Reentrance Phenomena in Noise Induced Transitions

Abstract

We analyze a model that has been shown to undergo a purely noise induced transition, from a monostable regime to a bistable one, when it is submitted to a white or colored noise source. We show, using a consistent interpolating Markovian approximation for the colored noise case, that for large values of the correlation time, the system undergoes a new transition to a monostable state, indicating a reentrancelike phenomenon in its phase diagram. Numerical results support our findings. PACS numbers: 05.40.+j The study of dynamical systems subject to a noise perturbation has become a recurrent theme in physics, chemistry, and biology, as well as in several other areas. Particularly for nonequilibrium systems, where the macrovariables obey nonlinear equations of motion, noise plays a crucial role. For instance, the system can overcome potential barriers and reach different macrostates due to only the presence of noise [1]. One aspect that attracted considerable attention was the fact that some systems, when far from thermodynamic equilibrium, dueto the inhuence of external noise sources, show the striking characteristic of undergoing transitions to new states that sometimes are not present in the deterministic description. These transition phenomena pose a fascinating problem as, contrary to all intuition, it is the environmental randomness that deeply influences the macroscopic system's properties, inducing a more structured behavior.These types of nonequilibrium transition phenomena have been called noise induced transitions [2,3].On the other hand, more realistic models of physical systems require considering noise sources with finite correlation times (i.e., colored noise). For example, in order to describe the static and dynamical properties of dye lasers, the usual model includes in its stochastic differential equations (SDE's) not only the standard internal white noise, but also an external colored noise [4]. The effect of time correlations in the fluctuations has also been taken into account in several models [2,3,5,6]. Some recent papers and reviews on the colored noise problem [7 -10] offer a view of the state of the art. Many efforts were oriented to obtaining Markovian approximations, with the aim of capturing the essential features of the original non-Markovian problem. Along this line, a recent approach was based on an interpolation procedure [11].In this paper, we present an analysis of a chemical reaction system and/or genic selection model [2,3,5], when it is subject to a colored noise source, by using the interpolation procedure. This approach allows us, at variance with previous studies, to obtain the complete phase diagram for the whole range of parameters. The choice of the above indicated genic model is due to the fact that it is an archetype of the kind of models (2) studied within the realm of noise induced transitions [3]. The result of our study, supported by numerical evidence, is that the phase diagram shows a novel feature corresponding to a reentrancelike phenomenon. The colored noise pr...

Time-Delay Induced Reentrance Phenomenon in a Triple-Well Potential System Driven by Cross-Correlated Noises

“…The reentrance phenomenon was firstly observed in an analysis of a chemical reaction system driven by one colored noise, which showed that the system undergoes a purely noiseinduced transition from a monostable regime to a bistable one, then to the monostable regime as the noise correlation time increases [2]. This phenomenon has also been found in a bistable system subjected to the cross-correlated noises [10].…”

Dynamics of Gompertzian tumour growth under environmental fluctuations

“…Studies have confirmed the role of noise in nonlinear stochastic systems [1]. Now it has been realized that the effects of noise have widely appeared in all kinds of nonlinear systems including bistable systems [2][3][4][5], laser systems [6,7] and biological systems [8][9][10][11][12]. Some of the notable examples of noise induced effects are stochastic resonance in biological systems.…”

Effect of non-Gaussian noise sources in a noise-induced transition

“…We believe that this will be a general behavior for long tail (power-law) and cut-off noise distributions, respectively. A closer look shows the persistence of the reentrance effect previously found in [13]: for some fixed values of the noise intensity D it is possible to obtain a transition, first from unimodal to bimodal and then back to unimodal, just by varying the time correlation τ of the noise.…”

“…In Ref. [13] the effect of a colored Gaussian (Ornstein-Uhlenbeck) noise on that model was analyzed in detail and a novel reentrance phenomenon in the phase diagram was found. In order to study the effect that the non-Gaussian nature of the noise can have in the transition, we have used here a particular distribution [14] whose departure from the Gaussian behavior is governed by a parameter q: while q = 1 corresponds to Gaussian noise, the value q > 1 produces a long tail distribution and q < 1 gives a cut-off distribution.…”