Effect of symmetry on volume conserving surface models

We study the effect of symmetry on volume conserving models without deposition and evaporation. By using the master equation approach, we identify two types of stochastic continuum equation with a conservative noise, depending on the symmetry of hopping rate in diffusion rules. In the model with symmetric hopping rate, a Laplacian term is essentially absent from the continuum equation. The dynamic scaling of this model is thus determined by the nonlinear fourth order equation with a conservative noise. When the symmetry is broken, a Laplacian term may be present, so the asymptotic scaling behavior is governed by the Laplacian term with nonzero coefficient. We verify this result by investigating a simple discrete model analytically.     

Long time tails in stationary random media II: Applications

In a previous paper we developed a mode-coupling theory to describe the long time properties of diffusion in stationary, statistically homogeneous, random media. Here the general theory is applied to deterministic and stochastic Lorentz models and several hopping models. The mode-coupling theory predicts that the amplitudes of the long time tails for these systems are determined by spatial fluctuations in a coarse-grained diffusion coefficient and a coarse-grained free volume. For one-dimensional models these amplitudes can be evaluated, and the mode-coupling theory is shown to agree with exact solutions obtained for these models. For higher-dimensional Lorentz models the formal theory yields expressions which are difficult to evaluate. For these models we develop an approximation scheme based upon projecting fluctuations in the diffusion coefficient and free volume onto fluctuations in the density of scatterers.Work supported by grant No. CHE 77-16308 from the National Science Foundation and by a Nato Travel Grant.     

Exact propagator of the Fokker-Planck equation with a space-dependent diffusion coefficient and a time-dependent mean-reverting force

We have investigated the algebraic structure of the Fokker-Planck equation with a variable diffusion coefficient and a time-dependent mean-reverting force. Such a model could be useful to study the general problem of a Brownian walker with a space-dependent diffusion coefficient. We also show that this model is related to the Fokker-Planck equation with a constant diffusion coefficient and a time-dependent anharmonic potential of the form V(x, t) = ?a(t)x ² + b ln x, which has been widely applied to model different physical and biological phenomena, e.g. the study of neuron models and stochastic resonance in monostable nonlinear oscillators. Using the Lie algebraic approach we have derived the exact diffusion propagators for the Fokker-Planck equations associated with different boundary conditions, namely (i) the case of a single absorbing barrier, and (ii) the case of two absorbing barriers. These exact diffusion propagators enable us to study the time evolution of the corresponding stochastic systems. Received 23 October 2001 and Received in final form 24 December 2001     

A Stochastic Differential Equation Model with Jumps for Fractional Advection and Dispersion

The path of a tracer particle through a porous medium is typically modeled by a stochastic differential equation (SDE) driven by Brownian noise. This model may not be adequate for highly heterogeneous media. This paper extends the model to a general SDE driven by a Lévy noise. Particle paths follow a Markov process with long jumps. Their transition probability density solves a forward equation derived here via pseudo-differential operator theory and Fourier analysis. In particular, the SDE with stable driving noise has a space-fractional advection-dispersion equation (fADE) with variable coefficients as the forward equation. This result provides a stochastic solution to anomalous diffusion models, and a solid mathematical foundation for particle tracking codes already in use for fractional advection equations.     

随机磁场对静电波驱动的速度扩散的影响

随机磁场改变了波与粒子之间的耦合关系。因而使波驱动的速度扩散受到影响。其结果是:波电场的横向分量可以对纵向速度扩散有贡献;∈<<1情形,扩散系数的共振峰被展宽;∈>>1情形,扩散系数的振荡效应被削弱。当随机磁场的关联时间与波的特征时间之间相对大小不同时,随机磁场产生影响的具体机制不完全相同,所造成的后果也有差别。对于接近于在垂直方向传播的波,随机磁场对速度扩散的影响一般是重要的。     

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.     

A theorem allowing to derive deterministic evolution equations from stochastic evolution equations, II: The non-Markovian extension

Deterministic evolution equations of classical as well as quantum mechanical models are derived from a set of non-Markovian stochastic evolution equations after an average over realization using a theorem. Examples are given, show that deterministic differential equations that contain derivatives with respect to time higher than or equal to two can be derived after a Taylor series expansion of the dynamical variables. It is shown that the derivation of such deterministic differential equations can be done by solving a set of linear equations that increase in number after increasing the number of previous time steps in the updating rules that define a given model. Two explicit examples, the first containing updating rules that depend on two previous time steps and the second on three, are worked in some detail in order to show some features of the linear transformation that allow one to obtain the deterministic differential equations.     

Large Scale Dynamics of the Persistent Turning Walker Model of Fish Behavior

This paper considers a new model of individual displacement, based on fish motion, the so-called Persistent Turning Walker (PTW) model, which involves an Ornstein-Uhlenbeck process on the curvature of the particle trajectory. The goal is to show that its large time and space scale dynamics is of diffusive type, and to provide an analytic expression of the diffusion coefficient. Two methods are investigated. In the first one, we compute the large time asymptotics of the variance of the individual stochastic trajectories. The second method is based on a diffusion approximation of the kinetic formulation of these stochastic trajectories. The kinetic model is a Fokker-Planck type equation posed in an extended phase-space involving the curvature among the kinetic variables. We show that both methods lead to the same value of the diffusion constant. We present some numerical simulations to illustrate the theoretical results.     

Four-dimensional mapping model for two-frequency electron cyclotron resonance heating

Two-frequency electron cyclotron resonance heating (ECRH) is modelled by a four-dimensional symplectic mapping derived from the nonrelativistic single particle equations of motion. The model includes changes in parallel energy due to the spatially separate resonance zones, not given by previous two-dimensional models. Fixed points are located and their linear stability limits determined. Resonances in action space are calculated along with their widths and used to obtain the adiabatic barrier to heating. Quasilinear diffusion coefficients are derived for the stochastic regime and found to agree well with numerical calculations. The primary diffusion in perpendicular energy can couple to the parallel motion, leading to diffusion in parallel energy. The resulting diffusion coefficient is calculated analytically and compared with numerical results. The much weaker Arnold diffusion along a resonance layer is also treated analytically, yielding diffusion coefficients in reasonable agreement with numerical values.     

Application of laser beam deflection technique to study the diffusion process in electrolyte solutions

A simple method based on laser beam deflection to study the variation of diffusion coefficient with concentration in a solution is presented. When a properly fanned out laser beam is passed through a rectangular cell filled with solution having concentration gradient, the emergent beam traces out a curved pattern on a screen. By taking measurements on the pattern at different concentrations, the variation of diffusion coefficient with concentration can be determined.     

Universality Class in Abelian Sandpile Models with Stochastic Toppling Rules

We present a stochastic critical slope sandpile model, where the amount of grains that fall in an overturning event is stochastic variable. The model is local, conservative, and Abelian. We apply the moment analysis to evaluate critical exponents and finite size scaling method to consistently test the obtained results. Numerical results show that this model, Oslo model, and one-dimensional Abelian Manna model have the same critical behavior although the three models have different stochastic toppling rules, which provides evidences suggesting that Abelian sandpile models with different stochastic toppling rules are in the same universality class.     

On a conjecture of Alley and Alder for fluids and Lorentz models

We discuss a conjecture of Alley and Alder predicting a relation between the four-point and the two-point velocity autocorrelation functions for fluids and Lorentz models at sufficiently long times. If the conjecture is correct a modified Burnett coefficient can be defined, which has a finite value, contrary to the ordinary Burnett coefficient, which is divergent. The conjecture is tested for four classes of models with different methods: for three-dimensional fluids mode-coupling theory yields a negative result. The conjecture is confirmed for thed-dimensional deterministic Lorentz gas (d 2) and for a class ofd-dimensional stochastic Lorentz models (d 1) by low-density kinetic theory, as well as by rigorous results, available for one dimension. For yet another class of one-dimensional stochastic Lorentz models, which are exactly solvable in one dimension, the result is negative again. All four classes of models show long-time tails in the velocity autocorrelation function and have a finite diffusion coefficient.     

Note on coefficient matrices from stochastic Galerkin methods for random diffusion equations

In a recent work by Xiu and Shen [D. Xiu, J. Shen, Efficient stochastic Galerkin methods for random diffusion equations, J. Comput. Phys. 228 (2009) 266–281], the Galerkin methods are used to solve stochastic diffusion equations in random media, where some properties for the coefficient matrix of the resulting system are provided. They also posed an open question on the properties of the coefficient matrix. In this work, we will provide some results related to the open question.     

Universality Class in Abelian Sandpile Models with Stochastic Toppling Rules

We present a stochastic critical slope sandpile model, where the amount of grains that fall in an overturning event is stochastic variable. The model is local, conservative, and Abelian. We apply the moment analysis to evaluate critical exponents and finite size scaling method to consistently test the obtained results. Numerical results show that this model, Oslo model, and one-dimensional Abelian Manna model have the same critical behavior although the three models have different stochastic toppling rules, which provides evidences suggesting that Abelian sandpile models with different stochastic toppling rules are in the same universality class.     

Thermo-Quantum Diffusion

A new approach to the thermo-quantum diffusion is proposed and a nonlinear quantum Smoluchowski equation is derived, which describes classical diffusion in the field of the Bohm quantum potential. A nonlinear thermo-quantum expression for the diffusion front is obtained, being a quantum generalization of the classical Einstein law. The quantum diffusion at zero temperature is also described and a new dependence of the position dispersion on time is derived. A stochastic Bohm-Langevin equation is also proposed.     

Fractional nonlinear diffusion equation and first passage time

We investigate the solutions and the first passage time for anomalous diffusion processes governed by the fractional nonlinear diffusion equation with a space- and time-dependent diffusion coefficient subject to absorbing boundaries and the initial condition. We obtain explicit analytical expression for the probability distribution, the first passage time distribution, the mean first passage time, and the mean squared displacement corresponding to different time-dependent diffusion coefficient. In addition, we compare our results for the first passage time distribution and the mean first passage time with the one obtained by usual linear diffusion equation with time-dependent diffusion coefficient.     

Stochastic quantization and detailed balance in Fokker-Planck dynamics

The path integral and operator formulations of the Fokker-Planck equation are considered as stochastic quantizations of underlying Euler-Lagrange equations. The operator formalism is derived from the path integral formalism. It is proved that the Euler-Lagrange equations are invariant under time reversal if detailed balance holds and it is shown that the irreversible behavior is introduced through the stochastic quantization. To obtain these results for the nonconstant diffusion Fokker-Planck equation, a transformation is introduced to reduce it to a constant diffusion Fokker-Planck equation. Critical comments are made on the stochastic formulation of quantum mechanics.     

Determination of binary diffusion coefficients of gases using photothermal deflection technique

A photothermal deflection (PD) technique was applied to measure the binary diffusion coefficients of various gases (CO₂–N₂, CO₂–O₂, N₂–He, O₂–He, and CO₂–He). With an in-house-made Loschmidt diffusion cell, a transverse PD system was employed to measure the time-resolved PD signal associated with the variation of the thermal diffusivity and the temperature coefficient of the refractive index of the gas mixture during the diffusion. The concentration evolution of the gas mixture was deduced from the PD amplitude and phase signals based on our diffraction PD model and was processed using two mass-diffusion models explored in this work for both short- and long-time diffusions to find the diffusion coefficient. An optical fiber oxygen sensor was also used to measure the concentration changes of the mixtures with oxygen. Experimental results demonstrated that the binary diffusion coefficients precisely measured with the PD technique were in agreement with the literature values. Moreover, the PD technique can measure the diffusion coefficients of various gas mixtures with both short- and long-time diffusions. In contrast, the oxygen sensor is only suitable for the long-time diffusion measurements of the gas mixtures with oxygen. PACS 78.20.Nv; 51.20.+d     

基于海杂波散射特性的微弱信号检测方法

从研究海杂波的电磁散射特性出发,利用随机微分理论对海杂波的物理特性进行了系统地分析.首先建立了海杂波电磁散射所满足的随机微分方程,然后利用Itô公式得到了海杂波散射信号幅度和相位的扩散过程模型,最后利用海杂波散射信号的规律给出了微弱信号检测的相关处理方法. 关键词： 随机微分方程 海杂波 散射 微弱信号     

Dependence of chaotic diffusion on the size and position of holes

A particle driven by deterministic chaos and moving in a spatially extended environment can exhibit normal diffusion, with its mean square displacement growing proportional to the time. Here, we consider the dependence of the diffusion coefficient on the size and the position of areas of phase space linking spatial regions ('holes') in a class of simple one-dimensional, periodically lifted maps. The parameter dependent diffusion coefficient can be obtained analytically via a Taylor-Green-Kubo formula in terms of a functional recursion relation. We find that the diffusion coefficient varies non-monotonically with the size of a hole and its position, which implies that a diffusion coefficient can increase by making the hole smaller. We derive analytic formulas for small holes in terms of periodic orbits covered by the holes. The asymptotic regimes that we observe show deviations from the standard stochastic random walk approximation. The escape rate of the corresponding open system is also calculated. The resulting parameter dependencies are compared with the ones for the diffusion coefficient and explained in terms of periodic orbits.