We study nonequilibrium steady states, phase transitions and critical phenomena in a d-dimensional lattice model which represents a magnetic system under the action of a field fluctuating very rapidly with time. This induces competing kinetics which produces a sort of (dynamical) frustration which might occur also in some natural disordered systems. The exact solution for d = 1, partial exact results for d ≥ 2, and a comparison with some related models are reported.
We study nonequilibrium steady states, phase transitions and critical phenomena in a d-dimensional lattice model which represents a magnetic system under the action of a field fluctuating very rapidly with time. This induces competing kinetics which produces a sort of (dynamical) frustration which might occur also in some natural disordered systems. The exact solution for d = 1, partial exact results for d ≥ 2, and a comparison with some related models are reported.
The authors study d-dimensional lattice model systems whose dynamics consist of competing independent processes which may induce nonequilibrium steady states. Interesting phase transitions and critical phenomena ensue which are influenced, even dominated, by a sort of dynamical bond disorder. The latter, which may occur in natural systems, reminds one of the disorder in more familiar bond-diluted and other impure models, for example, percolation-like phenomena arise in some of the model cases. Exact solutions for d=1, some exact results for d<1, and a comparison with more standard, either quenched or annealed, magnetically diluted models are reported.
We study a class of lattice interacting-spin systems evolving stochastically under the simultaneous operation of several spin-flip mechanisms, each acting independently and responding to a different applied magnetic field. This induces an extra randomness which may occur in real systems, e.g., a magnetic system under the action of a field varying with a much shorter period than the mean time between successive transitions. Such a situation--in which one may say in some sense that frustration has a dynamical origin-- may also be viewed as a nonequilibrium version of the random-field Ising model. By following a method of investigating stationary probability distributions in systems with competing kinetics [P. L. Garrido and J. Marro, Phys. Rev. Lett. 62, 1929 (1989)], we solve one-dimensional lattices supporting different field distributions and transition rates for the elementary kinetical processes, thus revealing a rich variety of phase transitions and critical phenomena. Some exact results for lattices of arbitrary dimension, and comparisons with the standard quenched and annealed random-field models, and with a nonequilibrium diluted antiferromagnetic system, are also reported.
We report on exact results for stochastic Ising systems that evolve in time due to a simultaneous action of several independent spin-flip mechanisms. The transition probability describes a competition between different values for the two parameters characterizing the involved (nearest-neighbour) Hamiltonian, namely, the interaction strength and the applied magnetic field. Such a conflicting dynamics aims to represent the sort of diffusing microscopic disorder that may occur in real systems. We determine the steady states for a class of transition probabilities for one-dimensional lattices; they exhibit amazing critical behaviour. Furthermore, it is shown that the steady state for a singular class of transition probabilities may be represented by an effective Hamiltonian that has the Ising structure with complex parameters for any lattice dimension.
We report first-order mean-field results for nonequilibrium random-exchange lattice systems, namely, Ising-like models whose kinetics involve a simultaneous, random competition between ferromagnetic and antiferromagnetic interactions that generally induces nonequilibrium steady states. We consider the competition between symmetric bonds. \ifmmode\pm\else\textpm\fi{}J0, symmetric bonds competing with broken ones, \ifmmode\pm\else\textpm\fi{}J0 and J=0, nonsymmetric bonds, \ensuremath{\Vert}J1\ensuremath{\Vert} and -\ensuremath{\Vert}J2\ensuremath{\Vert}, and some of those systems under an external magnetic field h. The time evolution and steady-state properties, including phase diagrams, are obtained for several lattice coordination numbers, e.g., the case of simple-cubic lattices of dimension d\ensuremath{\le}3. A comparison is made with existing exact results for d=1, h=0, and \ifmmode\pm\else\textpm\fi{}J0.
The ordinary kinetic versions of the Ising model may be modified to exhibit steady nonequilibrium states. This is illustrated in chapter 4 where a conflict between two canonical mechanisms ( diffusion and reaction ) drives the configuration away from equilibrium. A more systematic investigation of this possibility, when the conflict is between different reaction processes only, is described here. We focus on spin systems evolving by a superposition of independent local processes of the kind variously known as spin flips, birth/death or creation/annihilation. The restriction to spin flip dynamics does not prevent the systems in this class from exhibiting a variety of nonequilibrium phase transitions and critical phenomena. Their consideration may therefore help in developing nonequilibrium theory. In addition, they have some practical interest, e.g., conflicting dynamics may occur in disordered materials such as dilute magnetic systems, and some of these situations can be implemented in the laboratory. The present chapter describes some exact, mean-field and MC results that together render an intriguing picture encouraging further study. It is argued in §7.1 that some of the peculiar, emergent, macroscopic behavior of microscopically disordered materials may be related to diffusion of disorder. This provides a physical motivation for the nonequilibrium random-field Ising model (NRFM).
We uncover a finite-time dynamical phase transition in the thermal relaxation of a mean-field magnetic model. The phase transition manifests itself as a cusp singularity in the probability distribution of the magnetization that forms at a critical time. The transition is due to a sudden switch in the dynamics, characterized by a dynamical order parameter. We derive a dynamical Landau theory for the transition that applies to a range of systems with scalar, parity-invariant order parameters. Close to criticalilty, our theory reveals an exact mapping between the dynamical and equilibrium phase transitions of the magnetic model, and implies critical exponents of mean-field type. We argue that interactions between nearby saddle points, neglected at the mean-field level, may lead to critical, spatiotemporal fluctuations of the order parameter, and thus give rise to novel, dynamical critical phenomena.