Transport as a consequence of state-dependent diffusion

Overdamped particles subject to a drift in a force field with sinusoidal space dependence and also a sinusoidally modulated space-dependent diffusion, with the same period as the drift, experience a net driving force. The resulting current depends on the amplitude of the modulation of the diffusion and is a periodic function of the phase difference between the sinusoidal drift and the sinusoidal modulation of the diffusion. For small modulation amplitudes a particle subject to state-dependent noise behaves the same way as a particle subject to thermal noise but with a drift which, in addition to the sinusoidal term, contains a net force term [M. Büttiker,Z. Phys. B 68:161 (1987)]. A specific example of this behavior [N. G. van Kampen,IBM J. Res. Dev. 32:107 (1988); R. Landauer,J. Stat. Phys. 53:233 (1988).] is the motion of overdamped particles in a ring subject to a nonuniform temperature field. When the drift and the temperature, which are periodic with a period equal to the ring circumference, are not in phase a noise-induced circulating current results.This paper will appear in a forthcoming issue of theJournal of Statistical Physics.     

Effective long-time phase dynamics of limit-cycle oscillators driven by weak colored noise

An effective white-noise Langevin equation is derived that describes long-time phase dynamics of a limit-cycle oscillator driven by weak stationary colored noise. Effective drift and diffusion coefficients are given in terms of the phase sensitivity of the oscillator and the correlation function of the noise, and are explicitly calculated for oscillators with sinusoidal phase sensitivity functions driven by two typical colored Gaussian processes. The results are verified by numerical simulations using several types of stochastic or chaotic noise. The drift and diffusion coefficients of oscillators driven by chaotic noise exhibit anomalous dependence on the oscillator frequency, reflecting the peculiar power spectrum of the chaotic noise.     

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.     

One-Dimensional Steady Transport by Molecular Dynamics Simulation：Non-Boltzmann Position Distribution and Non-Arrhenius Dynamical Behavior

A non-equilibrium steady state can be characterized by a nonzero but stationary flux driven by a static external force. Under a weak external force, the drift velocity is difficult to detect because the drift motion is feeble and submerged in the intense thermal diffusion. In this article, we employ an accurate method in molecular dynamics simulation to determine the drift velocity of a particle driven by a weak external force in a one-dimensional periodic potential. With the calculated drift velocity, we found that the mobility and diffusion of the particle obey the Einstein relation, whereas their temperature dependences deviate from the Arrhenius law. A microscopic hopping mechanism was proposed to explain the non-Arrhenius behavior. Moreover, the position distribution of the particle in the potential well was found to deviate from the Boltzmann equation in a non-equilibrium steady state. The non-Boltzmann behavior may be attributed to the thermostat which introduces an effective "viscous" drag opposite to the drift direction of the particle.     

随机外磁场作用下Ising自旋体系的随机共振

通过采用数字仿真手段，研究了经平均场处理的Ising自旋体系在弱确定性周期外场和随机外场（噪声场）混合驱动下的动态响应行为.着重考察了不同强度混合驱动外场作用下，Is ing自旋体系的非平衡动态转变所表现出区别于单纯确定性周期场作用下的新特征——随机 共振.选定体系非平衡动态转变的动态序参量Q为表征参量，系统模拟计算了混合驱动外场在多种参数组合下体系的动态响应特征，高场低温下的非连续动态转变和低场高温下的连续动态转变.模拟计算表明在适当混合驱动外场的作用下，Ising自旋体系具有随机共振现象，并诱发形成非 关键词： Ising自旋体系 随机共振 动态相变 对称性     

A gas of Brownian particles in statistical dynamics

We consider a large number of particles on a one-dimensional latticel Z in interaction with a heat particle; the latter is located on the bond linking the position of the particle to the point to which it jumps. The energy of a single particle is given by a potentialV(x), x∈Z. In the continuum limit, the classical version leads to Brownian motion with drift. A quantum version leads to a local drift velocity which is independent of the applied force. Both these models obey Einstein's relation between drift, diffusion, and applied force. The system obeys the first and second laws of thermodynamics, with the time evolution given by a pair of coupled non linear heat equations, one for the density of the Brownian particles and one for the heat occupation number; the equation for a tagged Brownian particle can be written as a stochastic differential equation.     

Dynamic axial-position control of a laser-trapped particle by wave-front modification

The axial position of a laser-trapped particle has been controlled by modification of the wave front by means of a membrane deformable mirror. The mirror gives wave-front modulation in terms of Zernike polynomials. By modulation of the Zernike defocus term we can modulate the particle position under conditions of laser trapping. A polystyrene particle of 1-microm diameter was moved along the optical axis direction for a distance of 2370 nm in minimum steps of 55.4 nm. We also demonstrated particle oscillation along the optical axis by changing the focal position in a sinusoidal manner. From the frequency dependency of the amplitude of particle oscillation we determined the spring constant as 91.7 nN/m.     

Noise-induced transport with low randomness

We study the transport of overdamped Brownian particles in periodic potentials subject to a spatially modulated Gaussian white noise. We derive an analytical expression for the diffusion coefficient of particles. By means of velocity, diffusion coefficient, and their ratio (Péclet number) we discuss (a) symmetric potential and modulation of noise intensity and (b) a ratchet profile with strong noise modulation. It is shown that state dependent fluctuations may not only induce directed transport, but also a pronounced coherence of transport if the potential exhibits a strong asymmetry.     

Influence of a uniform driving force on tracer diffusion in a one-dimensional hard-core lattice gas

The influence of a uniform driving force on tracer diffusion is investigated for a one-dimensional lattice gas where particles jump stochastically to unoccupied neighboring sites. A new, simple calculation is presented for the diffusion coefficient of a tracer particle with respect to its average drift, obtained recently by rigorous methods by De Masi and Ferrari. A theoretical expression describing the tracer particle mean square displacement approximately for all times is derived and found to be in excellent agreement with the results of Monte Carlo simulations.     

Transport in a ratchet with spatial disorder and time-delayed feedback

A Brownian motor with Gaussian short-range correlated spatial disorder and time-delayed feedback is investigated. The effects of disorder intensity, correlation strength and delay time on the transport properties of an overdamped periodic ratchet are discussed for different driving force. For small driving force, the disorder intensity can induce a peak in the drift motion and a linear increasing function in diffusion motion. For large driving force, the disorder intensity can suppress the drift motion but enhance the diffusion motion. For both small and large driving forces, the correlation strength of the spatial disorder can enhance the drift motion but suppress the diffusion motion. While the delay time can reduce the drift motion to a small value and enhance the diffusion motion to a large value. The drift motion increases as the driving force increases. However, the diffusion motion is either decreases or only increases slightly when the driving force increases.     

Singular Limit of a Reaction-Diffusion Equation with a Spatially Inhomogeneous Reaction Term

We study reaction-diffusion equations with a spatially inhomogeneous reaction term. If the coefficient of these reaction term is much larger than the diffusion coefficient, a sharp interface appears between two different phases. We show that the equation of motion of such an interface involves a drift term despite the absence of drift in the original diffusion equations. In particular, we show that the same rich spatial patterns observed for a chemotaxis-growth model can be realized by a system without a drift term.     

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.     

Temporal interactions between pure tones and amplitude-modulated noise

An auditory interaction between the temporal fine structure of a low-frequency tone and the envelope of a high-frequency waveform was observed at very large frequency separations. Thresholds for detection of sinusoidal amplitude modulation of a high-frequency, narrow-band noise were measured as a function of the relative phase between the modulator and a pure tone with the same frequency as the modulator. These "phase functions" were determined at various intensities of the noise and tone for three different modulation frequencies. In general, the phase functions show that low-frequency stimulation has a cyclic effect on the sensitivity to amplitude modulation; over a limited range of relative phases, the modulation threshold is lower than that measured without low-frequency stimulation whereas over a broader range of relative phases, the modulation threshold is much higher. The difference between minimum and maximum modulation thresholds was observed to be as great as 23 dB. Despite this substantial degree of temporal interaction, little, if any, masking by the low-frequency tone of the high-frequency noise was observed.     

Nonlinear diffusion under a time dependent external force: q-maximum entropy solutions

Nonlinear diffusion equations provide useful models for a number of interesting phenomena, such as diffusion processes in porous media. We study here a family of nonlinear Fokker-Planck equations endowed both with a power-law nonlinear diffusion term and a drift term with a time dependent force linear in the spatial variable. We show that these partial differential equations exhibit exact time dependent particular solutions of the Tsallis maximum entropy (q-MaxEnt) form. These results constitute generalizations of previous ones recently discussed in the literature [C. Tsallis, D.J. Bukman, Phys. Rev. E 54, R2197 (1996)], concerning q-MaxEnt solutions to nonlinear Fokker-Planck equations with linear, time independent drift forces. We also show that the present formalism can be used to generate approximate q-MaxEnt solutions for nonlinear Fokker-Planck equations with time independent drift forces characterized by a general spatial dependence. Received 25 April 2001 and Received in final form 6 June 2001     

高斯白噪声相位调制的激光光谱展宽

激光器是现代光学中一种常用的光源,分析其谱宽特性对于激光的研究具有重要意义.由于高性能窄线宽激光的半高全宽通常在几十M Hz以内,难以直接满足宽光谱领域的应用需求,因而无法发挥其在成本和性能方面的优势,这在一定程度上限制了激光的发展.如果在激光内部直接进行调制实现线宽展宽,又会导致频率的严重漂移,破坏稳频特性.为了在激...     

Entropy and Vacuum Radiation

It is shown that entropy increase in thermodynamic systems can plausibly be accounted for by the random action of vacuum radiation. A recent calculation by Rueda using stochastic electrodynamics (SED) shows that vacuum radiation causes a particle to undergo a rapid Brownian motion about its average dynamical trajectory. It is shown that the magnitude of spatial drift calculated by Rueda can also be predicted by assuming that the average magnitudes of random shifts in position and momentum of a particle correspond to the lower limits of the uncertainty relation. The latter analysis yields a plausible expression for the shift in momentum caused by vacuum radiation. It is shown that when the latter shift in momentum is magnified in particle interactions, the fractional change in each momentum component is on the order of unity within a few collision times, for gases and (plausibly) for denser systems over a very broad range of physical conditions. So any system of particles in this broad range of conditions would move to maximum entropy, subject to its thermodynamic constraints, within a few collision times. It is shown that the spatial drift caused by vacuum radiation, as predicted by the above SED calculation, can be macroscopic in some circumstances, and an experimental test of this effect is proposed. Consistency of the above results with quantum mechanics is discussed, and it is shown that the diffusion constant associated with the above Brownian drift is the same as that used in stochastic interpretations of the Schrödinger equation.     

Quantifying the distortion products generated by amplitude-modulated noise

When sinusoidal amplitude modulation (SAM) is applied to noise or tone carriers, the stimuli can generate audible distortion products in the region of the modulation frequency. As a result, when bandpass-filtered SAM noise is used to investigate temporal processing, a band of unmodulated noise is typically positioned at the modulation frequency to mask any distortion products. This study was designed to investigate the distortion products for bandpass noise carriers, and so reduce ambiguity about the form of this distortion and its role in perception. The distortion consists of two distortion-noise bands and a distortion tone at the modulation frequency. In the first two experiments, the level and phase of the distortion tone are measured using two different experimental paradigms. In the third experiment, modulation-frequency difference limens are measured for filtered SAM noise and it is shown that performance deteriorates markedly when the distortion tone is canceled. In a fourth experiment, masked threshold is measured at low frequencies for bands of high-frequency, unmodulated noise with the same levels and spectra as the SAM noises in the earlier experiments. The results confirm that unmodulated noise also produces quadratic distortion which may explain some aspects of earlier reports on remote masking.     

Second-order temporal modulation transfer functions

Detection thresholds were measured for a sinusoidal modulation applied to the modulation depth of a sinusoidally amplitude-modulated (SAM) white noise carrier as a function of the frequency of the modulation applied to the modulation depth (referred to as f'm). The SAM noise acted therefore as a "carrier" stimulus of frequency fm, and sinusoidal modulation of the SAM-noise modulation depth generated two additional components in the modulation spectrum: fm-f'm and fm+f'm. The tracking variable was the modulation depth of the sinusoidal variation applied to the "carrier" modulation depth. The resulting "second-order" temporal modulation transfer functions (TMTFs) measured on four listeners for "carrier" modulation frequencies fm of 16, 64, and 256 Hz display a low-pass segment followed by a plateau. This indicates that sensitivity to fluctuations in the strength of amplitude modulation is best for fluctuation rates f'm below about 2-4 Hz when using broadband noise carriers. Measurements of masked modulation detection thresholds for the lower and upper modulation sideband suggest that this capacity is possibly related to the detection of a beat in the sound's temporal envelope. The results appear qualitatively consistent with the predictions of an envelope detector model consisting of a low-pass filtering stage followed by a decision stage. Unlike listeners' performance, a modulation filterbank model using Q values > or = 2 should predict that second-order modulation detection thresholds should decrease at high values of f'm due to the spectral resolution of the modulation sidebands (in the modulation domain). This suggests that, if such modulation filters do exist, their selectivity is poor. In the latter case, the Q value of modulation filters would have to be less than 2. This estimate of modulation filter selectivity is consistent with the results of a previous study using a modulation-masking paradigm [S. D. Ewert and T. Dau, J. Acoust. Soc. Am. 108, 1181-1196 (2000)].     

Modulated Weiner process

Summary We define and develop a novel process, related to stochastic resonance, in which a particle experiences three forces: a constant drift, a zero-mean white noise, and a time-periodic modulation. Upon reaching a threshold, the particle immediately returns to the starting point. The resulting process exhibits multiple maxima in the output power at the modulation frequency as a function of the white-noise variance, multimodal first-passage-time densities, and evidence of phase locking. Our model is an extension of the Gerstein-Mandelbrot model of neuron firing to the case of periodic stimuli, and therefore has applications in neural modeling. Paper presented at the International Workshop ?Fluctuations in Physics and Biology: Stochastic Resonance, Signal Processing and Related Phenomena?, Elba, 5–10 June 1994.