Kinetics of phase transitions with singular multiplicative noise

The behavior of the most probable values of the order parameter x and the amplitude p of conjugate force fluctuations is studied for a stochastic system with a noise amplitude depending on x as |x|^a. It is shown that the phase half-plane x>0 for the canonical pair x, p is divided into isolated regions of large, intermediate, and small values of x. In the first region, the trajectories converge to values of x, p → ∞ as the time t → ∞, and the probability of their realization is negligibly small. In the intermediate region, the configuration point tends to the attraction center corresponding to a stationary ordered state. In the region , the trajectories converge to the point x=p=0 for 0<a<1/2 and to x=0, p → ∞ for 1/2<a≤1. In the former case, the probability of realization of trajectories is finite, while, in the latter case, it is negligibly small, and an absorbing state can be formed.     

Nonequilibrium transitions driven by external dichotomous noise

The stationary probability densityP _s for a class of nonlinear one-dimensional models driven by a dichotomous Markovian process (DMP)I _t , can be calculated explicitly. For the specific case of the Stratonovich model, x=ax –-x ³ +I _t x, the qualitative shape ofP _s and its support is discussed in the whole parameter region. The location of the maxima ofP _s shows a behavior similar to order parameters in continuous phase transitions. The possibility of a noiseinduced change from continuous to a discontinuous transition in an extended model, in which the DMP couples also to the cubic term, is discussed. The time-dependent moments x_t ⁿ can be represented as an infinite series of terms, which are determined by a recursion formula. For negative even moments the series terminates and the long-time behavior can be obtained analytically. As a function of the physical parameters, qualitative changes of this behavior may occur which can be partially related to the behavior ofP _s . All results reproduce those for Gaussian white noise in the corresponding limit. The influence of the finite correlation time and the discreteness of the space of states of the DMP are discussed. An extensive list of references is contained in U. Behn, K. Schiele, and A. Teubel,Wiss. Z. Karl-Marx-Univ. Leipzig, Mathem.-Naturwiss. R. 34:602 (1985).Contribution to the symposium on the Statistical Mechanics of Phase Transitions —Mathematical and Physical Aspects, Tebo, Czechoslovakia, September 1–6, 1986.     

Kink propagation induced by multiplicative noise

The influence of spatio-temporal external multiplicative fluctuations on a single kink in a bistable distributed system is studied. For this purpose we derive a stochastic dynamic equation for the position of the shifted kink. An analytical estimate for spatio-temporally uncorrelated fluctuations is represented and discussed. We draw the conclusion that multiplicative noise induces a propagation of the most probable kink into the region of larger noise. This effect is demonstrated in numerical simulations.     

Nonequilibrium phase transitions and chemical instabilities

The effect of inhomogeneous fluctuations on instabilities in various nonlinear chemical models is studied in terms of concepts developed in the theory of equilibrium phase transitions.Chercheur qualifié au Fonds National de la Recherche Scientifique.     

Nonequilibrium phase transitions in chemical reactions

The far from equilibrium steady states of a simple nonlinear chemical system are analyzed. A standard macroscopic analysis shows that the nonlinearity introduces an instability which causes a transition analogous to a thermodynamic second-order phase transition. Fluctuations are introduced into this model through a stochastic master equation approach. The solution of this master equation in the steady state reveals that the system goes into a more ordered state above the transition point. An analogy is drawn with the nonequilibrium phase transition occurring in the laser at threshold.Supported by a New Zealand U.G.C. Postgraduate Scholarship.     

Nonequilibrium phase transitions into absorbing states

Systems with absorbing (trapped) states may exhibit a nonequilibrium phase transition from a noise-free inactive phase into an ever-lasting active phase. We briefly review the absorbing critical phenomena and universality classes, and discuss over the controversial issues on the pair contact process with diffusion (PCPD). Two different approaches are proposed to clarify its universality issue, which unveil strong evidences that the PCPD belongs to a new universality class other than the directed percolation class.     

Nonequilibrium noise-induced phase transitions in simple systems

The theory and the results of an investigation of nonequilibrium noise-induced phase transitions in the simple example of a physical pendulum with a randomly oscillating pivot are presented. It is shown that such transitions lead to the appearance of a more ordered state of the system. The possibility of a difference between noise-induced oscillations and chaotic oscillations of dynamic origin is discussed. Zh. éksp. Teor. Fiz. 111, 358–378 (January 1997)     

Nonequilibrium phase transitions associated with DNA replication

Thermodynamics governing the synthesis of DNA and RNA strands under a template is considered analytically and applied to the population dynamics of competing replicators. We find a nonequilibrium phase transition for high values of polymerase fidelity in a single replicator, where the two phases correspond to stationary states with higher elongation velocity and lower error rate than the other. At the critical point, the susceptibility linking velocity to thermodynamic force diverges. The overall behavior closely resembles the liquid-vapor phase transition in equilibrium. For a population of self-replicating macromolecules, Eigen's error catastrophe transition precedes this thermodynamic phase transition during starvation. For a given thermodynamic force, the fitness of replicators increases with increasing polymerase fidelity above a threshold.     

Nonequilibrium phase transitions in coordinated biological motion: critical fluctuations

We report the results of experiments on biological motion demonstrating the presence of critical order parameter fluctuations as the system evolves from one coordinated state to another at a critical control parameter value. This is a key feature of nonequilibrium phase transitions.     

Nonequilibrium quantum phase transitions in the Dicke model

We establish a set of nonequilibrium quantum phase transitions in the Dicke model by considering a monochromatic nonadiabatic modulation of the atom-field coupling. For weak driving the system exhibits a set of sidebands which allow the circumvention of the no-go theorem which otherwise forbids the occurrence of superradiant phase transitions. At strong driving we show that the system exhibits a rich multistable structure and exhibits both first- and second-order nonequilibrium quantum phase transitions.     

Integration of Langevin equations with multiplicative noise and the viability of field theories for absorbing phase transitions

Efficient and accurate integration of stochastic (partial) differential equations with multiplicative noise can be obtained through a split-step scheme, which separates the integration of the deterministic part from that of the stochastic part, the latter being performed by sampling exactly the solution of the associated Fokker-Planck equation. We demonstrate the computational power of this method by applying it to the most absorbing phase transitions for which Langevin equations have been proposed. This provides precise estimates of the associated scaling exponents, clarifying the classification of these nonequilibrium problems, and confirms or refutes some existing theories.     

Nonequilibrium first-order phase transition in semiconductor system driven by colored noise

We examine the mechanisms of action of colored multiplicative noise in a positionally disordered semiconductor with Moss–Burstein shift. It is shown that the action of multiplicative noise causes nonequilibrium first-order phase transition of the disorder–order-type in electron subsystem of semiconductor. There are relatively little changes in a condensed matter at such phase transitions, but the electron subsystem undergoes a strong reorganization. The steady photocarriers concentration distribution is studied within a unified colored-noise approximation.     

Nonequilibrium phase transitions in a simple three-state lattice gas

We investigate the dynamics of a three-state stochastic lattice gas consisting of holes and two oppositely charged species of particles, under the influence of an electric field at zero total charge. Interacting only through an excluded-volume constraint, particles exchange with holes and, on a slower time scale, with each other. Using a combination of Monte Carlo simulations and meanfield equations of motion, we study a set of suitably defined order parameters, their histograms and fluctuations, as well as the current through the system. With increasing particle density and drive, the system first orders into a charge-segregated state, and then disorders again near complete filling. The transition is first order at low densities and turns second order at higher ones. The finite-size and aspect-ratio dependence of characteristic quantities is discussed at the mean-field level.     

Nonequilibrium phase transitions in pattern recognition and associative memory

It is shown that a recently devised abstract model of parallel computing for pattern recognition and associative memory can be interpreted as describing nonequilibrium phase transitions in analogy to those occurring in fluids and lasers.     

Nano-pattern stabilization by multiplicative noise

Within the reaction-diffusion framework, a one-component system was confined by means of a multiplicative noise into the attraction basin of a patterning attractor in its (infinite-dimensional) configuration space. In this way, inhomogeneities that otherwise would have been “expelled” from this basin, and subsequently eliminated through the diffusive process, were stabilized. For the present study, a model describing the physics of adsorbed particles on a metallic surface has been used. In particular, an underlying deterministic inhomogeneity-building mechanism was exploited, that acts driven by lateral interactions among the adsorbed particles. This process cannot by itself sustain and stabilize the inhomogeneities, but together with the contribution of a particular form of multiplicative noise, it is able to confine the system into the region of configuration space where this mechanism is enabled, hence stabilizing the pattern. Although the proposal could be applied to more general situations, for the particular model studied here we have found that nanopatterns that without the indicated noise source would be eliminated by diffusion, under its effect can grow and be stabilized.     

Nonequilibrium phase transitions of vortex matter in three-dimensional layered superconductors

Large-scale simulations on the three-dimensional (3D) frustrated anisotropic XY model have been performed to study the nonequilibrium phase transitions of vortex matter in weak random pinning potential in layered superconductors. The first-order phase transition from the moving Bragg glass to the moving smectic is clarified, based on thermodynamic quantities. A washboard noise is observed in the moving Bragg glass in 3D simulations for the first time. It is found that the activation of the vortex loops plays the dominant role in the dynamical melting at high drive.     

Nonequilibrium phase transitions in stochastic lattice systems: Influence of the hopping rates

Two-dimensional lattice-gas models with attractive interactions and particle-conserving hopping dynamics under the influence of a very large external electric field E along a principal axis are studied in the case of different ratios between the jump rates in the field direction and perpendicular to it using different transition probabilities. We investigate the dependence of the non-equilibrium steady-state properties on the transition mechanism. We find self-similarity with respect to (T, ) and a coexistence curve critical exponent which, for small, seems independent of. There is some evidence that this exponent might be halfway between the equilibrium mean field and Onsager's values. A crossover toward mean-field behavior for large seems also identified.Partially supported by the US-Spanish Cooperative Research Program, Grant CCB-8402025.     

Nonequilibrium phase transitions and bifurcation of limit cycles and multiperiodic flows

In many nonequilibrium physical systems such as lasers, fluids, chemical reactions, semiconductors etc. new structures or patterns occur when external parameters are changed. Furthermore with further changing parameters a whole hierarchy of new pattern formations may occur where the first step starting from a homogeneous state is well understood. The present paper investigates what happens at the next steps where a system performs complicated motions in the form of a limit cycle or, still more generally, in the form of a multiperiodic flow. In this latter case the time dependence is composed of periodic motions of different frequencies. We study the case in which by change of an external parameter the stable motion (trajectory) becomes unstable and is replaced by a set of new trajectories (bifurcation). It is assumed that the bifurcating trajectories remain in the neighbourhood of the former stable trajectory. Our procedure allows for a number of applications to lasers, fluid dynamics, chemical waves etc. as will be shown in subsequent papers.     

External non-white noise and nonequilibrium phase transitions

Langevin equations with external non-white noise are considered. A Fokker Planck equation valid in general in first order of the correlation time of the noise is derived. In some cases its validity can be extended to any value of. The effect of a finite in the nonequilibrium phase transitions induced by the noise is analyzed, by means of such Fokker Planck equation, in general, for the Verhulst equation under two different kind of fluctuations, and for a genetic model. It is shown that new transitions can appear and that the threshold value of the parameter can be changed.