分数阶Brown马达及其定向输运现象

选取幂函数作为广义Langevin方程的阻尼核函数,采用闪烁棘轮势,建立了过阻尼分数阶Brown马达模型.结合分数阶微积分的记忆性,分析了粒子在过阻尼分数阶Brown马达作用下的运动特性.研究发现,较之整数阶情形,过阻尼分数阶Brown马达也会产生定向输运现象,并且在某些阶数下会产生整数阶情形所不具有的反向定向流.此外,还讨论了阶数和噪声强度对系统输运速度的影响,发现当阶数固定时,其平均输运速度会随噪声变化出现随机共振;当噪声强度固定时,其输运速度会随阶数变化而振荡,即出现多峰的广义随机共振现象.     

Finite Time Approach to Equilibrium in a Fractional Brownian Velocity Field

We consider the solution of the equation r(t) = W(r(t)), r(0) = r ₀ > 0 where W(⋅) is a fractional Brownian motion (f.B.m.) with the Hurst exponent α∈ (0,1). We show that for almost all realizations of W(⋅) the trajectory reaches in finite time the nearest equilibrium point (i.e. zero of the f.B.m.) either to the right or to the left of r ₀, depending on whether W(r ₀) is positive or not. After reaching the equilibrium the trajectory stays in it forever. The problem is motivated by studying the separation between two particles in a Gaussian velocity field which satisfies a local self-similarity hypothesis. In contrast to the case when the forcing term is a Brownian motion (then an analogous statement is a consequence of the Markov property of the process) we show our result using as the principal tools the properties of time reversibility of the law of the f.B.m., see Lemma 2.4 below, and the small ball estimate of Molchan, Commun. Math. Phys. 205 (1999) 97–111.     

Magnetic Field Line Random Walk in Two‐dimensional Turbulence: Markovian Diffusion versus Superdiffusion

In the present article previous work is complemented by investigating analytically the field line random walk in partially turbulent magnetic fields. By using the well‐established model of two‐dimensional turbulence with a general spectrum at large scales, we compute the field line diffusion coefficient for all length scales. This work will also clarify some confusion about the superdiffusive and diffusive regimes discovered earlier (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Anisotropic Diffusion in Driven Convection Arrays

We numerically investigate the transport of a Brownian colloidal particle in a square array of planar counter-rotating convection rolls at high Péclet numbers. We show that an external force produces huge excess peaks of the particle’s diffusion constant with a height that depends on the force orientation and intensity. In sharp contrast, the particle’s mobility is isotropic and force independent. We relate such a nonlinear response of the system to the advection properties of the laminar flow in the suspension fluid.     

Torsion Fields, Cartan–Weyl Space–Time and State-Space Quantum Geometries, their Brownian Motions, and the Time Variables

We review the relation between spacetime geometries with trace-torsion fields, the so-called Riemann–Cartan–Weyl (RCW) geometries, and their associated Brownian motions. In this setting, the drift vector field is the metric conjugate of the trace-torsion one-form, and the laplacian defined by the RCW connection is the differential generator of the Brownian motions. We extend this to the state-space of non-relativistic quantum mechanics and discuss the relation between a non-canonical quantum RCW geometry in state-space associated with the gradient of the quantum-mechanical expectation value of a self-adjoint operator given by the generalized laplacian operator defined by a RCW geometry. We discuss the reduction of the wave function in terms of a RCW quantum geometry in state-space. We characterize the Schroedinger equation in terms of the RCW geometries and Brownian motions. Thus, in this work, the Schroedinger field is a torsion generating field, both for the linear and non-linear cases. We discuss the problem of the many times variables and the relation with dissipative processes, and the role of time as an active field, following Kozyrev and a recent experiment in non-relativistic quantum systems. We associate the Hodge dual of the drift vector field with a possible angular-momentum source for the phenomenae observed by Kozyrev.     

On the Unification of Geometric and Random Structures through Torsion Fields: Brownian Motions,Viscous and Magneto-fluid-dynamics

We present the unification of Riemann–Cartan–Weyl (RCW) space-time geometries and random generalized Brownian motions. These are metric compatible connections (albeit the metric can be trivially euclidean) which have a propagating trace-torsion 1-form, whose metric conjugate describes the average motion interaction term. Thus, the universality of torsion fields is proved through the universality of Brownian motions. We extend this approach to give a random symplectic theory on phase-space. We present as a case study of this approach, the invariant Navier–Stokes equations for viscous fluids, and the kinematic dynamo equation of magnetohydrodynamics. We give analytical random representations for these equations. We discuss briefly the relation between them and the Reynolds approach to turbulence. We discuss the role of the Cartan classical development method and the random extension of it as the method to generate these generalized Brownian motions, as well as the key to construct finite-dimensional almost everywhere smooth approximations of the random representations of these equations, the random symplectic theory, and the random Poincaré–Cartan invariants associated to it. We discuss the role of autoparallels of the RCW connections as providing polygonal smooth almost everywhere realizations of the random representations.     

周期场中非各态历经布朗运动

将自由状态下呈弹道扩散的非各态历经系统置于周期场中,进而将非各态历经布朗运动分为两类.第一类是阻尼核的Laplace变换的低频为零的系统,当温度远大于势垒高度时,系统平均能量的动能部分依赖粒子的初始速度分布;随温度降低,系统的各态历经性得到恢复.然后将第一类系统的稳定速度变量作为一个内部噪声,再去驱动一个自由布朗粒子,则阻尼核的Laplace变换在零频时为无穷大.结果发现,粒子扩散系数随温度的增加而趋于零,显示一种经典局域化特征,系统的渐进分布依赖于初始坐标分布.这是第二类非各态历经性运动,不能通过外加势而恢复. 关键词： 非各态历经 非Markov布朗运动 扩散系数 噪声谱     

Pricing model for equity warrants in a mixed fractional Brownian environment and its algorithm

This paper deals with the problem of pricing equity warrants in a mixed fractional Brownian environment. Based on the quasi-conditional expectation and the Fourier transform, we present the pricing model for equity warrants. Moreover, a hybrid intelligent algorithm, which is based on the Genetic Algorithm, is employed to solve the nonlinear optimization problem. The performance of our model and the proposed algorithm have been illustrated with some numerical examples.     

Brownian diffusion in a nonuniform gas

An analysis is made of the effects on the diffusion of Brownian particles whose Knudsen number is large compared to unity, of nonuniformities in the host gas. As examples, in one type of nonuniformity of the host gas, the Chapman-Enskog velocity distribution function for the gas molecules is used; in the other, the host gas is a free-molecule Couette flow. In both cases, a new force on the Brownian particles appears. Two techniques are used (extending Kramers' method and utilizing the Chapman-Enskog method) to transform the new Fokker-Planck equation into generalized Smoluchowski and convective diffusion equations. In these equations, the diffusion coefficient appears as a second-order tensor. Thus, it is demonstrated that Brownian diffusion in a nonuniform gas is anisotropic.The work of Slinn was financially supported in part by Battelle Memorial Institute and in part by U.S. Atomic Energy Commission Contract AT(45-1)-1830. The work of Shen was supported in part by U.S. Air Force Office of Scientific Research Contract 49(638)-1346.     

Exact Solutions of a Generalized Multi-Fractional Nonlinear Diffusion Equation in Radical Symmetry

This paper is devoted to investigating exact solutions of a generalized fractional nonlinear anomalous diffusion equation in radical symmetry. The presence of external force and absorption is also considered. We first investigate the nonlinear anomalous diffusion equations with one-fractional derivative and then multi-fractional ones. In both situations, we obtain the corresponding exact solutions, and the solutions found here can have a compact behavior or a long tailed behavior.     

Effect of Coupling between Rotational and Translational Brownian Motions on NMR Spin Relaxation: Consideration Using Green Function of Rigid Body Diffusion

Using the Green function of arbitrary rigid Brownian diffusion (Goldstein,Biopolymers33, 409–436, 1993), it was analytically shown that coupling between translation and rotation diffusion degrees of freedom does not affect the correlation functions relevant to the NMR intramolecular relaxation. It follows that spectral densities usually used for the anisotropic rotation diffusion (Woessner,J. Chem. Phys.37, 647–654, 1962) can be regarded as exact in respect to the rotation–translation coupling for the spin system connected with a rigid body.     

Axisymmetric Fractional Diffusion with Mass Absorption in a Circle under Time-Harmonic Impact

The axisymmetric time-fractional diffusion equation with mass absorption is studied in a circle under the time-harmonic Dirichlet boundary condition. The Caputo derivative of the order 0<α≤2 is used. The investigated equation can be considered as the time-fractional generalization of the bioheat equation and the Klein–Gordon equation. Different formulations of the problem for integer values of the time-derivatives α=1 and α=2 are also discussed. The integral transform technique is employed. The outcomes of numerical calculations are illustrated graphically for different values of the parameters.     

Exact Solution of Fractional Diffusion Model with Source Term used in Study of Concentration of Fission Product in Uranium Dioxide Particle

The exact solution of fractional diffusion model with a location-independent source term used in the study of the concentration of fission product in spherical uranium dioxide (UO₂) particle is built. The adsorption effect of the fission product on the surface of the UO₂ particle and the delayed decay effect are also considered. The solution is given in terms of Mittag-Leffler function with finite Hankel integral transformation and Laplace transformation. At last, the reduced forms of the solution under some special physical conditions, which is used in nuclear engineering, are obtained and corresponding remarks are given to provide significant exact results to the concentration analysis of nuclear fission products in nuclear reactor.     

Critical mass of bacterial populations and critical temperature of self-gravitating Brownian particles in two dimensions

We show that the critical mass M_c=8π of bacterial populations in two dimensions in the chemotactic problem is the counterpart of the critical temperature T_c=GMm/4k_B of self-gravitating Brownian particles in two-dimensional gravity. We obtain these critical values by using the Virial theorem or by considering stationary solutions of the Keller-Segel model and Smoluchowski-Poisson system. We also consider the case of one-dimensional systems and develop the connection with the Burgers equation. Finally, we discuss the evolution of the system as a function of M or T in bounded and unbounded domains in dimensions d=1, 2 and 3 and show the specificities of each dimension. This paper aims to point out the numerous analogies between bacterial populations, self-gravitating Brownian particles and, occasionally, two-dimensional vortices.     

Current reversal and mass separation of inertial Brownian motors in a two-noise ratchet

Transport of an inertial Brownian motor moving in an asymmetric periodic potential driven by an external force and correlated noises is investigated. Using the numerical algorithm, the asymptotic mean velocity (AMV) for characterizing directed transport is obtained. The effects of the external driving force f and the correlation λ between the two noises on the AMV are discussed. The results manifest: (1) the external driving force and the correlation between the two noises can lead to the phenomena like current reversal. (2) The competitions among the external driving force and the correlation between the two noises are necessary for current reversal, i.e., fλ>0. (3) Different directions of transport are found for different masses of the Brownian particles under the condition fλ>0. Therefore a theoretical study is suggested for separating Brownian particles according to their different masses in the ratchet system.     

谐振势阱中的布朗运动——磁镊实验与模拟

从磁镊实验和模拟角度研究了处于谐振势阱中的布朗运动. 利用实验和模拟的结果验证了理论.然后通过理论与实验的对照, 对磁镊实验中DNA分子的持久长度大小对小球位移分布的影响, 以及磁镊实验中的测力误差作了相关分析.分析指出:持久长度的变化对沿 DNA链方向上的布朗运动影响更大;小的外力作用下力的测量会出现较大误差.     

MHD from a Microscopic Concept and Onset of Turbulence in Hartmann Flow

We derive higher order magneto-hydrodynamic （MHD） equations from a microscopic picture using pro-jection and perturbation formalism. In an application to Hartmann flow we find velocity profiles flattening towards the center at the onset of turbulence in hydrodynamic limit. Comparison with the system under the effect of a uniform magnetic field yields difference in the onset of turbulence consistent with observations, showing that the presence of magnetic field inhibits onset of instability or turbulence. The laminar-turbulent transition is demonstrated in a phase transition plot of the development in time of the relative average velocities vs. Reynolds number showing a sharp increase of the relative average velocity at the transition point as determined by the critical Reynolds number.     

Scaling and long-range dependence in option pricing II: Pricing European option with transaction costs under the mixed Brownian–fractional Brownian model

This paper deals with the problem of discrete-time option pricing by the mixed Brownian–fractional Brownian model with transaction costs. By a mean-self-financing delta hedging argument in a discrete-time setting, a European call option pricing formula is obtained. In particular, the minimal pricing c_min(t,s_t) of an option under transaction costs is obtained, which shows that timestep δt and Hurst exponent H play an important role in option pricing with transaction costs. In addition, we also show that there exists fundamental difference between the continuous-time trade and discrete-time trade and that continuous-time trade assumption will result in underestimating the value of a European call option.     

压缩拐角激波/湍流边界层干扰特性分析

用大涡模拟方法对Mach数3.0下的压缩拐角激波/湍流边界层干扰问题进行数值研究.对拐角上游平板区域边界层转捩及湍流进行模拟,设定平板区域长度,使得转捩过程于平板区域发生并充分完成,从而在拐角处产生激波/湍流边界层相互干扰,研究激波/湍流边界层的作用机理.研究表明:流场能够在非定常扰动激励下迅速转捩,并于平板区发展为完全湍流;湍流边界层与激波相互作用过程中,拐角附近分离区较层流情况明显减小;展向不同区域分离区大小差异较大,局部区域分离现象消失.     

Brownian Particle in a Poisson-Shot-Noise Active Bath: Exact Statistics,Effective Temperature,and Inference

The dynamics of an overdamped Brownian particle in a thermal bath that contains a dilute solution of active particles is studied. The particle moves in a harmonic potential and experiences Poisson shot-noise kicks with specified amplitude distribution due to moving active particles in the bath. From the Fokker–Planck equation for the particle dynamics, the stationary solution for the displacement distribution is derived along with the moments characterizing mean, variance, skewness, and kurtosis, as well as finite-time first and second moments. An effective temperature is also computed through the fluctuation–dissipation theorem and show that equipartition theorem holds for all zero-mean kick distributions, including those leading to non-Gaussian stationary statistics. For the case of Gaussian-distributed active kicks, a re-entrant behavior from non-Gaussian to Gaussian stationary states and a heavy-tailed leptokurtic distribution across a wide range of parameters are found as seen in recent experimental studies. Further analysis reveals statistical signatures of the irreversible dynamics of the particle displacement in terms of the time asymmetry of cross-correlation functions. Fruits of the work is the development of an compact inference scheme that may allow experimentalists to extract the rate and moments of underlying shot-noise solely from the statistics the particle position.