White Noise Limit Research Articles

Mean-field dynamo due to spatio-temporal fluctuations of the turbulent kinetic energy

In systems where the standard$\alpha$effect is inoperative, one often explains the existence of mean magnetic fields by invoking the ‘incoherent$\alpha$effect’, which appeals to fluctuations of the mean kinetic helicity at a mesoscale. Most previous studies, while considering fluctuations in the mean kinetic helicity, treated the mean turbulent kinetic energy at the mesoscale as a constant, despite the fact that both these quantities involve second-order velocity correlations. The mean turbulent kinetic energy affects the mean magnetic field through both turbulent diffusion and turbulent diamagnetism. In this work, we use a double-averaging procedure to analytically show that fluctuations of the mean turbulent kinetic energy at the mesoscale (giving rise to$\eta$-fluctuations at the mesoscale, where the scalar$\eta$is the turbulent diffusivity) can lead to the growth of a large-scale magnetic field even when the kinetic helicity is zero pointwise. Constraints on the operation of such a dynamo are expressed in terms of dynamo numbers that depend on the correlation length, correlation time and strength of these fluctuations. In the white-noise limit, we find that these fluctuations reduce the overall turbulent diffusion, while also contributing a drift term which does not affect the growth of the field. We also study the effects of non-zero correlation time and anisotropy. Turbulent diamagnetism, which arises due to inhomogeneities in the turbulent kinetic energy, leads to growing mean-field solutions even when the$\eta$-fluctuations are statistically isotropic.

Collisional-model quantum trajectories for entangled-qubit environments

We study the dynamics of quantum systems interacting with a stream of entangled-qubit pairs in the weak coupling limit. For a large class of two-qubit bath states, we present a detailed framework describing conditional dynamical maps for the system, known as quantum trajectories, that arise when the qubits are measured. Depending on the measurement basis, these quantum trajectories can be jump-type or diffusive-type, and can successively transfer entanglement from the bath qubits other quantum systems. They also exhibit features not present in standard quantum optical trajectories due to the fact that collisional models are not confined by the standard white-noise limit for optical-mode baths. As an example of this formalism, we consider the case of two remote two-level systems, where jump-type quantum trajectories herald the birth and death of entanglement.

Memory effects in spiral diffusion of rotary self-propellers

The coupling of deterministic rotary motion and stochastic orientational diffusion of a self-propeller leads to a spiral trajectory of the expected displacement. We extend our former analysis of spiral diffusion [Phys. Rev. E 94, 030601(R) (2016)10.1103/PhysRevE.94.030601] in the white-noise limit to a more realistic scenario of stochastic noise with Gaussian memory and orientational fluctuations driven by an Ornstein-Uhlenbeck process. A variety of dynamical regimes including crossovers from ballistic to diffusive to ballistic in the angular dynamics are determined by the inertial timescale, orientational diffusivity, and angular speed.

Irreversibility in linear systems with colored noise.

Time irreversibility is a distinctive feature of nonequilibrium dynamics and several measures of irreversibility have been introduced to assess the distance from thermal equilibrium of a stochastically driven system. While the dynamical noise is often approximated as white, in many real applications the time correlations of the random forces can actually be significantly long-lived compared to the relaxation times of the driven system. We analyze the effects of temporal correlations in the noise on commonly used measures of irreversibility and demonstrate how the theoretical framework for white-noise-driven systems naturally generalizes to the case of colored noise. Specifically, we express the autocorrelation function, the area enclosing rates, and mean phase space velocity in terms of solutions of a Lyapunov equation and in terms of their white-noise limit values.

Propagation of non-Gaussian voltage angle fluctuations in high-voltage power grids

Recent measurements have reported non-Gaussian tails in the distribution of frequency data in electric power grids. Large frequency deviations may induce grid instabilities and it is therefore crucial to understand how noise disturbances with long, non-Gaussian tails propagate. Here, we investigate how fluctuations in power feed-in, characterized by non-zero cumulants of their distribution, propagate through high-voltage power grids. Unlike previous investigations which focused on the white-noise limit, we consider the limit of long noise correlation time, where power feed-in fluctuates over times longer than the inherent dynamical time scales of the grid - the relevant regime for large-scale, high-voltage distribution grids. We show that in this limit, the skewness and kurtosis of the power feed-in distribution propagate similarly as its variance, independently of the distribution of inertia. Non-Gaussianities from individual sources of noise therefore persist throughout the entire network. This finding is corroborated by numerical results on a realistic model of the synchronous grid of continental Europe.

Emerging magnetic nutation

Nutation has been recognized as of great significance for spintronics; but justifying its presence has proven to be a hard problem. In this paper we show that nutation can be understood as emerging from a systematic expansion of a kernel that describes the history of the interaction of a magnetic moment with a bath of colored noise. The parameter of the expansion is the ratio of the colored noise timescale to the precession period. In the process we obtain the Gilbert damping from the same expansion. We recover the known results, when the coefficients of the two terms are proportional to one another, in the white noise limit; and show how colored noise leads to situations where this simple relation breaks down, but what replaces it can be understood by the appropriate generalization of the fluctuation--dissipation theorem. Numerical simulations of the stochastic equations support the analytic approach. In particular we find that the equilibration time is about an order of magnitude longer than the timescale set by the colored noise for a wide range of values of the latter and we can identify the presence of nutation in the non-uniform way the magnetization approaches equilibrium.

Synchronization transition in the two-dimensional Kuramoto model with dichotomous noise.

We numerically study the celebrated Kuramoto model of identical oscillators arranged on the sites of a two-dimensional periodic square lattice and subject to nearest-neighbor interactions and dichotomous noise. In the nonequilibrium stationary state attained after a long time, the model exhibits a Berezinskii-Kosterlitz-Thouless (BKT)-like transition between a phase at a low noise amplitude characterized by quasi long-range order (critically ordered phase) and an algebraic decay of correlations and a phase at a high noise amplitude that is characterized by complete disorder and an exponential decay of correlations. The interplay between the noise amplitude and the noise-correlation time is investigated, and the complete, nonequilibrium stationary-state phase diagram of the model is obtained. We further study the dynamics of a single topological defect for various amplitudes and correlation time of the noise. Our analysis reveals that a finite correlation time promotes vortex excitations, thereby lowering the critical noise amplitude of the transition with an increase in correlation time. In the suitable limit, the resulting phase diagram allows one to estimate the critical temperature of the equilibrium BKT transition, which is consistent with that obtained from the study of the dynamics in the Gaussian white noise limit.

Effective temperatures for single particle system under dichotomous noise

Three different definitions of effective temperature—, and related to kinetic theory, system entropy and response theory, respectively—are applied in the description of a non-equilibrium generalised massive Langevin model in contact with dichotomous noise. The differences between the definitions of naturally wade out as the reservoir reaches its white-noise limit, approaching Gaussian features. The same framework is employed in its overdamped version as well, showing the loss of inertial contributions to the dynamics of the system also makes the three mentioned approaches for effective temperature equivalent.

Coupling of paraxial and white-noise approximations of the Helmholtz equation in randomly layered media

We study the simultaneous paraxial and white-noise limit of the Helmholtz equation in randomly layered media where the refractive index fluctuations are in the direction of propagation. We consider the regime in which the wavelength is of the same order as the correlation length of the random fluctuations of the refractive index. We show that this simultaneous limit can be taken in this regime by introducing into the equation an arbitrarily small regularization parameter. The corresponding paraxial white-noise approximation that we derive is different from that of the previously studied high-frequency regime. Since the correlation length of the refractive index fluctuations due to atmospheric turbulence varies substantially, our results are relevant for numerous different propagation scenarios including microwave and radiowave propagation through various regions of the atmosphere.

Noise Sensitivities for an Atom Shuttled by a Moving Optical Lattice via Shortcuts to Adiabaticity.

We find the noise sensitivities (i.e., the quadratic terms of the energy with respect to the perturbation of the noise) of a particle shuttled by an optical lattice that moves according to a shortcut-to-adiabaticity transport protocol. Noises affecting different optical lattice parameters, trap depth, position, and lattice periodicity, are considered. We find generic expressions of the sensitivities for arbitrary noise spectra but focus on the white-noise limit as a basic reference, and on Ornstein–Uhlenbeck noise to account for the effect of non-zero correlation times.

External removal of endpoint-discontinuity artifacts in the reciprocal-space analysis of spectra

We present a systematic method of removing endpoint-discontinuity artifacts in the Fourier analysis of spectral segments, enabling the more accurate extraction of information. This principal-component-removal approach differs from a previous version by using extrapolated (or extended) data outside rather than inside the spectral range. This not only allows coefficients to be accessed to the white-noise limit with no distortion of the segment, but also generates interpolated coefficients for improved analytic insight. Examples are provided.

Emission spectrum of broadband quantum dot superluminescent diodes

We present a microscopic theory of the amplified spontaneous emission of a spectrally broadband quantum dot superluminescent diode with tilted end facets within the quantum white noise limit. From this multimode quantum theory, we have the ability to obtain all orders of temporal correlation functions. In particular, we derive rate equations for the optical power densities, the level occupation of inhomogeneous ensemble of quantum dots within the diode, as well as the emitted optical spectra. As the main result, we find the external power spectrum as a convolution of the intra-diode photon spectrum with a Lorentzian response. Assuming a Gaussian light-matter coupling results in a similar shaped Gaussian output spectrum, which agrees very well with available experimental data.

Josephson Junctions and SQUIDs Created by Focused Helium-Ion-Beam Irradiation ofYBa2Cu3O7

By scanning with a $30\, \mathrm{keV}$ focused He ion beam (He-FIB) across YBa$_2$Cu$_3$O$_7$ (YBCO) thin film microbridges, we create Josephson barriers with critical current density $j_\mathrm{c}$ adjustable by irradiation dose $D$. The dependence $j_\mathrm{c} (D)$ yields an exponential decay. At $4.2\, \mathrm{K}$, a transition from flux-flow to Josephson behavior occurs when $j_\mathrm{c}$ decreases below $\approx 2\, \mathrm{MA/cm^2}$. The Josephson junctions exhibit current-voltage characteristics (IVCs) that are well described by the resistively and capacitively shunted junction model, without excess current for characteristic voltages $V_\mathrm{c} \lesssim 1\, \mathrm{mV}$. Devices on MgO and LSAT substrates show non-hysteretic IVCs, while devices on SrTiO$_3$ show a small hysteresis. For all junctions an approximate scaling $V_\mathrm{c} \propto j_\mathrm{c}^{1/2}$ is found. He-FIB irradiation with high dose produces barriers with $j_\mathrm{c}=0$ and high resistances of $10\, \mathrm{k\Omega} \ldots 1\, \mathrm{G\Omega}$. This provides the possibility to write highly resistive walls or areas into YBCO using a He-FIB. Transmission electron microscopy reveals an amorphous phase within the walls, whereas for lower doses the YBCO stays crystalline. We have also ``drawn'' superconducting quantum interference devices (SQUIDs) by using a He-FIB for definition of the SQUID hole and the junctions. The SQUIDs show high performance, with flux noise $< 500\, \mathrm{n \Phi_0/Hz^{1/2}}$ in the thermal white noise limit for a device with $19\, \mathrm{pH}$ inductance.

Generation of large-scale magnetic fields due to fluctuating in shearing systems

We explore the growth of large-scale magnetic fields in a shear flow, due to helicity fluctuations with a finite correlation time, through a study of the Kraichnan–Moffatt model of zero-mean stochastic fluctuations of the$\unicode[STIX]{x1D6FC}$parameter of dynamo theory. We derive a linear integro-differential equation for the evolution of the large-scale magnetic field, using the first-order smoothing approximation and the Galilean invariance of the$\unicode[STIX]{x1D6FC}$-statistics. This enables construction of a model that is non-perturbative in the shearing rate$S$and the$\unicode[STIX]{x1D6FC}$-correlation time$\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}$. After a brief review of the salient features of the exactly solvable white-noise limit, we consider the case of small but non-zero$\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}$. When the large-scale magnetic field varies slowly, the evolution is governed by a partial differential equation. We present modal solutions and conditions for the exponential growth rate of the large-scale magnetic field, whose drivers are the Kraichnan diffusivity, Moffatt drift, shear and a non-zero correlation time. Of particular interest is dynamo action when the$\unicode[STIX]{x1D6FC}$-fluctuations are weak; i.e. when the Kraichnan diffusivity is positive. We show that in the absence of Moffatt drift, shear does not give rise to growing solutions. But shear and Moffatt drift acting together can drive large-scale dynamo action with growth rate$\unicode[STIX]{x1D6FE}\propto |S|$.

Comparison of Sensitivity and Low-Frequency Noise Contributions in Giant-Magnetoresistive and Tunneling-Magnetoresistive Spin-Valve Sensors with a Vortex-State Free Layer

Magnetoresistive spin valve sensors based on the giant- (GMR) and tunnelling- (TMR) magnetoresisitve effect with a flux-closed vortex state free layer design are compared by means of sensitivity and low frequency noise. The vortex state free layer enables high saturation fields with negligible hysteresis, making it attractive for applications with a high dynamic range. The measured GMR devices comprise lower pink noise and better linearity in resistance but are less sensitive to external magnetic fields than TMR sensors. The results show a comparable detectivity at low frequencies and a better performance of the TMR minimum detectable field at frequencies in the white noise limit.

The Small-Mass Limit and White-Noise Limit of an Infinite Dimensional Generalized Langevin Equation

We study asymptotic properties of the Generalized Langevin Equation (GLE) in the presence of a wide class of external potential wells with a power-law decay memory kernel. When the memory can be expressed as a sum of exponentials, a class of Markovian systems in infinite-dimensional spaces is used to represent the GLE. The solutions are shown to converge in probability in the small-mass limit and the white-noise limit to appropriate systems under minimal assumptions, of which no Lipschitz condition is required on the potentials. With further assumptions about space regularity and potentials, we obtain $L^1$ convergence in the white-noise limit.

Fast shuttling of a particle under weak spring-constant noise of the moving trap

We investigate the excitation of a quantum particle shuttled in a harmonic trap with weak springconstant colored noise. The Ornstein-Uhlenbeck model for the noise correlation function describes a wide range of possible noises, in particular for short correlation times the white-noise limit examined in Lu et al, Phys. Rev. A 89, 063414 (2014) and, by averaging over correlation times, "1/f ficker noise". We find expressions for the excitation energy in terms of static (independent of trap motion) and dynamical sensitivities, with opposite behavior with respect to shuttling time, and demonstrate that the excitation can be reduced by proper process timing and design of the trap trajectory.

Nonequilibrium dynamics of probe filaments in actin-myosin networks.

Active dynamic processes of cells are largely driven by the cytoskeleton, a complex and adaptable semiflexible polymer network, motorized by mechanoenzymes. Small dimensions, confined geometries, and hierarchical structures make it challenging to probe dynamics and mechanical response of such networks. Embedded semiflexible probe polymers can serve as nonperturbing multiscale probes to detect force distributions in active polymer networks. We show here that motor-induced forces transmitted to the probe polymers are reflected in nonequilibrium bending dynamics, which we analyze in terms of spatial eigenmodes of an elastic beam under steady-state conditions. We demonstrate how these active forces induce correlations among the mode amplitudes, which furthermore break time-reversal symmetry. This leads to a breaking of detailed balance in this mode space. We derive analytical predictions for the magnitude of resulting probability currents in mode space in the white-noise limit of motor activity. We relate the structure of these currents to the spatial profile of motor-induced forces along the probe polymers and provide a general relation for observable currents on two-dimensional hyperplanes.

Stochastic Resonance in Simple Electrical Circuits Driven by Quadratic Gaussian Noise

In this brief, we report exact analytical and numerical evidence of the occurrence of the stochastic resonance (SR) phenomenon in simple electrical circuits driven by a quadratic Gaussian colored noise and by a Gaussian white noise. As an example, we have used an RL configuration. However, it is clear that the phenomenon occurs in any other configuration governed by the same type of evolution equation. The main result is that the phenomenon occurs with a quadratic colored noise. When noise enters only linearly we have verified that the SR occurs both with colored noise and white noise. The robustness of our results were verified by calculating the variance as a function of time in each case. We believe that the results obtained could be observed in a laboratory experiment using some kind of digital resistance. The white noise limit with quadratic noise was excluded.

Fractional White-Noise Limit and Paraxial Approximation for Waves in Random Media

This work is devoted to the asymptotic analysis of high frequency wave propagation in random media with long-range dependence. We are interested in two asymptotic regimes, that we investigate simultaneously: the paraxial approximation, where the wave is collimated and propagates along a privileged direction of propagation, and the white-noise limit, where random fluctuations in the background are well approximated in a statistical sense by a fractional white noise. The fractional nature of the fluctuations is reminiscent of the long-range correlations in the underlying random medium. A typical physical setting is laser beam propagation in turbulent atmosphere. Starting from the high frequency wave equation with fast non-Gaussian random oscillations in the velocity field, we derive the fractional Ito–Schrodinger equation, that is, a Schrodinger equation with potential equal to a fractional white noise. The proof involves a fine analysis of the backscattering and of the coupling between the propagating and evanescent modes. Because of the long-range dependence, classical diffusion-approximation theorems for equations with random coefficients do not apply, and we therefore use moment techniques to study the convergence.