On drift,diffusion and geometry

We present some reflections on the links between drift, diffusion and geometry. For this purpose, we examine different sources of “diffusion models”, in physics and in mathematics. We observe that diffusion processes may arise from original models either deterministic, or random but where dynamics and noise are clearly delineated. In the end, we get a diffusion process where noise and dynamics (“drift”) are generally intimately entangled in a second-order partial differential operator. We focus on the following questions. Are there implicit geometric structures to properly define a diffusion? How are drift/dynamics and diffusion mixed? Are there geometric structures needed to separate drift and diffusion? We stress the importance of recurrent differential geometric structures – connections and Riemannian metrics – needed to properly define a “diffusion term” and also to separate drift from diffusion.     

Symbolic dynamics of one-dimensional maps: Entropies,finite precision,and noise

In the study of nonlinear physical systems, one encounters apparently random or chaotic behavior, although the systems may be completely deterministic. Applying techniques from symbolic dynamics to maps of the interval, we compute two measures of chaotic behavior commonly employed in dynamical systems theory: the topological and metric entropies. For the quadratic logistic equation, we find that the metric entropy converges very slowly in comparison to maps which are strictly hyperbolic. The effects of finite precision arithmetric and external noise on chaotic behavior are characterized with the symbolic dynamics entropies. Finally, we discuss the relationship of these measures of chaos to algorithmic complexity, and use algorithmic information theory as a framework to discuss the construction of models for chaotic dynamics.     

On the prediction of impact noise,V: The noise from drop hammers

In earlier papers in this series, the concepts of “acceleration” and “ringing” noise have been studied in relation to impact machines, and values of radiation efficiency have been obtained for the various types of structural components. In the work reported in this paper the predicted and measured noise radiation from a drop hammer, both in full-scale and in $\text{1}\text{3}\text{-}\text{scale}$ model form, were examined. It is found that overall noise levels (L_eq per event) can be predicted from vibration measurements to within ± 1·5 dB, and to within ±2·5 dB in one-third octave bands. In turn this has permitted noise reduction techniques to be examined by studies of local component vibration levels rather than overall noise, a method which provides considerable enlightenment at the design stage. It is shown that on one particular drop hammer, the noise energy is shared surprisingly uniformly over four or five sources, and that when these have been reduced, the overall noise reduction is severely limited by the “acceleration” noise from the “tup” or “hammer” itself. As this is difficult to eliminate without a basic change in forging technology, it follows that “tup” enclosure or modification of the sharpness of the final “hard” impact are the only means available for any serious noise reduction. Also indicated is the reliability of using model techniques, suitably scaled in frequency and impulse magnitude, in developing machinery with impact characteristics.     

The lumping mass effect on frequencies of beam-columns

The results of two experiments are reported in which 55 university staff and students carried out a variety of mental tasks and were required to adjust their ambient noise to a “comfortable working level”. Preferred listening levels whilst carrying out mental tasks in noise show very large between-subject differences. Some subjects prefer to work in the quiet and others in “deafening” noise levels (over 90 dB(A)) and yet the latter suffer no deterioration in task performance compared with the former. Preferred listening levels are determined in part by (1) the type of noise, (2) the degree of task difficulty and (3) the personality traits of extroversion and psychoticism, and possibly by noise sensitivity. Such personal differences as sex, age and “status” of the subject do not apparently affect preferred listening levels. These levels are shown to be those at which subjects judge a noise over which they have no control to be between “quiet” and “noticeable”.     

The influence of random fluctuations on the behaviour of an aerosol

The purpose of the paper is to assess in a general manner the influence of random fluctuations in the local velocity, deposition rate, growth rate and coagulation constant on the behaviour of an aerosol. Using stochastic methods of probability theory we have shown that fluctuations in the local velocity lead to a broadening of the volume distribution which increases with time. The fluctuation in the deposition rate has a destabilizing effect and reduces the deterministically calculated rate of decay. Fluctuations in growth rate lead to the broadening of an initially monodisperse distribution and allow for evaporation and condensation. The rate of increase of suspended mass arising from this condensation-evaporation process is greater than the corresponding deterministic value.The effect of fluctuations in the coagulation constant leads to a decrease in the usual rate of removal of particles, although this is a small effect.The fluctuations are assumed to be stationary, random functions of time or position and in most cases are taken to be Gaussian variables. However, it is found that the infinite range of the Gaussian can lead to certain unphysical results and other distributions are examined.     

Velocity-dependent Lyapunov exponents as a measure of chaos for open-flow systems

Many flows in nature are “open flows” (e.g. pipe flow). We study two open-flow systems driven by low-level external noise: the time-dependent generalized Ginzburg-Landau equation and a system of coupled logistic maps. We find that a flow which gives every appearance of being chaotic may nonetheless have no positive Lyapunov exponents. By generalizing the notions of convective instability and Lyapunov exponents we define a measure of chaos for these flows.     

Understanding synchronization induced by “common noise”

Noise-induced synchronization refers to the phenomenon where two uncoupled, independent nonlinear oscillators can achieve synchronization through a “common” noisy forcing. Here, “common” means identical. However, “common noise” is a construct which does not exist in practice. Noise by nature is unique and two noise signals cannot be exactly the same. How to justify and understand this central concept in noise-induced synchronization? What is the relation between noise-induced synchronization and the usual chaotic synchronization? Here we argue and demonstrate that noise-induced synchronization is closely related to generalized synchronization as characterized by the emergence of a functional relation between distinct dynamical systems through mutual interaction. We show that the same mechanism applies to the phenomenon of noise-induced (or chaos-induced) phase synchronization.     

Overall railway noise impact in the U.K.

As part of a British Rail (BR) “Environmental and Social Impact” study in 1975, an attempt at assessing the relative noise impact of rail and road transport was made; 24 hour L_eq in dB(A) units was adopted for the noise measure, as this appeared to give good correlation with “dissatisfaction” and permitted simple estimation of levels from traffic and location data. Five train types were defined, two classes of line, three regions of population density and three standard topographies. The base L_eq value for each traffic mix, line and population region could be established from BR survey data, and the propagation from topography and population (house) densities, to give the population subjected to each L_eq. The percentage “dissatisfied” at each L_eq was then applied to arrive at the total population “dissatisfied”, which was found to be 106 000. A similar approach applied to motorways and principal “A” roads gave 4480 000 “dissatisfied”.     

Numerical simulations of acoustic cavitation noise with the temporal fluctuation in the number of bubbles

Numerical simulations of cavitation noise have been performed under the experimental conditions reported by Ashokkumar et al. (2007) [26]. The results of numerical simulations have indicated that the temporal fluctuation in the number of bubbles results in the broad-band noise. “Transient” cavitation bubbles, which disintegrate into daughter bubbles mostly in a few acoustic cycles, generate the broad-band noise as their short lifetimes cause the temporal fluctuation in the number of bubbles. Not only active bubbles in light emission (sonoluminescence) and chemical reactions but also inactive bubbles generate the broad-band noise. On the other hand, “stable” cavitation bubbles do not generate the broad-band noise. The weaker broad-band noise from a low-concentration surfactant solution compared to that from pure water observed experimentally by Ashokkumar et al. is caused by the fact that most bubbles are shape stable in a low-concentration surfactant solution due to the smaller ambient radii than those in pure water. For a relatively high number density of bubbles, the bubble–bubble interaction intensifies the broad-band noise. Harmonics in cavitation noise are generated by both “stable” and “transient” cavitation bubbles which pulsate nonlinearly with the period of ultrasound.     

Correlated noise-based switches and stochastic resonance in a bistable genetic regulation system

The correlated noise-based switches and stochastic resonance are investigated in a bistable single gene switching system driven by an additive noise (environmental fluctuations), a multiplicative noise (fluctuations of the degradation rate). The correlation between the two noise sources originates from on the lysis-lysogeny pathway system of the λ phage. The steady state probability distribution is obtained by solving the time-independent Fokker-Planck equation, and the effects of noises are analyzed. The effects of noises on the switching time between the two stable states (mean first passage time) is investigated by the numerical simulation. The stochastic resonance phenomenon is analyzed by the power amplification factor. The results show that the multiplicative noise can induce the switching from “on” → “off” of the protein production, while the additive noise and the correlation between the noise sources can induce the inverse switching “off” → “on”. A nonmonotonic behaviour of the average switching time versus the multiplicative noise intensity, for different cross-correlation and additive noise intensities, is observed in the genetic system. There exist optimal values of the additive noise, multiplicative noise and cross-correlation intensities for which the weak signal can be optimal amplified.     

Defensive activation toward noise

Long lasting sounds cause physiological reactions referred to as “ergotropic”, “orienting” or “defensive” reactions, the components of which have been demonstrated in laboratory experiments. In a study of residents exposed to aircraft noise in Germany, about 400 subjects were given white noise exposure of 85 and 100 dB in the laboratory. About 70% demonstrated vasoconstriction at the finger tips and temple, increase in muscle activity, initial deceleration of the heart rate and increase in tracking error and bodily movements. The number and degree of these reactions to laboratory noise increase with the number and loudness of aircraft movements at the homes of the subjects.     

Estimating the fractal dimension,K 2 -entropy,and the predictability of the atmosphere

The series of mean daily temperature of air recorded over a period of 215 years is used for analysing the dimensionality and the predictability of the atmospheric system. The total number of data points of the series is 78527. Other 37 versions of the original series are generated, including “seasonally adjusted” data, a smoothed series, series without annual course, etc. Modified methods of Grassberger &; Procaccia are applied. A procedure for selection of the “meaningful” scaling region is proposed. Several scaling regions are revealed in the lnC(r) versus Inr diagram. The first one in the range of larger lnr has a gradual slope and the second one in the range of intermediate lnr has a fast slope. Other two regions are settled in the range of small lnr. The results lead us to claim that the series arises from the activity of at least two subsystems. The first subsystem is low-dimensional (d _f =1.6) and it possesses the potential predictability of several weeks. We suggest that this subsystem is connected with seasonal variability of weather. The second subsystem is high-dimensional (d _f >17) and its error-doubling time is about 4–7 days. It is found that the predictability differs in dependence on season. The predictability time for summer, winter and the entire year (T ₂ ≈4.7 days) is longer than for transitionseasons (T ₂ ≈4.0 days for spring,T ₂ ≈3.6 days for autumn). The role of random noise and the number of data points are discussed. It is shown that a 15-year-long daily temperature series is not sufficient for reliable estimations based on Grassberger &; Procaccia algorithms.     

Time-evolving distribution of time lags between commercial airline disasters

We have studied the time lags between commercial line airplane disasters and their occurrence frequency till 2002, as obtained from a freely available website. We show that the time lags seem to be well described by Poisson random events, where the average events rate is itself a function of time, i.e., time-dependent Poisson events. This is likely due to the unsteady growth of the industry. The time lag distribution is compared with a truncated Tsallis distribution, thereby showing that the “phenomenon” has similarities with a Brownian particle with time-dependent mass. We distinguish between “other causes” (or natural causes) and “terrorism acts”, the latter amounts to about 5%, but we find no drastic difference nor impact due to the latter on the overall distribution.     

Noise-assisted traffic of spikes through neuronal junctions

The presence of noise, i.e., random fluctuations, in the nervous system raises at least two different questions. First, is there a constructive role noise can play for signal transmission in a neuron channel? Second, what is the advantage of the power spectra observed for the neuron activity to be shaped like 1/f(k)? To address these questions a simple stochastic model for a junction in neural spike traffic channels is presented. Side channel traffic enters main channel traffic depending on the spike rate of the latter one. The main channel traffic itself is triggered by various noise processes such as Poissonian noise or the zero crossings of Gaussian 1/f(k) noise whereas the variation of the exponent k gives rise to a maximum of the overall traffic efficiency. It is shown that the colored noise is superior to the Poissonian and, in certain cases, to deterministic, periodically ordered traffic. Further, if this periodicity itself is modulated by Gaussian noise with different spectral exponents k, then such modulation can lead to noise-assisted traffic as well. The model presented can also be used to consider car traffic at a junction between a main and a side road and to show how randomness can enhance the traffic efficiency in a network. (c) 2001 American Institute of Physics.     

Frictionless random dynamics: hydrodynamical formalism

We investigate an undamped random phase-space dynamics in deterministic external force fields (conservative and magnetic ones). By employing the hydrodynamical formalism for those stochastic processes we analyze microscopic kinetic-type “collision invariants” and their relationship to local conservation laws (moment equations) in the fully nonequilibrium context.     

Subjective response to sound conditioning in a landscaped office

A sound conditioning system was installed in a landscaped office accomodating 44 staff, where speech privacy was poor. Twelve different conditions of artificial background noise ranging from 36 dB(A) to 46 dB(A) were tested, each for a period of three weeks. The occupant's reactions were monitored at the end of each three-week test period by means of self-completion questionnaires. Eight conditions based on random noise were, on balance, considered acceptable, but were judged to be ineffective and were not considered by the occupants to give any overall improvement. Four conditions based on “natural” recorded noise were also tested, but two of these (office noise and seashore noise) were rejected as unacceptable and ineffective. None of the conditions tested was particularly successful. Such parameters as frequency spectrum and overall level appeared to have little influence, although acceptability and ease of conversation tended to decline with increase in level of the introduced sound.     

Unbiased estimate of forces from measured correlation functions,including the case of strong multiplicative noise

Starting from appropriate short-time correlation function measurements, we propose a dynamical “learning” method to derive the deterministic and stochastic forces underlying an observed process, even if this process contains strong multiplicative noise. To do this we extend the ideas of our previous paper [1] to establish mathematical relationships in this more general case between the joint distribution function of the process and its corresponding Ito-Langevin equation. A numerical example for a simulated process containing strong multiplicative noise shows good agreement with the theory.