Noise-induced Phase Transition Research Articles

Random walk approach to the analytic solution of random systems with multiplicative noise—The Anderson localization problem

We discuss here in detail a new analytical random walk approach to calculating the phase diagram for spatially extended systems with multiplicative noise. We use the Anderson localization problem as an example. The transition from delocalized to localized states is treated as a generalized diffusion with a noise-induced first-order phase transition. The generalized diffusion manifests itself in the divergence of averages of wavefunctions (correlators). This divergence is controlled by the Lyapunov exponent γ , which is the inverse of the localization length, ξ = 1 / γ . The appearance of the generalized diffusion arises due to the instability of a fundamental mode corresponding to correlators. The generalized diffusion can be described in terms of signal theory, which operates with the concepts of input and output signals and the filter function. Delocalized states correspond to bounded output signals, and localized states to unbounded output signals, respectively. The transition from bounded to unbounded signals is defined uniquely by the filter function H ( z ) .

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.

The Role of Noise-Induced Nonequilibrium Phase Transitions in the Structurization of Hydrodynamic Turbulence

The aim of this paper is to develop a continual theory of developed structurized turbulence in shear flows in a compressible fluid modeled by a superposition of two mutually penetrating continuums, where the first continuum refers to the averaged field of turbulent motion, and the second, to the turbulent spacetime chaos associated with the fine-grained fluctuation motion. Incorporating into the thermohydrodynamic description of the subsystem of turbulent chaos a set of internal stochastic coordinates q k (like the locally averaged rate of turbulent energy dissipation), which characterize the structure and temporal evolution of the vorticity of the pulsational hydrodynamic field, has made it possible to use methods of statistical nonequilibrium thermodynamics to derive stochastic differential equations for these parameters and the corresponding Fokker-Planck-Kolmogorov equations for the probability density of transition. In accordance with the modified Kolmogorov similarity theory, positive fluctuating parameters q k are believed to obey the lognormal distribution in the stationary state of chaos. The Gaussian white noise with zero memory, which provides an idealized description of the real noise of vortex chaos with a very short but still finite memory, is used to model the force effect of the noise of chaos (whose fluctuations are due to the cumulative effect of numerous factors determining the state of the turbulized medium). A general concept of the birth of coherent structures in the thermodynamically open subsystem of turbulent chaos is formulated, which attributes the formation of such structures to the phenomenon of nonequilibrium phase transitions induced by the multiplicative noise of chaos during an increase of supercriticality. The interrelation between such transitions and the process of self-organization—the development of ordered “multimolecular” formations with lower symmetry as compared to that of the initial state—is discussed. The ultimate aim of this study is to refine a number of representative hydrodynamic models of natural space environments, including the formation of galaxies and galactic clusters; the birth of stars from the diffuse medium of gas and dust clouds; the formation of accretion disks and subsequent accumulation of planetary systems, and the formation of gaseous envelopes (atmospheres) of planets; variously scaled flows in the atmospheres and in the circumplanetary plasma, etc. The study continues the stochastically thermodynamic approach to the synergetic description of structurized turbulence in astrogeophysical systems that the author has been developing in a series of previous papers (Kolesnichenko, 2002, 2003, 2004).

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.

Noise-Induced Phase Transition in Traffic Flow**The project supported by National Natural Science Foundation of China under Grant No. 70225005 and the Research Award Program (2001) for the Outstanding Young Teachers in Higher Education Institutions of the Ministry of Education of China

One of the dynamic phases of the traffic flow is the traffic jam. It appears in traffic flow when the vehicle density is larger than the critical value. In this paper, a new method is presented to investigate the traffic jam when the vehicle density is smaller than the critical value. In our method, we introduce noise into the traffic system after sufficient transient time. Under the effect of noise, the traffic jam appears, and the phase transition from free to synchronized flow occurs in traffic flow. Our method is tested for the deterministic NaSch traffic model. The simulation results demonstrate that there exist a broad range of lower densities at which the noise effect leading to traffic jam can be observed.

Development of turbulence in subsonic submerged jets

The development of turbulence in subsonic submerged jets is reviewed. It is shown that the turbulence results from a strong amplification of the weak input noise that is always present in the jet nozzle exit section. At a certain distance from the nozzle the amplification becomes essentially nonlinear. This amplified noise leads to a transition of the system to a qualitatively new state, which depends only slightly on the characteristics of the input noise, such as its power spectrum. Such a transition has much in common with nonequilibrium noise-induced phase transitions in nonlinear oscillators with multiplicative and additive noise. The Krylov–Bogolyubov method for spatially extended systems is used to trace the evolution of the power spectra, the root-mean-square amplitude of the turbulent pulsations, and the mean velocity, with increasing distance from the nozzle. It is shown that, as turbulence develops, its longitudinal and transverse scales increase. The results coincide qualitatively and also, in specific cases, quantitatively, with known experimental data.

Macroscopic limit cycle via pure noise-induced phase transitions.

Bistability generated via a pure noise-induced phase transition is reexamined from the view of bifurcations in macroscopic cumulant dynamics. It allows an analytical study of the phase diagram in more general cases than previous methods. In addition, using this approach we investigate spatially extended systems with two degrees of freedom per site. For this system, the analytic solution of the stationary Fokker-Planck equation is not available and a standard mean field approach cannot be used to find noise-induced phase transitions. A different approach based on cumulant dynamics predicts a noise-induced phase transition through a Hopf bifurcation leading to a macroscopic limit cycle motion, which is confirmed by numerical simulation.

Stability analysis of stationary states of mean field models described by Fokker–Planck equations

So far, the stability of stationary solutions of mean field models described by Fokker–Planck equations has only been determined analytically in some special cases. Following two earlier studies [J. Korean Phys. Soc. 40 (2002) 1037; Prog. Theoret. Phys. Suppl. 150 (2003) 48], we discuss a stability analysis for a large class of mean field models based on Fokker–Planck equations. To this end, we use linear stability analysis in addition to the well-known approaches by means of transcendent equations and Lyapunov’s direct method. We demonstrate that all three methods yield consistent results for systems that exhibit free energy functionals (e.g., equilibrium systems). We show that the simple transcendent equation analysis fails for systems that do not exhibit free energy functionals (e.g., nonequilibrium systems) and show how to solve this problem by means of a more sophisticated transcendent equation analysis. Furthermore, we propose a norm for the perturbations of stationary states and illustrate some of our results by a model that exhibits a reentrant noise-induced phase transition.

Noise-induced excitability in oscillatory media.

A noise-induced phase transition to excitability is reported in oscillatory media with FitzHugh-Nagumo dynamics. This transition takes place via a noise-induced stabilization of a deterministically unstable fixed point of the local dynamics, while the overall phase-space structure of the system is maintained. Spatial coupling is required to prevent oscillations through suppression of fluctuations (via clustering in the case of local coupling). Thus, the joint action of coupling and noise leads to a different type of phase transition and results in a stabilization of the system. The resulting regime is shown to display characteristic traits of excitable media, such as stochastic resonance and wave propagation. This effect thus allows the transmission of signals through an otherwise globally oscillating medium.

Intrinsic noise-induced phase transitions: beyond the noise interpretation.

We discuss intrinsic noise effects in stochastic multiplicative-noise partial differential equations, which are qualitatively independent of the noise interpretation (Itô vs Stratonovich), in particular in the context of noise-induced ordering phase transitions. We study a model which, contrary to all cases known so far, exhibits such ordering transitions when the noise is interpreted not only according to Stratonovich, but also to Itô. The main feature of this model is the absence of a linear instability at the transition point. The dynamical properties of the resulting noise-induced growth processes are studied and compared in the two interpretations and with a reference Ginzburg-Landau-type model. A detailed discussion of a different numerical algorithm valid for both interpretations is also presented.

Fluctuation induced reconstruction of phase transition

Within mean field approximation, a procedure is elaborated to consider noise induced phase transitions with arbitrary relations between the noises of different degrees of freedom. The proposed approach is applied to investigate effects of cross correlation between noises in the generalized synergetic model of Lorenz type. This cross correlation is shown to induce phase transitions of the dynamical system under consideration. Additionally, we find the correlation between noises transforms a synergetic behavior to a thermodynamic one.

Noise-driven mechanism for pattern formation.

We extend the mechanism for noise-induced phase transitions proposed by Ibañes et al. [Phys. Rev. Lett. 87, 020601 (2001)] to pattern formation phenomena. In contrast with known mechanisms for pure noise-induced pattern formation, this mechanism is not driven by a short-time instability amplified by collective effects. The phenomenon is analyzed by means of a modulated mean field approximation and numerical simulations.

Recombination instability and double S-switching in p-Ge(Au)

We report on the experimental investigation of p-Ge(Au) samples and compare the results qualitatively to a one-dimensional theoretical model describing the recombination instability of current in gold-compensated germanium in a two-parameter (voltage-emission) space. Experiments showed the existence of three regimes of the system functioning in the parametric space. The second S-switching region was found on the current-voltage characteristic, which is probably related to a noise-induced nonequilibrium phase transition.

Stochastic analysis of a non-normal dynamical system mimicking a laminar-to-turbulent subcritical transition.

The effects of stochastic perturbations on a non-normal dynamical system mimicking a laminar-to-turbulent subcritical transition are investigated both analytically and numerically. It is found that a nonlinear dynamical system with non-normal transient linear growth is very sensitive to the presence of weak random perturbations. The effect of non-normality on the exit probability from the zero fixed point is analyzed numerically for small values of the noise intensity parameter. It is found that an increase in the intensity of the noise, or a decrease of the non-normality parameter leads to qualitative changes in the behavior of the trajectories that can be interpreted as noise-induced phase transitions. By using the Itô formula and the adiabatic elimination procedure a stochastic equation governing the slow evolution of the energy of the non-normal system is derived.

Disordering effects of colour in a system of coupled Brownian motors: phase diagram and anomalous to normal hysteresis transition

A system of periodically coupled nonlinear phase oscillators—submitted to both additive and multiplicative white noises—has been recently shown to exhibit ratchetlike transport, negative zero-bias conductance, and anomalous hysteresis. These features stem from the asymmetry of the stationary probability distribution function, arising through a noise-induced non-equilibrium phase transition which is re-entrant as a function of the multiplicative noise intensity. Using an explicit mean-field approximation we analyse the effect of the multiplicative noises being coloured, finding a contraction of the ordered phase (and a re-entrance as a function of the coupling) on one hand, and a shift of the transition from anomalous to normal hysteresis inside this phase on the other.

Vibrational resonance in a noise-induced structure.

We report on the effect of vibrational resonance in a spatially extended system of coupled noisy oscillators under the action of two periodic forces, a low-frequency one (signal) and a high-frequency one (carrier). Vibrational resonance manifests itself in the fact that for optimally selected values of high-frequency force amplitude, the response of the system to a low-frequency signal is optimal. This phenomenon is a synthesis of two effects, a noise-induced phase transition leading to bistability, and a conventional vibrational resonance, resulting in the optimization of signal processing. Numerical simulations, which demonstrate this effect for an extended system, can be understood by means of a zero-dimensional "effective" model. The behavior of this "effective" model is also confirmed by an experimental realization of an electronic circuit.

Noise-Induced Scenario for Inverted Phase Diagrams

We introduce a class of exactly solvable models exhibiting an ordering noise-induced phase transition in which order arises as a result of a balance between the relaxing deterministic dynamics and the randomizing character of the fluctuations. A finite-size scaling analysis of the phase transition reveals that it belongs to the universality class of the equilibrium Ising model. All these results are analyzed in the light of the nonequilibrium probability distribution of the system, which can be obtained analytically. Our results could constitute a possible scenario of inverted phase diagrams in the so-called lower critical solution temperature transitions.

Transition from anomalous to normal hysteresis in a system of coupled Brownian motors: a mean-field approach.

We address a recently introduced model describing a system of periodically coupled nonlinear phase oscillators submitted to multiplicative white noises, wherein a ratchetlike transport mechanism arises through a symmetry-breaking noise-induced nonequilibrium phase transition. Numerical simulations of this system reveal amazing novel features such as negative zero-bias conductance and anomalous hysteresis, explained by performing a strong-coupling analysis in the thermodynamic limit. Using an explicit mean-field approximation, we explore the whole ordered phase finding a transition from anomalous to normal hysteresis inside this phase, estimating its locus, and identifying (within this scheme) a mechanism whereby it takes place.

Dynamics and scaling of noise-induced domain growth

The domain growth processes originating from noise-induced nonequilibrium phase transitions are analyzed, both for non-conserved and conserved dynamics. The existence of a dynamical scaling regime is established in the two cases, and the corresponding growth laws are determined. The resulting universal dynamical scaling scenarios are those of Allen-Cahn and Lifshitz-Slyozov, respectively. Additionally, the effect of noise sources on the behaviour of the pair correlation function at short distances is studied.

Nonequilibrium phase transitions induced by multiplicative noise: effects of self-correlation

A recently introduced lattice model, describing an extended system which exhibits a reentrant (symmetry-breaking, second-order) noise-induced nonequilibrium phase transition, is studied under the assumption that the multiplicative noise leading to the transition is colored. Within an effective Markovian approximation and a mean-field scheme it is found that when the self-correlation time tau of the noise is different from zero, the transition is also reentrant with respect to the spatial coupling D. In other words, at variance with what one expects for equilibrium phase transitions, a large enough value of D favors disorder. Moreover, except for a small region in the parameter subspace determined by the noise intensity sigma and D, an increase in tau usually prevents the formation of an ordered state. These effects are supported by numerical simulations.