Effects of Violating Detailed Balance on Critical Dynamics

Abstract

We present an overview of the effects of detailed-balance violating perturbations on the universal static and dynamic scaling behavior near a critical point. It is demonstrated that the standard critical dynamics universality classes are generally quite robust: In systems with non-conserved order parameter, detailed balance is effectively restored at criticality. This also holds for models with conserved order parameter, and isotropic non-equilibrium perturbations. Genuinely novel features are found only for models with conserved order parameter and spatially anisotropic noise correlations.PACS numbers: 05.40.+j, 64.60.Ak, 64.60.Ht.One of the major goals in theoretical non-equilibrium physics has been the identification and classification of universality classes for the long-wavelength, long-time scaling behavior both near continuous dynamical phase transitions, and for systems displaying generic scale invariance. Indeed, through investigations of certain specific models, a number of prototypical non-equilibrium universality classes have been identified. A complementary approach is to study the influence of non-equilibrium perturbations on the known universality classes for equilibrium dynamical critical phenomena [5]. Equilibrium critical dynamics is concerned with the relaxational and reversible kinetics near a thermodynamic critical point at temperature T c , as generically described by the Landau-Ginzburg-Wilson (LGW) model for an n-component order parameter vector field S in d space dimensions [6]. In addition to the two independent static critical exponents, e.g. the correlation length exponent ν defined via ξ ∝ |τ | −ν (τ = T − T c ) and Fisher's exponent η for the algebraic decay of the two-point correlation function at criticality (T = T c ),, the order parameter relaxation is governed by a dynamic exponent z that describes critical slowing down: The characteristic time scale diverges as t ch ∝ |τ | −zν upon approaching the transition. This allows for time scale separation and thus a formulation of critical dynamics in terms of non-linear Langevin equations: The relevant 'slow' modes consist of the order parameter and all conserved quantities to which it is statically or dynamically coupled. All remaining 'fast' degrees of freedom are captured through an effective Gaussian white noise. Different values for z ensue depending on whether the order parameter is a conserved quantity or not, and on the additional conserved quantities present. The diffusive relaxation of the latter near criticality can either be characterized by the same exponent z as for the order parameter ('strong' dynamic scaling), or be given by different power laws ('weak' scaling) [5].In order to ensure relaxation towards thermal equilibrium at long times, as given by a Gibbs distribution, one has to carefully implement detailed-balance conditions. In the language of non-linear Langevin equations, these are (i) the Einstein relation between the relaxation constants and the noise strengths, and (ii) the condition that the probability cur...

Some recent developments in models with absorbing states

“…The effects of violating detailed-balance on the universal static and dy-namic scaling behavior has been investigated independently by Grinstein et al [15] and Täuber et al [16]. It turns out that the standard critical dynamics universality classes are rather robust and that detailed-balance can be effectively restored at criticality in some cases.…”

Field-Theory Approaches to Nonequilibrium Dynamics

“…A one-loop RG analysis yields a runaway flow, and no stable RG fixed point is found [38]. Similar behaviour ensues in other anisotropic nonequilibrium variants of critical dynamics models with conserved order parameter; the precise interpretation of the apparent instability is as yet unclear [33].…”

“…Next we address the question [33], What happens if the detailed balance conditions (7.17) and (7.93) are violated? To start, we change the noise strength D → D in the purely relaxational models A and B, which (in our units) violates the Einstein relation (7.17).…”

“…Again, since the dynamic critical behaviour is governed by the universal fixed point (7.107), thermal equilibrium becomes effectively restored at criticality. More generally, it has been established that isotropic detailed balance violations do not affect the universal properties in other models for critical dynamics that contain additional conserved variables either: the equilibrium RG fixed points tend to be asymptotically stable [33].…”

“…We may now evaluate the Gaussian integrals over the noise ζ α , which yields 33) with the statistical weight now governed by the Janssen-De Dominicis "response" functional [9,18,19]…”