature on ‘glassy’ kinetics. In this context, we derive the critical initial slip exponent for the order parameter, and briefly discuss interesting persistence properties. Second, we explore the long-time scaling laws in phase ordering kinetics and coarsening for the relaxational models, following a fast temperature ‘quench’ into the ordered phase, in systems with either non-conserved or conserved order parameters. Aside from a few explicit computations in the spherical model limit (wherein the number of order parameter components n → ∞), we largely employ phenomenological considerations and scaling theory, stressing the importance of topological defects for energy dissipation during coarsening. Next we address the question how violations of Einstein's relation that links relaxation coefficients and noise correlations might affect the asymptotic dynamic critical scaling behavior. Whereas in systems with non-conserved order parameter typically the equilibrium scaling laws are recovered in the vicinity of the critical point, in contrast genuinely novel universality classes may emerge in the case of a conserved parameter, provided it is driven out of equilibrium in a spatially anisotropic fashion.
We investigate the critical dynamics of the n-component relaxational models C and D, which incorporate the coupling of a nonconserved and conserved order parameter S, respectively, to the conserved energy density rho, under nonequilibrium conditions by means of the dynamical renormalization group. Detailed balance violations can be implemented isotropically by allowing for different effective temperatures for the heat baths coupling to the slow modes. In the case of model D with conserved order parameter, the energy density fluctuations can be integrated out, leaving no trace of the nonequilibrium perturbations in the asymptotic regime. For model C with scalar order parameter, in equilibrium governed by strong dynamic scaling (z(S)=z(rho)), we find no genuine nonequilibrium fixed point either. The nonequilibrium critical dynamics of model C with n=1 thus follows the behavior of other systems with nonconserved order parameter wherein detailed balance becomes effectively restored at the phase transition. For n> or =4, the energy density generally decouples from the order parameter. However, for n=2 and n=3, in the weak dynamic scaling regime (z(S)< or =z(rho)) entire lines of genuine nonequilibrium model C fixed points emerge to one-loop order, which are characterized by continuously varying static and dynamic critical exponents. Similarly, the nonequilibrium model C with spatially anisotropic noise and n<4 allows for continuously varying exponents, yet with strong dynamic scaling. Subjecting model D to anisotropic nonequilibrium perturbations leads to genuinely different critical behavior with softening only in subsectors of momentum space and correspondingly anisotropic scaling exponents. Similar to the two-temperature model B (randomly driven diffusive systems) the effective theory at criticality can be cast into an equilibrium model D dynamics, albeit incorporating long-range interactions of the uniaxial dipolar or ferroelastic type.
The growth of order in Ising models with nonconserved order parameter is considered for quenches to final temperatures T f =0 and T f =T c . The results of numerical simulations in spatial dimension d=2 are presented. In all cases a scaling regime is entered for sufficiently long times, where the characteristic length scale is the 'domain size' L(t) approximately t 1 2/, for T f =0, and the 'nonequilibrium correlation length', xi (t) approximately t 1 z/, for T f =T c . The equal-time correlation function has the expected scaling forms f(r/L(t)) and r -(d-2+ eta ) f c (r/ xi (t)) for T f =0 and T c respectively. The scaling function f c (x) has interesting short-distance behaviour which is elucidated using scaling arguments and by in - and 1/n-expansions. The T=0 scaling function f(x) depends on whether the spin correlations present in the initial conditions are of long or short range, as does the exponent which describes the decay of the autocorrelation function, A(t)=((S i (t))S i (O)) approximately L(t) - . Results for a quench from the equilibrium critical state to T f =0 are consistent with theoretical predictions.
Phase transitions in non-equilibrium steady states of O(n)-symmetric models with reversible mode couplings are studied using dynamic field theory and the renormalization group. The systems are driven out of equilibrium by dynamical anisotropy in the noise for the conserved quantities, i.e., by constraining their diffusive dynamics to be at different temperatures T^\par and T^\perp in d_\par- and d_\perp-dimensional subspaces, respectively. In the case of the Sasv'ari-Schwabl-Sz'epfalusy (SSS) model for planar ferro- and isotropic antiferromagnets, we assume a dynamical anisotropy in the noise for the non-critical conserved quantities that are dynamically coupled to the non-conserved order parameter. We find the equilibrium fixed point (with isotropic noise) to be stable with respect to these non-equilibrium perturbations, and the familiar equilibrium exponents therefore describe the asymptotic static and dynamic critical behavior. Novel critical features are only found in extreme limits, where the ratio of the effective noise temperatures T^\par/T^\perp is either zero or infinite. On the other hand, for model J for isotropic ferromagnets with a conserved order parameter, the dynamical noise anisotropy induces effective long-range elastic forces, which lead to a softening only of the d_\perp-dimensional sector in wavevector space with lower noise temperature T^\perp < T^\par. The ensuing static and dynamic critical behavior is described by power laws of a hitherto unidentified universality class, which, however, is not accessible by perturbational means for d_\par \geq 1. We obtain formal expressions for the novel critical exponents in a double expansion about the static and dynamic upper critical dimensions and d_\par, i.e., about the equilibrium theory.