Diffusion Coefficient of a Brownian Particle with a Friction Function Given by a Power Law

Nonequilibrium biological systems like moving cells or bacteria have been phenomenologically described by Langevin equations of Brownian motion in which the friction function depends on the particle’s velocity in a nonlinear way. An important subclass of such friction functions is given by power laws, i.e., instead of the Stokes friction constant γ ₀ one includes a function γ(v)∼v ^2α . Here I show using a recent analytical result as well as a dimension analysis that the diffusion coefficient is proportional to a simple power of the noise intensity D like D ^{(1−α)/(1+α)} (independent of spatial dimension). In particular the diffusion coefficient does not depend on the noise intensity at all, if α=1, i.e., for a cubic friction F _fric=−γ(v)v∼v ³. The exact prefactor is given in the one-dimensional case and a fit formula is proposed for the multi-dimensional problem. All results are confirmed by stochastic simulations of the system for α=1, 2, and 3 and spatial dimension d=1, 2, and 3. Conclusions are drawn about the strong noise behavior of certain models of self-propelled motion in biology.     

Stochastic resonance for signal-modulated pump noise in a single-mode laser

By adopting the gain-noise model of the single-mode laser in which with bias and periodical signals serve as inputs, combining with the effect of coloured pump noise, we use the linear approximation method to calculate the power spectrum and signal-to-noise ratio (SNR) of the laser intensity under the condition of pump noise and quantum noise cross-related in the form of δ function. It is found that with the change of pump noise correlation time, both SNR and the output power will occur stochastic resonance (SR). If the bias signal α is very small, changing the intensities of pump noise and quantum noise respectively does not lead to the appearance of SR in the SNR; while α increases to a certain number, SR appears.     

白交叉关联色噪声驱动的线性振子的扩散

研究了由白交叉相关的两色噪声所驱动的线性振子的扩散,其中一个噪声受偏置的周期信号所调制.揭示了乘性色噪声的自关联时间τ1和加性色噪声的自关联时间τ2对d-Q,d-D及d-Ω曲线的影响,其中d,Q,D和Ω分别为振子的扩散系数、受信号调制的噪声强度、未受调制的噪声强度和信号的频率.发现d-Q和d-D曲线均存在一个极小值,而d-Ω曲线则在Ω=～Ω处呈现极大值,～Ω为振子频率.推导了使扩散系数d=0的条件.     

Stochastic resonance of bias signal-modulated noise in a single-mode laser＊

Stochastic resonance (SR) for bias signal modulation is studied in a single-mode laser system. By investigating a gain-noise model driven by correlated pump noise and quantum noise, we find that, whether the correlation coefficient between both the noises is positive or negative, SR always appears in the dependence of signal-to-noise ratio (SNR) upon the noise correlation time and the frequency of the modulation signal. However, only when the correlation coefficient between both noises is negative can SR occur in the dependence of SNR upon the quantum noise intensity and pump noise intensity, while when the correlation coefficient between both noises is positive, it shows monotonically.     

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.     

Effects of time delay on transport processes in an active Brownian particle

The transport properties of an active Brownian particle with a time-delayed feedback and an external bias are investigated theoretically. By virtue of the perturbation theory for small delay, analytical expressions for the mean velocity and effective diffusion coefficient are derived. There exists a critical absolute value of the bias, below and above which the delay, respectively, enhances and weakens the diffusion, for a fixed noise intensity. The effects of delay observed above are more pronounced for weaker noise. These results are further verified via direct numerical simulations.     

Suppression of diffusion by a weak external field in periodic potentials

Diffusion of overdamped Brownian particles in sawtooth potentials subject to a spatially uniform tilt is studied focusing on the influence of a small bias. It is shown that in the potentials with an asymmetry in the direction of tilting force, the application of a weak external force leads to a suppression of diffusive motion: the diffusion coefficient as a function of bias passes through a minimum which precedes an increase of diffusion caused by delocalization of particles. In the weak noise limit the effect can be understood as a competition between the forward and backward escape rates over potential barriers determining the behaviour of the diffusion coefficient in a weak external field. The asymptotic lower border for the reduction of spreading of particles at fixed temperature is established. The decrease of diffusion is accompanied by more rapid increase of current.     

Automatic segmentation for brain MR images via a convex optimized segmentation and bias field correction coupled model

Accurate segmentation of magnetic resonance (MR) images remains challenging mainly due to the intensity inhomogeneity, which is also commonly known as bias field. Recently active contour models with geometric information constraint have been applied, however, most of them deal with the bias field by using a necessary pre-processing step before segmentation of MR data. This paper presents a novel automatic variational method, which can segment brain MR images meanwhile correcting the bias field when segmenting images with high intensity inhomogeneities. We first define a function for clustering the image pixels in a smaller neighborhood. The cluster centers in this objective function have a multiplicative factor that estimates the bias within the neighborhood. In order to reduce the effect of the noise, the local intensity variations are described by the Gaussian distributions with different means and variances. Then, the objective functions are integrated over the entire domain. In order to obtain the global optimal and make the results independent of the initialization of the algorithm, we reconstructed the energy function to be convex and calculated it by using the Split Bregman theory. A salient advantage of our method is that its result is independent of initialization, which allows robust and fully automated application. Our method is able to estimate the bias of quite general profiles, even in 7T MR images. Moreover, our model can also distinguish regions with similar intensity distribution with different variances. The proposed method has been rigorously validated with images acquired on variety of imaging modalities with promising results.     

Effects of signal modulation and coloured cross-correlation of coloured noises on the diffusion of a harmonic oscillator

The diffusion in a harmonic oscillator driven by coloured noises $\xi(t)$ and $\eta(t)$ with coloured cross-correlation in which one of the noises is modulated by a biased periodic signal is investigated. The exact expression of diffusion coefficient d as a function of noise parameter, signal parameter, and oscillator frequency is derived. The findings in this paper are as follows. 1) The curves of d versus noise intensity D and d versus noises cross-correlation time $\tau_3$ exist as two different phases. The transition between the two phases arises from the change of the cross-correlation coefficient $\lambda$ of the two Ornstein--Uhlenbeck (O-U) noises. 2) Changing the value of $\tau_3$, the curves of d versus Q, the intensity of colored noise that is modulated by the signal, can transform from a phase having a minimum to a monotonic phase. 3) Changing the value of signal amplitude A, d versus Q curves can transform from a phase having a minimum to a monotonic phase. The above-mentioned results demonstrate that a like noise-induced transition appears in the model.     

偏置信号调制下色关联噪声驱动的单模激光的光强相对涨落

采用线性化近似，计算了由具有色关联的受偏置信号调制的色泵噪声和色量子噪声驱动 的单模激光增益模型的光强相对涨落，发现相对涨落随噪声强度的变化曲线中存在极小值，分析了偏置信号的强度、周期信号频率、噪声间的互关联强度和互关联时间对曲线的影响； 对单模激光增益模型在偏置信号调制和直接信号调制两种方式下的输出光强相对涨落进行了比较，发现光强涨落与偏置信号的强度密切相关. 关键词： 偏置信号调制 光强相对涨落 线性化近似     

Escape process and stochastic resonance under noise intensity fluctuation

We study the effects of noise intensity fluctuations on the stationary and dynamical properties of an overdamped Langevin model with a bistable potential and external periodical driving force. We calculated the stationary distributions, mean-first passage time (MFPT) and the spectral amplification factor using a complete set expansion (CSE) technique. We found resonant activation (RA) and stochastic resonance (SR) phenomena in the system under investigation. Moreover, the strength of RA and SR phenomena exhibit non-monotonic behavior and their trade-off relation as a function of the squared variation coefficient of the noise intensity process. The reliability of CSE is verified with Monte Carlo simulations.     

Adatom dynamics in a periodic potential under time-periodic bias

We study the dynamics of a Brownian particle in a 1D external potential under the influence of a time-periodic bias with an amplitude small with respect to the potential barriers. We consider both a periodic potential corresponding to a smooth crystal surface and a regular array of steps with an extra Ehrlich–Schwoebel barrier for step crossing. For the smooth surface, we extend our previous work in the high friction limit to the low friction case and find that the oscillating bias enhances the diffusion coefficient D_T due to the broadening of the jump length distribution. In the case of a stepped surface with terraces of length L, the bias induces a non-zero average current J_ave in the direction of descending steps as long as the driving frequency is smaller than a threshold frequency Ω_T≈L⁻¹. The current shows a maximum as a function of temperature for fixed L. However, no evidence of stochastic resonance type of enhancement can be found either in D_T or J_ave.     

Diffusion tensor imaging at low SNR: nonmonotonic behaviors of tensor contrasts

Diffusion tensor imaging (DTI) provides measurements of directional diffusivities and has been widely used to characterize changes in the tissue microarchitecture of the brain. DTI is gaining prominence in applications outside of the brain, where resolution, motion and short T2 values often limit the achievable signal-to-noise ratio (SNR). Consequently, it is important to revisit the topic of tensor estimation in low-SNR regimes. A theoretical framework is developed to model noise in DTI, and by using simulations based on this theory, the degree to which the noise, tensor estimation method and acquisition protocol affect tensor-derived quantities, such as fractional anisotropy and apparent diffusion coefficient, is clarified. These results are then validated against clinical data. It is shown that reliability of tensor contrasts depends on the noise level, estimation method, diffusion-weighting scheme and underlying anatomy. The propensity for bias and errors does not monotonically increase with noise. Comparative results are shown in both graphical and tabular forms, so that decisions about suitable acquisition protocols and processing methods can be made on a case-by-case basis without exhaustive experimentation.     

Effect on Intensity Correlation Time by Input Signal in a Single-Mode Laser with Bias Signal Modulation

The effect on intensity correlation time T by input signal is studied for gain-noise model of a single-mode laser driven by colored pump noise and colored quantum noise with colored cross-correlation with a bias signal modulation in this paper. By using the linear approximation method, we detect that there exists maximum （i.e., resonance） in the curve of the intensity correlation time T upon bias current io when the noise correlation coefficient λ is positive; and there exists minimum （i.e., suppression） in the T-io curve when λ is negative. And whenλ is zero, T increases monotonously with increasing io. Furthermore, the curve of T upon the signal frequency Ω is also studied. Our study shows that no matter what the value of λ is, there exists minimum （i.e., suppression） in the T-Ω curve.     

Curious behaviour of the diffusion coefficient and friction force for the strongly inhomogeneous HMF model

We present first elements of kinetic theory appropriate to the inhomogeneous phase of the Hamiltonian Mean Field (HMF) model. In particular, we investigate the case of strongly inhomogeneous distributions for T→0 and exhibit curious behaviour of the force auto-correlation function and friction coefficient. The temporal correlation function of the force has an oscillatory behaviour which averages to zero over a period. By contrast, the effects of friction accumulate with time and the friction coefficient does not satisfy the Einstein relation. On the contrary, it presents the peculiarity to increase linearly with time. Motivated by this result, we provide analytical solutions of a simplified kinetic equation with a time dependent friction. Analogies with self-gravitating systems and other systems with long-range interactions are also mentioned.     

Effects of solute-solvent and solvent-solvent attractive interactions on solute diffusion

A study is reported on the diffusion process of a solute molecule in a Lennard-Jones-like liquid near the triple point by a molecular dynamics simulation. Systematic changes were made to the strength of the solute-solvent or solvent-solvent attractive interaction in order to elucidate its effects on the diffusion coefficient. When the solute-solvent attractive interaction is enhanced, the diffusion coefficient of the solute becomes much smaller than that predicted by the Stokes-Einstein relationship with a stick boundary condition. The generalized friction coefficient on the solute molecule was investigated, and the attractive force between solute and solvent is found to be the main cause for the enhancement of the friction. When the attractive interaction between solvent molecules is weakened, the diffusion coefficient of a solute does not change, whereas that of a solvent does. Compared with the shear viscosity of the solvent, the diffusion coefficient of the solute breaks the Stokes-Einstein relationship, whereas the Stokes-Einstein relationship appears to hold in the case of the solvent molecule.     

Lifetime of the superconductive state in short and long Josephson junctions

We study the transient statistical properties of short and long Josephson junctions under the influence of thermal and correlated fluctuations. In particular, we investigate the lifetime of the superconductive metastable state finding the presence of noise induced phenomena. For short Josephson junctions we investigate the lifetime as a function both of the frequency of the current driving signal and the noise intensity and we find how these noise-induced effects are modified by the presence of a correlated noise source. For long Josephson junctions we integrate numerically the sine-Gordon equation calculating the lifetime as a function of the length of the junction both for inhomogeneous and homogeneous bias current distributions. We obtain a non-monotonic behavior of the lifetime as a function of the frequency of the current driving signal and the correlation time of the noise. Moreover we find two maxima in the non-monotonic behaviour of the mean escape time as a function of the correlated noise intensity.     

Stochastic resonance in a bias linear system with multiplicative and additive noise

In this paper, the stochastic resonance in a bias linear system subjected multiplicative and additive dichotomous noise is investigated. Using the linear-response theory and the properties of the dichotomous noise, this paper finds the exact expressions for the first two moments and the signal-to-noise ratio (SNR). It is shown that the SNR is a non-monotonic function of the correlation time of the multiplicative and additive noise, and it varies non-monotonously with the intensity and asymmetry of the multiplicative noise as well as the external field frequency. Moreover, the SNR depends on the system bias, the intensity of the cross noise between the multiplicative and additive noise, and the strength and asymmetry of the additive noise.     

Magnetic Resonance Imaging Segmentation via Weighted Level Set Model Based on Local Kernel Metric and Spatial Constraint

Magnetic resonance imaging (MRI) segmentation is a fundamental and significant task since it can guide subsequent clinic diagnosis and treatment. However, images are often corrupted by defects such as low-contrast, noise, intensity inhomogeneity, and so on. Therefore, a weighted level set model (WLSM) is proposed in this study to segment inhomogeneous intensity MRI destroyed by noise and weak boundaries. First, in order to segment the intertwined regions of brain tissue accurately, a weighted neighborhood information measure scheme based on local multi information and kernel function is designed. Then, the membership function of fuzzy c-means clustering is used as the spatial constraint of level set model to overcome the sensitivity of level set to initialization, and the evolution of level set function can be adaptively changed according to different tissue information. Finally, the distance regularization term in level set function is replaced by a double potential function to ensure the stability of the energy function in the evolution process. Both real and synthetic MRI images can show the effectiveness and performance of WLSM. In addition, compared with several state-of-the-art models, segmentation accuracy and Jaccard similarity coefficient obtained by WLSM are increased by 0.0586, 0.0362 and 0.1087, 0.0703, respectively.     

Coarse grained approach for volume conserving models

Volume conserving surface (VCS) models without deposition and evaporation, as well as ideal molecular-beam epitaxy models, are prototypes to study the symmetries of conserved dynamics. In this work we study two similar VCS models with conserved noise, which differ from each other by the axial symmetry of their dynamic hopping rules. We use a coarse-grained approach to analyze the models and show how to determine the coefficients of their corresponding continuous stochastic differential equation (SDE) within the same universality class. The employed method makes use of small translations in a test space which contains the stationary probability density function (SPDF). In case of the symmetric model we calculate all the coarse-grained coefficients of the related conserved Kardar–Parisi–Zhang (KPZ) equation. With respect to the symmetric model, the asymmetric model adds new terms which have to be analyzed, first of all the diffusion term, whose coarse-grained coefficient can be determined by the same method. In contrast to other methods, the used formalism allows to calculate all coefficients of the SDE theoretically and within limits numerically. Above all, the used approach connects the coefficients of the SDE with the SPDF and hence gives them a precise physical meaning.