Noise-induced Drift Research Articles

Biases in ice sheet models from missing noise-induced drift

Biases in ice sheet models from missing noise-induced drift

Coherent feedback enhanced quantum-dense metrology in a lossy environment.

Quantum dense metrology (QDM) performs high-precision measurements by a two-mode entangled state created by an optical parametric amplifier (PA), where one mode is a meter beam and the other is a reference beam. In practical applications, the photon losses of meter beam are unavoidable, resulting in a degradation of the sensitivity. Here, we employ coherent feedback that feeds the reference beam back into the PA by a beam splitter to enhance the sensitivity in a lossy environment. The results show that the sensitivity is enhanced significantly by adjusting the splitting ratio of the beam splitter. This method may find its potential applications in QDM. Furthermore, such a strategy that two non-commuting observables are simultaneous measurements could provide a new way to individually control the noise-induced random drift in phase or amplitude of the light field, which would be significant for stabilizing the system and long-term precision measurement.

Improving Atmospheric Models by Accounting for Chaotic Physics

Abstract It is well known that randomly perturbing an atmospheric model’s diabatic tendencies can increase its probabilistic forecast skill, mainly by increasing the spread of ensemble forecasts and making it more consistent with the errors of ensemble-mean forecasts. Less obvious and less well established is that such perturbations can also reduce the errors of the ensemble-mean forecasts and improve the model’s mean climate, variability, and sensitivity to forcing. A clear reduction in ensemble-mean forecast errors is demonstrated here in large ensembles of 15-day forecasts made with NOAA’s Global Forecast System model. The nearly ubiquitous reduction around the globe, obtained throughout the forecast range, is interpreted as arising in effect from a modification of the model’s deterministic evolution operator by a stochastic noise-induced drift. The effect is general in systems with state-dependent noise, and occurs even if the noise is not white. In the atmospheric context considered here, the effect is suggested to arise largely from noise-induced reductions of mechanical and thermal damping by chaotic boundary layer and cloud-radiative processes, which also tend to increase model sensitivity to forcing. The results presented here are consistent with many previous studies performed with models ranging from simple stochastically forced models to comprehensive global weather and climate models. They suggest that the diabatic interactions in most current global atmospheric models may not be sufficiently chaotic and this deficiency could be partly remedied by specifying additional stochastic terms. Using some empirical guidance in such specifications may be unavoidable, given the generally intractable complexity of the diabatic interactions.

Homogenization of Dissipative Hamiltonian Systems Under Lévy Fluctuations

We study the small mass limit for a class of Hamiltonian systems with multiplicative non-Gaussian Lévy noise. Derivation of the limiting equation depends on the structure of the stochastic Hamiltonian systems, in which a discontinuous noise-induced drift term arises. Firstly, we show that the momentum in the stochastic Hamiltonian system converges to zero when the kinetic energy has polynomial growth. Then, we prove that the stochastic Hamiltonian system with classical kinetic energy converges to the limiting equation in probability, with respect to Skorokhod topology as the mass tends to zero.

Anisotropic active Brownian particle with a fluctuating propulsion force.

The active Brownian particle (ABP) model describes a swimmer, synthetic or living, whose direction of swimming is a Brownian motion. The swimming is due to a propulsion force, and the fluctuations are typically thermal in origin. We present a two-dimensional model where the fluctuations arise from nonthermal noise in a propelling force acting at a single point, such as that due to a flagellum. We take the overdamped limit and find several modifications to the traditional ABP model. Since the fluctuating force causes a fluctuating torque, the diffusion tensor describing the process has a coupling between translational and rotational degrees of freedom. An anisotropic particle also exhibits a mass-dependent noise-induced drift, which does not disappear in the overdamped limit. We show that these effects have measurable consequences for the long-time diffusivity of active particles, in particular adding a contribution that is independent of where the force acts.

The Smoluchowski–Kramers limits of stochastic differential equations with irregular coefficients

In this paper, we study the small mass limit of the dynamic of stochastic system with state dependent friction and diffusion. We prove the strong convergence of the original system under weak assumptions on the coefficients, where the limiting equation admits an additional noise-induced drift term. Furthermore, the rate of convergence is also obtained. Our results even provide non-trivial improvement of the previous limit theorems for the classical Langevin equation.

A Smoluchowski–Kramers approximation for an infinite dimensional system with state-dependent damping

We study the validity of a Smoluchowski–Kramers approximation for a class of wave equations in a bounded domain of Rn subject to a state-dependent damping and perturbed by a multiplicative noise. We prove that in the small mass limit the solution converges to the solution of a stochastic quasilinear parabolic equation where a noise-induced extra drift is created.

Graph-Based Horizon Line Detection for UAV Navigation

Perceiving the horizon line is a critical alternative for unmanned aerial vehicle (UAV) autonomous navigation, especially in the presence of noise-induced drift, unavailability of satellite navigation, and multipath errors. However, its quite tough to detect the horizon line, due to the remotely sensed big data, the dynamic changes in flight, and the serious consequences of failure. To address these problems, we propose a graph-based horizon line detection technique that is composed of graph-based image segmentation, connected domain cascade filtering, horizon line extraction, and UAV attitude estimation. We improve the graph-based image segmentation algorithm so that the segmentation results are more conducive to horizon line detection. We then determine the sky-component by cascade filtering and extract the horizon line based on the boundaries of the sky-component. Furthermore, we directly compute the roll and pitch according to the extracted horizon line and eliminate the ambiguity of the angles. To validate our approach qualitatively and quantitatively, we designed a fixed-wing UAV system. We then validated our algorithm through extensive flights under various conditions and compared the estimated rolls and pitches to the IMU ones. Statistical results show that the proposed technique provides unbiased attitude angles with error variance of about 2, which verify the validity and robustness of our method. For engineering, our program runs at approximately 60 frames per second on the fly after optimizing.

An Averaging Approach to the Smoluchowski–Kramers Approximation in the Presence of a Varying Magnetic Field

We study the small mass limit of the equation describing planar motion of a charged particle of a small mass $\mu$ in a force field, containing a magnetic component, perturbed by a stochastic term. We regularize the problem by adding a small friction of intensity $\e>0$. We show that for all small but fixed frictions the small mass limit of $q_{\mu, \e}$ gives the solution $q_\e$ to a stochastic first order equation, containing a noise-induced drift term. Then, by using a generalization of the classical averaging theorem for Hamiltonian systems by Freidlin and Wentzell, we take the limit of the slow component of the motion $q_\e$ and we prove that it converges weakly to a Markov process on the graph obtained by identifying all points in the same connected components of the level sets of the magnetic field intensity function.

Synchronisation under shocks: The Lévy Kuramoto model

We study the Kuramoto model of identical oscillators on Erdős–Rényi (ER) and Barabasi–Alberts (BA) scale free networks examining the dynamics when perturbed by a Lévy noise. Lévy noise exhibits heavier tails than Gaussian while allowing for their tempering in a controlled manner. This allows us to understand how ‘shocks’ influence individual oscillator and collective system behaviour of a paradigmatic complex system. Skewed α-stable Lévy noise, equivalent to fractional diffusion perturbations, are considered, but overlaid by exponential tempering of rate λ. In an earlier paper we found that synchrony takes a variety of forms for identical Kuramoto oscillators subject to stable Lévy noise, not seen for the Gaussian case, and changing with α: a noise-induced drift, a smooth α dependence of the point of cross-over of synchronisation point of ER and BA networks, and a severe loss of synchronisation at low values of α. In the presence of tempering we observe both analytically and numerically a dramatic change to the α<1 behaviour where synchronisation is sustained over a larger range of values of the ‘noise strength’ σ, improved compared to the α>1 tempered cases. Analytically we study the system close to the phase synchronised fixed point and solve the tempered fractional Fokker–Planck equation. There we observe that densities show stronger support in the basin of attraction at low α for fixed coupling, σ and tempering λ. We then perform numerical simulations for networks of size N=1000 and average degree d̄=10. There, we compute the order parameter r as a function of σ for fixed α and λ and observe values of r≈1 over larger ranges of σ for α<1 and λ≠0. In addition we observe drift of both positive and negative slopes for different α and λ when native frequencies are equal, and confirm a sustainment of synchronisation down to low values of α. We propose a mechanism for this in terms of the basic shape of the tempered stable Lévy densities for various α and how it feeds into Kuramoto oscillator dynamics and illustrate this with examples of specific paths.

Noise-induced drift in two-dimensional anisotropic systems.

We study the isothermal Brownian dynamics of a particle in a system with spatially varying diffusivity. Due to the heterogeneity of the system, the particle's mean displacement does not vanish even if it does not experience any physical force. This phenomenon has been termed "noise-induced drift," and has been extensively studied for one-dimensional systems. Here, we examine the noise-induced drift in a two-dimensional anisotropic system, characterized by a symmetric diffusion tensor with unequal diagonal elements. A general expression for the mean displacement vector is derived and presented as a sum of two vectors, depicting two distinct drifting effects. The first vector describes the tendency of the particle to drift toward the high diffusivity side in each orthogonal principal diffusion direction. This is a generalization of the well-known expression for the noise-induced drift in one-dimensional systems. The second vector represents a novel drifting effect, not found in one-dimensional systems, originating from the spatial rotation in the directions of the principal axes. The validity of the derived expressions is verified by using Langevin dynamics simulations. As a specific example, we consider the relative diffusion of two transmembrane proteins, and demonstrate that the average distance between them increases at a surprisingly fast rate of several tens of micrometers per second.

Homogenization of dissipative, noisy, Hamiltonian dynamics

We study the dynamics of a class of Hamiltonian systems with dissipation, coupled to noise, in a singular (small mass) limit. We derive the homogenized equation for the position degrees of freedom in the limit, including the presence of a noise-induced drift term. We prove convergence to the solution of the homogenized equation in probability and, under stronger assumptions, in an Lp-norm. Applications cover the overdamped limit of particle motion in a time-dependent electromagnetic field, on a manifold with time-dependent metric, and the dynamics of nuclear matter.

Small Mass Limit of a Langevin Equation on a Manifold

We study damped geodesic motion of a particle of mass $m$ on a Riemannian manifold, in the presence of an external force and noise. Lifting the resulting stochastic differential equation to the orthogonal frame bundle, we prove that, as $m \to 0$, its solutions converge to solutions of a limiting equation which includes a {\it noise-induced drift} term. A very special case of the main result presents Brownian motion on the manifold as a limit of inertial systems.

Noise-induced bifurcations in magnetization dynamics of uniaxial nanomagnets

Stochastic magnetization dynamics in uniformly magnetized nanomagnets is considered. The system is assumed to have rotational symmetry as the anisotropy axis, the applied field, and the spin polarization are all aligned along an axis of symmetry. By appropriate integration of the Fokker-Planck equation associated to the problem, the stochastic differential equation governing the evolution of the angle between the magnetization orientation and the symmetry axis is derived. The drift terms present in this equation contain a noise-induced drift term, which, in combination with drift terms of deterministic origin, can be written as the derivative of an effective potential. Superparamagnetic-like transitions are studied in connections with the bifurcations of the effective potential as temperature and excitation conditions are varied.

Noise-induced speed up in repetitively firing neurons occurs far from spike threshold

Injecting white noise currents has been shown to increase mean firing rate in several types of neuron. This speed up is generally attributed to the fact that depolarizing currents on the tails of the noise distribution push cells past spike threshold sooner than they would have otherwise. As part of a larger effort to understand the dynamic properties of rat subthalamic (STN) neurons, we have performed perforated patch recordings of these tonically active neurons in brain slices, exposing them to contiguous sequences of brief current pulses drawn from a Gaussian distribution. Pulsed noise disrupted firing pattern with no significant change in rate over a wide range of noise amplitudes, so long as the pulse durations were brief (<2 ms). With longer pulse durations, neurons increased their firing rate with noise, with greater increases for pulses of larger magnitude and longer duration. However, individual neurons displayed varying degrees of noise-induced speed up. We used phase resetting methods to understand the noise-induced changes in the mean and variance of the recorded interspike interval distributions. For each cell, a phase response curves (PRC) was derived by using linear interpolation to estimate the phase of the neuron between spikes, followed by a linear regression to estimate the sensitivity of the cell to input perturbations at each phase. We then compared Monte Carlo simulations of a one-dimensional phase model to predict the response patterns of individual neurons. Surprisingly, simulation results produced a good match to experimental mean and variances even at noise levels leading to autocorrelation histograms with no visible periodicity (CVs ~ 0.4). We then investigated simulations results in detail to understand the dynamic mechanism that underlies the noise-induced speed up in these neurons. Plots of the mean phase as a function of time revealed that the noise-induced phase advance occurred early in an interspike interval, well before trajectories approach the time of the next spike. This phase advance can be understood as ‘noise-induced drift’ in the Stratanovich interpretation of stochastic differential equations, with possible supralinear contributions for pulses of finite duration. Current research uses artificially constructed PRCs to examine how the increasing spread of the phase distribution during each intervals leads to an asymmetric sampling of the noise-induced drift, preventing the early noise-induced speed up from being canceled by an expected noise-induced slowing as the PRC approaches zero near the time of the next spike.

Fundamental limits on persistent activity in networks of noisy neurons

Neural noise limits the fidelity of representations in the brain. This limitation has been extensively analyzed for sensory coding. However, in short-term memory and integrator networks, where noise accumulates and can play an even more prominent role, much less is known about how neural noise interacts with neural and network parameters to determine the accuracy of the computation. Here we analytically derive how the stored memory in continuous attractor networks of probabilistically spiking neurons will degrade over time through diffusion. By combining statistical and dynamical approaches, we establish a fundamental limit on the network's ability to maintain a persistent state: The noise-induced drift of the memory state over time within the network is strictly lower-bounded by the accuracy of estimation of the network's instantaneous memory state by an ideal external observer. This result takes the form of an information-diffusion inequality. We derive some unexpected consequences: Despite the persistence time of short-term memory networks, it does not pay to accumulate spikes for longer than the cellular time-constant to read out their contents. For certain neural transfer functions, the conditions for optimal sensory coding coincide with those for optimal storage, implying that short-term memory may be co-localized with sensory representation.

Noise-Induced Drift in Stochastic Differential Equations with Arbitrary Friction and Diffusion in the Smoluchowski-Kramers Limit

We consider the dynamics of systems with arbitrary friction and diffusion. These include, as a special case, systems for which friction and diffusion are connected by Einstein fluctuation-dissipation relation, e.g. Brownian motion. We study the limit where friction effects dominate the inertia, i.e. where the mass goes to zero (Smoluchowski-Kramers limit). {Using the It\^o stochastic integral convention,} we show that the limiting effective Langevin equations has different drift fields depending on the relation between friction and diffusion. {Alternatively, our results can be cast as different interpretations of stochastic integration in the limiting equation}, which can be parametrized by $\alpha \in \mathbb{R}$. Interestingly, in addition to the classical It\^o ($\alpha=0$), Stratonovich ($\alpha=0.5$) and anti-It\^o ($\alpha=1$) integrals, we show that position-dependent $\alpha = \alpha(x)$, and even stochastic integrals with $\alpha \notin [0,1]$ arise. Our findings are supported by numerical simulations.

The Impact of Rapid Wind Variability upon Air–Sea Thermal Coupling

Abstract The basic effect of extratropical atmosphere–ocean thermal coupling is to enhance the variance of both anomalous sea surface temperatures (SSTs) and air temperatures (AIRT) due to a decreased energy flux between the atmosphere and ocean, called reduced thermal damping. In this paper it is shown that rapidly varying surface winds, through their influence upon the turbulent surface heat fluxes that drive this coupling, act to effectively weaken the coupling and thus partially counteract the reduced thermal damping. In effect, rapid fluctuations in wind speed somewhat insulate the atmosphere and ocean from each other. The nonlinear relationship between the rapidly varying wind speed anomalies and SST and AIRT anomalies results in a rapidly varying component of the surface heat fluxes. The clear separation between the dynamical time scales of the ocean and atmosphere allows this rapidly varying flux to be simply approximated by a stochastic process in which rapidly varying wind speed is represented as Gaussian white noise whose amplitude is modulated by the more slowly evolving thermal anomalies. Such state-dependent (multiplicative) noise can alter the dynamics of atmosphere–ocean coupling because it induces an additional heat flux term, the noise-induced drift, that effectively acts to weaken both coupling and dissipation. Another key implication of the outlined theory is that air–sea coupling includes both deterministic and stochastic components. The theory is tested by examining daily observations during extended winter (November–April) at several ocean weather stations (OWSs). Two important results are found. First, multiplicative noise at OWS P effectively decreases the coupling by about one-third, with about a 10% (20%) decrease in the damping of SST (AIRT). This suggests that multiplicative noise may be responsible for roughly half of the AIRT variability at OWS P on subseasonal time scales. Second, OWS observations reveal that joint probability distribution functions of daily averaged SST and AIRT anomalies are significantly non-Gaussian. It is shown that treating the rapidly varying boundary layer heat fluxes as state-dependent noise can reproduce this observed non-Gaussianity. It is concluded that the effect of state-dependent noise is crucial to understand and model atmosphere–ocean coupling.

Electromagnetic and gravitational self-force on a relativistic particle from quantum fields in curved space

We provide a quantum field theoretical derivation of the Abraham-Lorentz-Dirac (ALD) equation, describing the motion of an electric point charge sourcing an electromagnetic field, which back-reacts on the charge as a self-force, and the Mino-Sasaki-Tanaka-Quinn-Wald (MSTQW) equation describing the motion of a point mass with self-force interacting with the linearized metric perturbations caused by the mass off an otherwise vacuous curved background spacetime. We regularize the formally divergent self-force by smearing the direct part of the retarded Green's function and using a quasilocal expansion. We also derive the ALD-Langevin and the MSTQW-Langevin equations with a classical stochastic force accounting for the effect of the quantum fluctuations in the field, which causes small fluctuations on the particle trajectory. These equations will be useful for studying the stochastic motion of charges and small masses under the influence of both quantum and classical noise sources, derived either self-consistently or put in by hand phenomenologically. We also show that history-dependent noise-induced drift motions could arise from such stochastic sources on the trajectory that could be a hidden feature of gravitational wave forms hitherto unknown.

Demonstration of a Controllable Three-Dimensional Brownian Motor in Symmetric Potentials

We demonstrate a Brownian motor, based on cold atoms in optical lattices, where isotropic random fluctuations are rectified in order to induce controlled atomic motion in arbitrary directions. In contrast to earlier demonstrations of ratchet effects, our Brownian motor operates in potentials that are spatially and temporally symmetric, but where spatiotemporal symmetry is broken by a phase shift between the potentials and asymmetric transfer rates between them. The Brownian motor is demonstrated in three dimensions and the noise-induced drift is controllable in our system.