Noise-Induced Linearisation and Delinearisation

It is demonstrated, by means of analogue electronic simulation and theoretically, that external noise can markedly change the character of the response of a nonlinear system to a low-frequency periodic field. In general, noise of sufficient intensity {\it linearises} the response. For certain parameter ranges in particular cases, however, an increase in the noise intensity can sometime have the opposite effect and is shown to {\it delinearise} the response. The physical origins of these contrary behaviours are discussed.Comment: 17 pages. No special macros. Figures on reques

The Geometry of Most Probable Trajectories in Noise-Driven Dynamical Systems

This paper presents a heuristic derivation of a geometric minimum action method that can be used to determine most-probable transition paths in noise-driven dynamical systems. Particular attention is focused on systems that violate detailed balance, and the role of the stochastic vorticity tensor is emphasized. The general method is explored through a detailed study of a two-dimensional quadratic shear flow which exhibits bifurcating most-probable transition pathways.Comment: 8 pages, 7 figure

Evolution of opinions on social networks in the presence of competing committed groups

Public opinion is often affected by the presence of committed groups of individuals dedicated to competing points of view. Using a model of pairwise social influence, we study how the presence of such groups within social networks affects the outcome and the speed of evolution of the overall opinion on the network. Earlier work indicated that a single committed group within a dense social network can cause the entire network to quickly adopt the group's opinion (in times scaling logarithmically with the network size), so long as the committed group constitutes more than about 10% of the population (with the findings being qualitatively similar for sparse networks as well). Here we study the more general case of opinion evolution when two groups committed to distinct, competing opinions $A$ and $B$, and constituting fractions $p_A$ and $p_B$ of the total population respectively, are present in the network. We show for stylized social networks (including Erd\H{o}s-R\'enyi random graphs and Barab\'asi-Albert scale-free networks) that the phase diagram of this system in parameter space $(p_A,p_B)$ consists of two regions, one where two stable steady-states coexist, and the remaining where only a single stable steady-state exists. These two regions are separated by two fold-bifurcation (spinodal) lines which meet tangentially and terminate at a cusp (critical point). We provide further insights to the phase diagram and to the nature of the underlying phase transitions by investigating the model on infinite (mean-field limit), finite complete graphs and finite sparse networks. For the latter case, we also derive the scaling exponent associated with the exponential growth of switching times as a function of the distance from the critical point.Comment: 23 pages: 15 pages + 7 figures (main text), 8 pages + 1 figure + 1 table (supplementary info

What Is Stochastic Resonance? Definitions, Misconceptions, Debates, and Its Relevance to Biology

Stochastic resonance is said to be observed when increases in levels of unpredictable fluctuations—e.g., random noise—cause an increase in a metric of the quality of signal transmission or detection performance, rather than a decrease. This counterintuitive effect relies on system nonlinearities and on some parameter ranges being “suboptimal”. Stochastic resonance has been observed, quantified, and described in a plethora of physical and biological systems, including neurons. Being a topic of widespread multidisciplinary interest, the definition of stochastic resonance has evolved significantly over the last decade or so, leading to a number of debates, misunderstandings, and controversies. Perhaps the most important debate is whether the brain has evolved to utilize random noise in vivo, as part of the “neural code”. Surprisingly, this debate has been for the most part ignored by neuroscientists, despite much indirect evidence of a positive role for noise in the brain. We explore some of the reasons for this and argue why it would be more surprising if the brain did not exploit randomness provided by noise—via stochastic resonance or otherwise—than if it did. We also challenge neuroscientists and biologists, both computational and experimental, to embrace a very broad definition of stochastic resonance in terms of signal-processing “noise benefits”, and to devise experiments aimed at verifying that random variability can play a functional role in the brain, nervous system, or other areas of biology

Fluctuational escape and related phenomena in nonlinear optical systems

In this chapter the authors discuss the application of simulation techniques to the study of fluctuational escape and related phenomena in nonliner optical systems: that is, situations where a large deviation of the system from an equilibrium state occurs under the influence of relatively weak noise present in the system. The authors are interested primarily in the analysis of situations where large deviations lead to new nontrivial behaviour or to a transition to a different state

Noise-induced escape on time scales preceding quasistationarity:new developments in the Kramers problem

Noise-induced escape from the metastable part of a potential is considered on time scales preceding the formation of quasiequilibrium within that part of the potential. It is shown that, counterintuitively, the escape flux may then depend exponentially strongly, and in a complicated manner, on time and friction

The role of noise in determining selective ionic conduction through nano-pores

The problem of predicting selective transport of ions through nano-pores from their structure in the biological and nano-technological systems is addressed. We use a molecular dynamics simulation to provide insight into the key physical parameters of nano-pores and develop a self-consistent analytic theory describing ionic conduction and selectivity through these devices. We analyse the ion's dehydration and excess chemical potential, derive an expression for the conductivity of the nanopore, and emphasize the role of fluctuations in its performance. The theory is verified by comparison of the predicted currentvoltage characteristics with the molecular dynamics results and experimental data obtained for a graphene nano-pore and the KcsA biological channel

Thermally activated escape of driven systems: the activation energy

Thermally activated escape in the presence of a periodic external field is investigated theoretically and through analogue experiments and digital simulations. The observed variation of the activation energy for escape with driving force parameters is accurately described by the logarithmic susceptibility (LS). The frequency dispersion of the LS is shown to differ markedly from the standard linear susceptibility. Experimental data on the dispersion are in quantitative agreement with the theory. Switching between different branches of the activation energy is demonstrated for a nonsinusoidal (biharmonic) force

Experiments on critical phenomena in a noisy exit problem

We consider a noise-driven exit from a domain of attraction in a two-dimensional bistable system lacking detailed balance. Through analog and digital stochastic simulations, we find a theoretically predicted bifurcation of the most probable exit path as the parameters of the system are changed, and a corresponding nonanalyticity of the generalized activation energy. We also investigate the extent to which the bifurcation is related to the local breaking of time-reversal invariance. [S0031-9007(97)04333-0]

Solution of the boundary value problem for optimal escape in continuous stochastic systems and maps

Topologies of invariant manifolds and optimal trajectories are investigated in stochastic continuous systems and maps. A topological method is introduced that simplifies the solution of boundary value problems: The activation energy is calculated as a function of a set of parameters characterizing the initial conditions of the escape path. The method is applied explicitly to compute the optimal escape path and the activation energy for a variety of dynamical systems and maps