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 nonconserved order parameter, detailed balance is effectively restored at criticality. This also holds for models with conserved order parameter, and isotropic nonequilibrium perturbations. Genuinely novel features are found only for models with conserved order parameter and spatially anisotropic noise correlations.
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 nonconserved order parameter, detailed balance is effectively restored at criticality. This also holds for models with conserved order parameter, and isotropic nonequilibrium perturbations. Genuinely novel features are found only for models with conserved order parameter and spatially anisotropic noise correlations.
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.
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 problem of which universality classes of dynamic surface critical behavior exist for semi-infinite systems whose dynamic bulk critical behavior and static surface critical behavior are representative of a given dynamic bulk universality class and a given static surface universality class. To this end systems whose bulk dynamics are described by either model B or model A of Halperin, Hohenberg, and Ma are considered, where the dynamics are allowed to be modified near the surface as follows: In the case of model B, surface terms that locally break the conservation law for the order-parameter density \ensuremath{\varphi} are allowed; in the case of model A, \ensuremath{\varphi} is assumed to be locally conserved near the surface. Semi-infinite field-theory models are constructed that are (i) compatible with the requested bulk dynamics and the requirements of causality, detailed balance, and relaxation to thermal equilibrium and (ii) minimal in the sense that no redundant or irrelevant surface terms are included. The boundary conditions to which the surface terms of the action correspond are worked out. For model B it is shown that nonconservative surface terms are relevant. Their strength can be parametrized by a coupling constant c${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{0}$\ensuremath{\ge}0 whose inverse plays the role of an extrapolation length for the auxiliary (Martin-Siggia-Rose) field \ensuremath{\varphi}\ifmmode \tilde{}\else \~{}\fi{}. Thus two distinct semi-infinite extensions of model B exist---one with c${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{0}$>0 called model BA, and one with c${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{0}$=0 called model BB---and each static surface universality class splits up into two dynamic surface universality classes.The static and dynamic surface critical exponents of these dynamic surface universality classes as well as the crossover exponent associated with the nonconservative surface term are shown to be expressible in terms of static bulk and surface exponents. Model BA and its results should apply, e.g., to uniaxial ferromagnets whose rotational spin symmetry is broken in the vicinity of a surface plane. For model A the considered local modification of the dynamics near the surface is irrelevant in the renormalization-group sense and hence does not lead to new dynamic surface universality classes.
In this talk/paper I discuss the concept of universality in phase transitions and the question of whether universality classes are more robust in equilibrium than away from it. In both of these situations, the main ingredients determining universality are symmetries, conservation laws, the dimension of the space and of the order‐parameter and the presence of long‐range interactions or quenched disorder. The existence of detailed‐balance and fluctuation‐dissipation theorems imposes severe constraints on equilibrium systems, allowing to define universality classes in a very robust way; instead, non‐equilibrium allows for more variability. Still, quite robust non‐equilibrium universality classes have been identified in the last decades. Here, I discuss some examples in which (i) non‐equilibrium phase transitions are simply controlled by equilibrium critical points, i.e. non‐equilibrium ingredients turn out to be irrelevant in the renormalization group sense and (ii) non‐equilibrium situations in which equilibrium seems to come out of the blue, generating an adequate effective description of intrinsically non‐equilibrium problems. Afterwards, I shall describe different genuinely non‐equilibrium phase transitions in which introducing small, apparently innocuous changes (namely: presence or absence of an underlying lattice, parity conservation in the overall number of particles, existence of an un‐accessible vacuum state, deterministic versus stochastic microscopic rules, presence or absence of a Fermionic constraint), the critical behavior is altered, making the case for lack of robustness. However, it will be argued that in all these examples, there is an underlying good reason (in terms of general principles) for universality to be altered. The final conclusions are that: (i) robust universality classes exist both in equilibrium and non‐equilibrium; (ii) symmetry and conservation principles are crucial in both, (iii) non‐equilibrium allows for more variability (i.e. it is less constrained).
In this talk/paper I discuss the concept of universality in phase transitions and the question of whether universality classes are more robust in equilibrium than away from it. In both of these situations, the main ingredients determining universality are symmetries, conservation laws, the dimension of the space and of the order‐parameter and the presence of long‐range interactions or quenched disorder. The existence of detailed‐balance and fluctuation‐dissipation theorems imposes severe constraints on equilibrium systems, allowing to define universality classes in a very robust way; instead, non‐equilibrium allows for more variability. Still, quite robust non‐equilibrium universality classes have been identified in the last decades. Here, I discuss some examples in which (i) non‐equilibrium phase transitions are simply controlled by equilibrium critical points, i.e. non‐equilibrium ingredients turn out to be irrelevant in the renormalization group sense and (ii) non‐equilibrium situations in which equilibrium seems to come out of the blue, generating an adequate effective description of intrinsically non‐equilibrium problems. Afterwards, I shall describe different genuinely non‐equilibrium phase transitions in which introducing small, apparently innocuous changes (namely: presence or absence of an underlying lattice, parity conservation in the overall number of particles, existence of an un‐accessible vacuum state, deterministic versus stochastic microscopic rules, presence or absence of a Fermionic constraint), the critical behavior is altered, making the case for lack of robustness. However, it will be argued that in all these examples, there is an underlying good reason (in terms of general principles) for universality to be altered. The final conclusions are that: (i) robust universality classes exist both in equilibrium and non‐equilibrium; (ii) symmetry and conservation principles are crucial in both, (iii) non‐equilibrium allows for more variability (i.e. it is less constrained).