White Noise Limit Research Articles

A modified pitchfork bifurcation as a model of the TI?TII transition in liquid helium counterflow with external noise

An amended pitchfork bifurcation is introduced to model recent experiments by Griswold and Tough on superfluid turbulence in liquid helium counterflow subject to strong external noise. We adopt the generalized white noise limit of Blankenship and Papanicolaou to take a short-correlation-time limit of the nonlinear noise which enters into the model, and we implement this limit by means of the wideband perturbation expansion. Novel boundary conditions are applied to the resultant diffusion process in order to obtain behavior in qualitative agreement with the observations at low vortex line density. We are able to account for the sharp peak in probability observed experimentally at a small positive line density. The drift and diffusion of our diffusion process may be estimated experimentally; we describe how to do this.

Analog simulation of underdamped stochastic systems driven by colored noise: Spectral densities.

Stochastic relaxation in underdamped nonlinear potentials driven by exponentially time-correlated noise is studied thoroughly by means of analog simulation. The experimental data reported herein should help to tie down future theoretical investigations. A novel dependence of the observed quantities on the noise correlation time \ensuremath{\tau} is revealed: all of the quantities measured are a function of ${\ensuremath{\tau}}^{2}$. In particular, the escape rate in a quartic double-well potential decreases exponentially with increasing ${\ensuremath{\tau}}^{2}$. The relevant spatial spectral densities are also determined for both the monostable and the bistable quartic potential on varying \ensuremath{\tau}. Previous theoretical predictions for the white-noise (\ensuremath{\tau}=0) limit are thus checked. The frequency of the resonance peak is shown to shift depending on ${\ensuremath{\tau}}^{2}$. Finally, some effort has been paid at justifying theoretically the results obtained.

Bistability driven by weakly colored Gaussian noise: The Fokker-Planck boundary layer and mean first-passage times.

We develop a singular perturbation approach to the problem of first-passage times for non-Markovian processes driven by Gaussian colored noise near the white-noise limit. In particular we treat the problem of overdamped tunneling in a bistable quartic potential in the presence of Gaussian noise of short correlation time \ensuremath{\tau}. The correct treatment of the absorbing boundary yields a lowest-order correction proportional to ${\mathrm{\ensuremath{\tau}}}^{1/2}$ with a proportionality constant involving the Milne extrapolation length for the Fokker-Planck equation, given in terms of the Riemann \ensuremath{\zeta} function.

Tests of numerical simulation algorithms for the Kubo oscillator

Numerical simulation algorithms for multiplicative noise (white or colored) are tested for accuracy against closed-form expressions for the Kubo oscillator. Direct white noise simulations lead to spurious decay of the modulus of the oscillator amplitude. A straightforward colored noise algorithm greatly reduces this decay and also provides highly accurate results in the white noise limit.

Stochastic analysis of environmental fluctuations in a compartmental system

In this paper the general two-compartment system with environmental stochasticity is investigated. The transfer rates and outputs are assumed to be random telegraph processes. The Laplace-transform of the mean-value function of the amount of substance present in the two compartments is evaluated. The Gaussian white noise limit case is also discussed and the stability of the system is examined. It is shown that while the deterministic model is stable, environmental stochasticity may induce in the mean-value function all sorts of behaviour — stable, unstable and oscillatory. This is in contrast to the intrinsic stochasticity in linear models wherefor the mean-value function is the same as the solution of the deterministic counterpart.

Exact probability distribution for soluble model with quadratic noise

A one-dimensional evolution equation transformable into a linear one coupled to a quadratic Smoluchowski (an Ornstein-Uhlenbeck) noise is considered. A one-dimensional probability distribution is obtained by way of a characteristic function which is expressed by functionals of the Smoluchowski process. It is shown that in the frame of the presented approach the probability density can be found only for a particular value of the damping constant in the linear-type relaxation equation. It is also shown that in a special case the white noise limit may be performed.

Thermal and parametric noise in dispersive optical bistability I General formalism

We formulate a running-wave, single-mode model for purely dispersive optical bistability that incorporates amplitude and frequency fluctuations in the incident-field, cavity-length fluctuations, and thermal noise in the radiation field and in the material. This model is given by a set of three Langevin-type equations for the real and the imaginary parts of the slowly varying electric field and for the material variable. In the white-noise limit, it is equivalent to a Fokker–Planck equation in three variables. Even if our model can also describe a Kerr medium or a two-level system in a suitable range of parameters, we are interested mainly in the case of miniaturized all-optical bistable devices that utilize semiconductor dispersive media. In this situation, we can adiabatically eliminate the field variables and therefore reduce the problem to a single stochastic differential equation in one variable, which contains several terms of additive and multiplicative noise. We find that noise in the imaginary part of the slowly varying electric field does not contribute to this equation. In the white-noise case, our stochastic equation is equivalent to a Fokker–Planck equation in one variable, whereas in the case of colored noise we obtain a one-dimensional Fokker–Planck equation only in the limit of short correlation times. Finally, we calculate the stationary solution of the resulting Fokker–Planck equation. We show that the presence of colored noise narrows the peaks of the probability distribution at steady state with respect to the white-noise case.

Theory of external two-state Markov noise in the presence of internal fluctuations

A theory is presented to take into account internal fluctuations in the study of stochastically driven systems. Internal fluctuations are modeled by a master equation in which external noise is introduced. External noise is modeled by a two-state Markov process. A unified theory of internal and external fluctuations is described in terms of an effective integrodifferential master equation or its equivalent generating function representation. Two examples for which exact analytical results can be obtained are presented. A discussion of the white noise limit of the theory is also given.

Asymptotic analysis of P.D.E.s with wide–band noise disturbances, and expansion of the moments

We present an asymptotic analysis -in the “ white-noise limit”- of a linear parabolic partial differential equation, whose coefficients are perturbed by a wide-band noise. After having studied some ergodic properties of a class of diffusion processes, we prove the convergence in law towards the solution of an Ito stochastic P.D.E. We then establish an expansion in powers of Δ ( 1/Δ being a measure of the bandwith of the driving noise) of the first moment of the solution

Partition function of a particle subject to Gaussian noise

We study the averaged partition function for a quantum particle subjected to Gaussian noise using the path integral representation. The noise is characterized by a covariance function with a strength and a range. It falls off rapidly with distance but the analytic form at short distances and the dimensionality are important. The remaining parameter is the thermal length of the particle. For a finite range we study the behavior of the partition function over the entire domain of strengths and thermal lengths. The techniques used are successively more accurate upper and lower bounds that include contributions from configurations involving traps. Particular attention is paid to a self-consistent field analysis lower bound and to a nonlocal quadratic action bound. We also study the white noise limit, i.e., vanishing range with finite values of the other parameters. In one dimension the white noise limit leads to convergent results. In three or higher dimensions the divergent terms can be isolated and computed. In two dimensions the degree of divergences changes at a finite value of the product of the strength and thermal length squared.

External non-white noise: Theory and experiment

Abstract We consider a system under the influence of an external gaussian noise with a flat spectrum and finite cut-off frequency. The stationery properties of the system are studied as a function of the correlation time of the noise. Theoretical predictions based on an expansion in powers of the correlation time are corroborated by experimental measurements on an electric circuit with a digital noise generator. The two main qualitative properties experimentally observed which can not be explained in a white noise limit theory are: (i) appearance of relative extrema of the stationery distribution for high noise intensity when the correlation time is not too small; (ii) dependence on the noise characteristics of the position of the maximum of the stationary distribution for low noise intensity.

Diffusion of a Quantum Particle in a Fluctuating Medium

Quantum motion of a particle in a fluctuating lattice is studied on the basis of a time-dependent Hamiltonian with site energies being randomly modulated by stochastic noises. The case of the two-state jump noise is studied in details. It is found that the particle moves diffusively on a long time scale. When the noise has a non-white spectrum, the diffusion constant is substantially enhanced in comparison with the white noise limit owing to the local coherence of the wavefunction of the particle. A numerical study is also carried out for one-dimensional cases. The analytical result is in rough agreement with the numerical calculation.

Stochastic Calculus and Some Models of Irreversible Processes

Each type, It()' s or Stratonovich' s, of the stochastic differential equations (SDE) has its own role to play in the theory of irreversible processes. It is argued generally that colored-noise random forces in mechanical equations of motion should, in the white noise limit, be interpreted in the sense of the Stratonovich type SDE. Its transformation into the It8 type facilitates the study of ensemble averages. By way of illustration, the problems of the random frequency modulation and the Brownian motion of a spin are treated to show how straightforwardly the SDE can bring out basic features of the irreversible processes.

Dielectric noise in membranes

Most theoretical and experimental studies of electrical fluctuations in membranes so far have been devoted to noise associated with conduction processes. In this paper a different type of noise is described which results from dipolar transitions in the membrane. Two mechanisms for the generation of such dielectric noise are analyzed: (a) conformational transitions of membrane proteins involving changes in dipolar moment and/or polarizibility, and (b) rotation of dipolar molecules dissolved in the lipid. The spectral intensity of current noise calculated for the two models exhibits a characteristic dependence on frequency ω with a decrease proportional to ω 2 towards low frequencies and an approach to a frequency-independent (white noise) limit at high frequencies. For a given number of dipolar molecules in the membrane, the spectral intensity is inversely proportional to the square of the membrane thickness.

1/f noise with a low frequency white noise limit

ONE of us (A.A.V.), who had recently been studying channel models for electrical fluctuations caused by potassium transport through nerve membrane, noticed a possible analogy with the flow pattern of grain through an hourglass of unusual shape designed by F. Vaudan, Paris. The hourglass was filled with small grains (0.1 to 1.0 mm diameter) of coloured glass and differed from ordinary hourglasses in the length of the pore connecting the two compartments which was drawn out to about 25 cm; about 80 times the smallest inner diameter (3 mm). When the hourglass was turned, the grain did not flow steadily but exhibited slow irregular density fluctuations (Fig. 1). With a view to the analogy with potassium channels in excitable membranes, we decided to examine comparative power spectra1–3.