White Noise Limit Research Articles

Path-integral approach to the electron density of states at the interface of a single modulation-doped heterojunction

The electronic density of states (DOS) at the interface of a single modulation-doped heterojunction is calculated both in the fluctuation band tail and in the semiclassical limit using the path-integral method. Due to charge density inhomogeneities in the heavily doped barrier region, random potential fluctuations are generated in whose minima carriers are localized, resulting in a band-tail density spectrum. The screening of the long-range potential fluctuations, which are important for the problem considered, is accounted for by using the two-dimensional Thomas-Fermi model. The statistical properties of the random impurity charge distribution are taken into account using the binary correlation function of the random potential for the specific geometry of the problem in two limiting cases of the general correlation function. Analytical expressions for the dimensionless functions of the exponential and preexponential of the band-tail DOS, as a function of the energy, are obtained in the weak and strong screening limit for the quasi-2D case under consideration and are compared to the respective functions for the general d-dimensional case. The dependence of the band-tail DOS behavior on the relevant parameters of the system, namely, spacer layer thickness, doping layer thickness, 2D EG thickness, and 3D-impurity concentration is studied numerically for ${\mathrm{Al}}_{x}{\mathrm{Ga}}_{1\ensuremath{-}x}\mathrm{A}\mathrm{s}\ensuremath{-}\mathrm{G}\mathrm{a}\mathrm{A}\mathrm{s}$ modulation doped heterostructures and numerical results for the 2D-DOS are presented. The band-tail results for the 2D-DOS are compared with the Kane approximation derived by taking the limit $\stackrel{\ensuremath{\rightarrow}}{t}0$ in order to determine $\ensuremath{\rho}(E)$ in the whole energy range. We compare the results from the two computed cases of the path-integral expression for the DOS corresponding to the two limits of the general correlation function with each other. On the other hand, we compare the semiclassical limit and the white Gaussian noise limit of the above results with many other theoretical methods such as the generalized semiclassical method, multiple scattering method and the simulations resulting from the tight-binding model.

Nonlinear Noise-Induced Transitions in Active Rotator Model

We investigate noise-induced transitions in active rotator model with a fluctuating threshold in the presence of an additive noise. The fluctuation of the threshold depends on the additive noise in a nonlinear fashion. In the white-noise limit of the fluctuation, the Fokker-Planck equation of the system reduces to that of the system with correlated linear fluctuation implying that the nonlinearity may be transformed into the correlation of linear noises. We also investigate the system with a nonlinear colored noise which depends on the additive noise as its square. The system shows a single peak, two peaks, and three peaks in its steady state probability distribution according to the noise intensities and the correlation time whose change leads to peak-creating, peak-splitting, and peak-merging transitions.

Noise-induced transport at zero temperature

We consider a particle in the over-damped regime at zero temperature under the influence of a sawtooth potential and of a noisy force, which is correlated in time. A current occurs, even if the mean of the noisy force vanishes. We calculate the stationary probability distribution and the stationary current. We discuss, how these items depend on the characteristic parameters of the underlying stochastic process. A formal expansion of the current around the white-noise limit not always gives the correct asymptotic behaviour. We improve the expansion for some simple but representative cases.

Dichotomously switched phase flows

The general formalism for periodic dichotomous noise on nonpotential flows is considered. This uncorrelated noise process switches suddenly at integer values of period $\ensuremath{\tau}$. The effect of additive noise of this kind on the planar FitzHugh-Nagumo ordinary differential equations [R. FitzHugh, Biophys. J. 1, 445 (1961); J. Nagumo, S. Arimoto, and Y. Yoshikawa, Proc. IRE 50, 2061 (1962)] is examined. For large $\ensuremath{\tau}$, quasifractal attractors are observed, whereas for the white-noise limit, where $\ensuremath{\tau}$ is small, a Fokker-Planck equation describes the evolution. The magnitude of $\ensuremath{\tau}$ determines the smoothness of the transient evolution and equilibrium density of the system. Typically the stochastic equations give rise to two regions of high density near the stable fixed points of the underlying autonomous system. The stiffness parameter $\ensuremath{\varepsilon}$ in the differential equations determines the fast variable, its associated nullcline, and the resulting flow structure. For small $\ensuremath{\varepsilon}$ the cubic nullcline controls the motion and transitions between the high-density peaks occur along segments of a noisy limit cycle. For large $\ensuremath{\varepsilon}$ the linear nullcline governs the transitions and the peaks are joined by a single band. The statistical behavior of the oscillatory and direct transitions is examined.

Photothermal measurements at picowatt resolution using uncooled micro-optomechanical sensors

Deflections of bimaterial microcantilever beams were optically detected with 400 fm resolution at room temperature. This enabled photothermal radiation detection with resolutions of 40 pW for power and 10 fJ for energy. The resolution was improved by an order of magnitude by optimizing the thickness ratio of the two beam materials, as well as by modulating the incident radiation at sufficiently high frequency to be in the range of the thermal white noise limit of the cantilever vibrations. Radiative power was detected with a noise spectral density of and 250 pW/Hz and detectivity, D*, of 4.6×107 cm Hz/W.

The Fokker-Planck equation for arbitrary nonlinear noise

We present the Fokker-Planck equation for arbitrary nonlinear noise terms. The white noise limit is taken as the zero correlation time limit of the Ornstein-Uhlenbeck process. The drift and diffusion coefficients of the Fokker-Planck equation are given by triple integrals of the fluctuations. We apply the Fokker-Planck equation to the active rotator model with a fluctuating potential barrier which depends nonlinearly on an additive noise. We show that the nonlinearity may be transformed into the correlation of linear noise terms.

Absorbing boundary conditions for inertial random processes.

A recent paper by J. Heinrichs [Phys. Rev. E 48, 2397 (1993)] presents analytic expressions for the first-passage times and the survival probability for a particle moving in a field of random correlated forces. We believe that the analysis there is flawed due to an improper use of boundary conditions. We compare that result, in the white noise limit, with the known exact expression of the mean exit time.

A stochastic model of IndoPacific sea surface temperature anomalies

It is often desirable to represent a rapidly varying physical process as stochastic forcing of some slower dynamical system. A review of this approximation is presented. The white-noise limit of rapidly varying processes reduces the dynamical description of the affected slower systems to stochastic differential equations, the properties of which are summarized. As an application of stochastic differential equations, we review the evidence that IndoPacific sea surface temperature anomalies (SSTAs) can be represented as a stable linear process driven by spatially coherent stochastic forcing. An “inversemodeling” approach is used. That is, the relevant parameters of the best-fit stable linear process are obtained from observations and, given these parameters, the assumptions of stability and linearity are subsequently tested. An “optimal initial structure” for growth, predicted by the model, is found to occur approximately seven months before most of the major warm and cold (El Niño/La Niña) events in the data record. The optimal structure preferentially occurs in the boreal spring, with the mature phase of the extreme event occurring in the subsequent late fall/winter. The best model to fit the observations, including the dependence of the dynamical system on the annual cycle, is driven by stochastic forcing with periodic statistics. These statistics are inferred using a time-dependent fluctuation-dissipation relation. The variation of the stochastic forcing with the annual cycle is not the same as that of the optimal initial structure. Yet, when a linear numerical model is driven with the derived stochastic forcing, the optimal structure, including its variation with the annual cycle, is recovered. This implies that deterministic dynamics play an important role in setting up the optimal structure and, therefore, a mature phase of El Niño may be predicted before the optimal structure actually appears.

Bistable kinetic model driven by correlated noises: Unified colored-noise approximation.

A Fokker-Planck equation for a general one-dimensional non-Markovian system driven by correlated Gaussian noises is derived by means of an extended unified colored-noise approximation. The general stationary probability distribution (SPD) is obtained. The SPD contains three important limits: the uncorrelated noise limit, the white noise limit, and the usual uncorrelated white noise limit. The following important physical aspects have been revealed by virtue of the above-mentioned SPD. (1) In contrast to the well known case of uncorrelated white noises where the parameter of additive noise cannot enter the extremal equation of SPD, now the additive noise parameter does enter the extremal equation as a non-Markovian effect even if the system is driven by uncorrelated noises. (2) When the correlation between the noises does exist, the SPD contains information caused by both correlation and color of the noises. The general results obtained in this Brief Report are applied to a bistable kinetic model. We find for the steady state of the model that in the case of correlated noises, the symmetry of SPD under the reflection of the state variable x with respect to the origin is destroyed. However in the case of non-Markovian processes driven by uncorrelated noises, the above symmetry is preserved.

Zero White Noise Limit through Dirichlet forms, with application to diffusions in a random medium

We study the Zero White Noise Limit for diffusions in a continuous multidimensional medium: given a continuous function on ℝ n ,W, we consider diffusions whose drift term is the gradient ofW and whose diffusion coefficient is constant equal to e. We describe the asymptotics of the exit time from a domain and of the law of the process when e tends to zero. By applying these results to a random self-similar mediumW we prove limit theorems for a diffusion in a random medium. Our theorems agree with results usually proved through the large deviation principle, although, in our setup, this last tool is not available. We extend to the multidimensional case properties of diffusions in a random medium already known in one dimension.

Turbulent mixing of a passive scalar

The statistically stationary state of a turbulently advected passive scalar is studied, with an imposed linear mean gradient in two dimensions, via a number of numerical experiments. For a synthetic Gaussian velocity field, which is generated by a linear stochastic process, and whose spectra and Eulerian correlation time follow Kolmogorov scaling on all scales, the exponents of the scalar spectra are consistent with 5/3 or 17/3 depending on the diffusivity. For large Péclet numbers (Pe), the probability density function (PDF) of the scalar gradients perpendicular to the mean is well fit, from about 0.1–10 times the root-mean-square value, by a stretched exponential with exponent ∼0.6. The PDF for gradients parallel to the mean has similar tails and a 𝒪(1) skewness for all Pe studied. The scalar has a ramp-and-cliff structure similar to that first seen in shear-flow experiments with scalars. A physical picture of the mechanism by which the ramp-and-cliff features form is given. A second model with the velocity evolving under the Euler equations restricted to a band of wave numbers produces the k−1 Batchelor spectrum when the scalar is dissipated with a hyperdiffusivity (∝k4). For physical dissipation (∝k2), the PDF of the scalar has exponential tails, and for gradients less than the cutoff set by the maximum strain, the PDF of the gradients is similar to that obtained with the stochastic velocity model. The PDF of the dissipation is approximately stretched exponential like the gradient PDFs and not lognormal. The skewness of the gradients parallel to the mean decreases with decreasing autocorrelation time of the velocity, and the gradient PDFs assume a limiting form in the white-noise limit.

Time Delay in Random Scattering

This work treats the time delay in scattering for the Schrodinger equation with a random potential. The time delay is related to the energy derivative of the phase shift. In the white noise limit, the phase shift and time delay satisfy a system of stochastic differential equations. In the high-frequency limit, the time delay is governed by a single stochastic differential equation. In this limit, the expected time delay is zero, but the variance of the time delay is the square of a kinematical time multiplied by a function of the ratio of the length of the scatterer to the localization length.

Stability of a fluid surface in a microgravity environment

We introduce a stochastic model to analyze in quantitative detail the effect of the high-frequency components of the residual accelerations onboard spacecraft (often called g jitter) on the motion of a fluid surface. The resid- ual acceleration field is modeled as a narrow-band noise characterized by three independent parameters: its intensity G2, a dominant frequency Q, and a characteristic spectral width T1. The white noise limit corresponds to QT —> 0, with G2T finite, and the limit of a periodic g jitter (or deterministic limit) can be recovered for QT —> , G2 finite. Analysis of the linear response of a fluid surface subjected to a fluctuating gravitational field leads to the stochastic Mathieu equation driven by both additive and multiplicative noise. We discuss the stability of the solutions of this linear equation in the two limits of white noise and deterministic forcing, and in the general case of narrow-band noise. The results are then applied to typical microgravity conditions. STOCHASTIC model is introduced to describe the high- frequency components of the residual accelerations onboard spacecraft (often called g jitter). The model is incorporated into the equations governing fluid motion, and the stability of a surface of discontinuity between two fluids of different density is analyzed. The linear stability of this surface is governed by the stochastic Mathieu equation, driven by both multiplicative and additive noise. We study the range of parameters in which the solutions of the stochastic Mathieu equation are stable and apply the results to a water-air surface in typical microgravity conditions. We find that the component of g jitter normal to the surface at rest couples non- linearly to the surface displacement and can lead to parametric in- stability. On the other hand, the two parallel components appear additively, and we show that they do not modify the linear stability boundaries associated with the normal component. A certain amount of attention has been paid recently to model- ing g jitter as a periodic function of time,13 but little attention has been paid to the more realistic case in which the effective accelera- tion spectrum contains a band of frequency components and is ran- dom in nature. Early work in this direction by Antar 4 considered the stability of the Rayleigh-Benard configuration under a random gravitational field with a uniform frequency spectrum (white noise). Later, Fichtl and Holland5 used a stochastic description to study the distribution of impulses that would exceed a prescribed threshold. We introduce a general model for g jitter, also based on a sto- chastic description,6 with a spectrum that is similar to residual ac- celeration spectra measured in space missions. The model, which we call narrow-band noise, is characterized by three parameters: the mean intensity of the fluctuations G2; their characteristic angu- lar frequency Q; and the characteristic width of the spectrum (peaked at Q) ir1 (Ref. 7). In the limit ^T -> 0, with D = G2T finite, narrow-band noise reduces to white noise of intensity Z), whereas

A Lateral-Effect Photodiode Scanning Sensor for Autonomous Vehicle Manoeuvres

A large field of view laser scanning sensor, based on the lateral-effect photodiode, and designed for close range mobile robot manoeuvres, is described. Laser modulation and synchronous detection are used to approach the theoretical white noise limit of sensor performance. Noise in the system, which determines ranging accuracy, is determined from logarithmic plots of signal strength and image resolution against range. The results suggest that sensor accuracy is around 82% above the theoretical performance limit. Finally, some typical scans are presented

Spectral properties of linear chain with one-point dynamical disorder

We have investigated the one-dimensional model of an exciton moving along a (finite or semi-infinite) chain of sites with the local energy at one specific site being randomly modulated. The modulation is described by the Markoff stochastic process and we do not invoke the white-noise assumption. For this model, we give an approximation-free calculation of the density of states, the projected density of states and the optical absorption spectrum. The exact solution is found for a broad family of stochastic processes which bridges (and includes) the dichotomic process and the Gaussian one. The results are discussed in various physical limits such as whitenoise limit, weak-perturbation limit and weak-tunneling limit. Particularly, in the slow-modulation (or static) limit, our model represents a chain with a static distribution of energy at the distinguished impurity site and the spectral characteristics reveal a nontrivial dependence on the width of this energy distribution.

Computer simulations of multiplicative stochastic differential equations

A class of robust algorithms for the computer simulation of stochastic differential equations with multiplicative noise is investigated. Excellent agreement is obtained with the known analytic behaviour of the Kubo oscillator in the white noise limit. The algorithms include a known first-order one-dimensional explicit method, as well as implicit methods of increased stability. A distinction is drawn between classes of stochastic differential equations depending on the type of spatial variation or curvature defined by the diffusion tensor. This allows greatly simplified numerical implementations of the new algorithms in certain cases. The results of different techniques are compared for the case of the Kubo oscillator, where a semi-implicit technique gives the greatest accuracy.

On a kinetic equation for the transport of particles in turbulent flows

The work described is part of a long-term study in pursuit of a kinetic equation for particles suspended in a turbulent flow. This equation represents the transport of the average particle phase space density and can be used to derive the continuum equations and constitutive relations for a two-fluid model of a dispersed particle flow and to establish the correct form of boundary conditions. A key role in this study is played by the equation of state of the dispersed particles, which is derived here for a Langevin equation of motion with a random driving force not limited to white noise. A suitable form of kinetic equation is then sought that reproduces this equation of state and predicts Gaussian spatial diffusion in the long-term limit with the correct form for the diffusion coefficient in homogeneous stationary turbulence. It is shown that both these requirements are automatically satisfied by constructing forms for the phase space diffusion coefficient that preserve invariance to a random Galilean transformation. The most general form is shown to be an expansion in successively higher-order cumulants of the aerodynamic driving force. When this force is a Gaussian process all terms apart from the first term contract to zero, i.e., the phase space diffusion current is of the form −[μ⋅(∂/∂v) +λ⋅(∂/∂x)]〈W〉 , where 〈W〉 is the average phase space density of particles with velocity v and position x at time t and μ and λ are phase space diffusion tensors whose components are memory integrals involving the correlation of the aerodynamic force along a particle trajectory. With both this form and the more general form of phase space diffusion current, the kinetic equation contracts to the classical Fokker–Planck equation in the white noise limit. Finally with the Gaussian form for the diffusion current, the form of the average phase space density is examined for the important case of a point source diffusing in an unbounded homogeneous stationary flow.

Escape over a fluctuating barrier: the white noise limit

The authors examine the problem of diffusion over a fluctuating barrier in the limit where the barrier fluctuations are extremely fast compared with all other timescales in the problem. In the white noise limit, the decay of probability from the metastable state is exponential with a characteristic timescale exhibiting Arrhenius behaviour. For small but finite correlation times for the fluctuating part of the potential, the effective barrier increases with the correlation time. Implications for liquids above the glass transition are discussed.

Lyapunov Exponents and Rotation number of Two-Ddimensional Systems with Telegraphic Noise

Exact expressions for the exponential growth rates (Lyapunov exponents) and rotation numbers of two-dimensional systems perturbed by telegraphic noise are derived. Asymptotic expansions for small, large, slow, and fast noise and the white noise limit are constructed. Then results are applied to the random oscillator, stochastic stability (in particular, stabilization by noise) and stochastic bifurcation.

Effects of laser phase fluctuations on squeezing in intracavity second-harmonic generation.

Excellent squeezing in intracavity second-harmonic generation has been predicted to occur on cavity resonance in the output intensity fluctuations. Cavity detunings cause laser phase noise to couple in and reduce the squeezing observable. Here we consider the effects of laser phase fluctuations on the output-squeezing spectrum. Laser phase noise is modeled as an Ornstein-Uhlenbeck (colored-noise) Gaussian stochastic process and its effects are compared with the white-noise limit. This indicates that the white-noise model may qualitatively overestimate the deleterious effects of laser fluctuations on sideband squeezing. We compare our results with the recently reported experiment of Pereira /ital et/ /ital al/. (Phys. Rev. A 38, 4931 (1988)) and present an analysis of the empty cavity for comparison.