Dmitrii O. Kharchenko

Stochastic effects at ripple formation processes in anisotropic systems with multiplicative noise

We study pattern formation processes in anisotropic system governed by the Kuramoto-Sivashinsky equation with multiplicative noise as a generalization of the Bradley-Harper model for ripple formation induced by ion bombardment. For both linear and nonlinear systems we study noise-induced effects at ripple formation and discuss scaling behavior of the surface growth and roughness characteristics. It was found that the secondary parameters of the ion beam (beam profile and variations of an incidence angle) can crucially change the topology of patterns and the corresponding dynamics.

Phase transitions induced by noise cross-correlations

A general approach for treating the spatially extended stochastic systems with the nonlinear damping and correlations between additive and multiplicative noises is developed. Within the modified cumulant expansion method, we derive an effective Fokker-Planck equation with stationary solutions that describe the character of the ordered state. We find that the fluctuation cross-correlations lead to a symmetry breaking of the distribution function even in the case of zero-dimensional system. In a general case, continuous, discontinuous and reentrant noise induced phase transitions take place. It appears that the cross-correlations play the role of bias field which can induce a chain of phase transitions of different nature. Within the mean field approach, we give an intuitive explanation of the system behavior by an effective potential of the thermodynamic type. This potential is written in the form of an expansion with coefficients defined by the temperature, intensity of spatial coupling, autocorrelation and cross-correlation times and intensities of both additive and multiplicative noises.

Evolution of the system with multiplicative noise

The governed equations for the order parameter, one- and two-time correlators are obtained for systems with white multiplicative noise. We consider the noise whose amplitude depends on stochastic variable as xa where 0 12, when the system is disordered, the correlator behaves in the course of time non-monotonically, whereas the autocorrelator increases monotonically. At a<12 the phase portrait of the system divides into two domains: at small initial values of the order parameter, the system evolves to a disordered state, as above; within the ordered domain it is attracted to the point with finite values of the autocorrelator and order parameter. The long-time asymptotes are defined to show that, within the disordered domain, the autocorrelator decays hyperbolically and the order parameter behaves as a power-law function with fractional exponent −2(1−a). Correspondingly, within the ordered domain, the behaviour of both dependencies is exponential with an index proportional to −tlnt.

Coloured noise influence on system evolution

Abstract:Within the power-law approach for noise amplitude dependence on stochastic variables, we present a picture of noise-induced transitions in systems affected by coloured multiplicative noise. The governed equations for main statistical moments are obtained and investigated in detail. We show that a reentrant noise-induced transition is realized within a window of the control parameter.

Path integral solution of the system with coloured multiplicative noise

The behaviour of most probable values of the order parameter x and the amplitude of the conjugate force fluctuations φ is studied for a stochastic system with a coloured multiplicative noise and absorbing state. The phase diagrams introduced as dependencies of the noise self-correlation time vs. temperature and noise growth velocity are defined. It is shown that phase plane (x,φ) is divided into isolated domains of large, intermediate and small values of x. The behaviour of the system in these domains is studied in terms of the path integral. In the region x⪡1, the trajectories converge to the point x=φ=0 at 0<a<12 and to x=0, φ→∞ at 12<a⩽1. In the former case, the probability of realization of trajectories is finite, while in the latter case it is vanishing small, and an absorbing state can be formed.

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.

SCALING LAWS IN STOCHASTIC SYSTEMS WITH ANOMALOUS DIFFUSION

We consider the stochastic system with an anomalous diffusion. According to the obtained relations between characteristics of diffusion processes the special class of models which exhibit the anomalous behaviour is considered. It was shown that indexes of super- and subdiffusion are related to the Hurst exponent which defines the properties of the phase space inherent to the proposed model of stochastic system.

Nonequilibrium phase transitions in stochastic systems with coloured fluctuations

We study the behaviour of a class of stochastic spatially extended systems exhibiting transition to absorbing configurations, reentrant noise induced phase transitions and phase transitions induced by noise crosscorrelations. We discuss the behaviour of the system in the presence of multiplicative fluctuations: a possibility of escaping from the absorbing state and the nature of disordered phase appearing beyond the second critical point of the reentrant phase transition. Making use of the mean field approach we have shown that noise cross-correlations lead to continuous, discontinuous and reentrant phase transitions.

Self-Organized Criticality within the Framework of the Variational Principle

Abstract In the framework of Euclidean field theory the system with self-organized criticality regime is studied. We consider a self-similar behavior introduced as a fractional feedback in a three-parameter model of Lorenz type. The main modes to govern the system dynamics are: an avalanche size, an energy of moving grains and a complexity (entropy) of the avalanche ensemble. We take account of the additive noise of the energy to investigate the process of the nondriven avalanche-formation process in the presence of the energy noise, which plays a crucial role. The kinetics of the system is studied in detail on the basis of the variational principle. This distribution is shown to be a solution of both fractional and linear Fokker–Planck equations. Relations between the exponent of the size distribution, fractal dimension of phase space, characteristic exponent of multiplicative noise, number of governing equations are obtained.