Random walks moments

The one-dimensional random walk of the last section is readily adapted to this problem once we recognize the following connection. As before, we imagine that one end of the chain is anchored at the origin of a three-dimensional coordinate system. Our interest is in knowing, on the average, what will be the distance of the other end of the chain from this origin. A moment s reflection will convince us that the x, y, and z directions are all equally probable as far as the perfectly flexible chain is concerned. Therefore one-third of the repeat units will be associated with each of the three perpendicular directions... 

The Helfand moment is the center of mass, energy or momentum of the moving particles, depending on whether the transport property is diffusion, heat conductivity, or viscosity. The Helfand moments associated with the different transport properties are given in Table III. Einstein formula shows that the Helfand moment undergoes a diffusive random walk, which suggests to set up a... 

The diffusive random walk of the Helfand moment is mled by a diffusion equation. If the phase-space region is defined by requiring Ga(t) < x/2, the escape rate can be computed as the leading eigenvalue of the diffusion equation with these absorbing boundary conditions for the Helfand moment [37, 39] ... 

To account for the effect of a sufficiently broad, statistical distribution of heterogeneities on the overall transport, we can consider a probabilistic approach that will generate a probability density function in space (5) and time (t), /(i, t), describing key features of the transport. The effects of multiscale heterogeneities on contaminant transport patterns are significant, and consideration only of the mean transport behavior, such as the spatial moments of the concentration distribution, is not sufficient. The continuous time random walk (CTRW) approach is a physically based method that has been advanced recently as an effective means to quantify contaminant transport. The interested reader is referred to a detailed review of this approach (Berkowitz et al. 2006). 

In this section, we begin the description of Brownian motion in terms of stochastic process. Here, we establish the link between stochastic processes and diffusion equations by giving expressions for the drift velocity and diffusivity of a stochastic process whose probability distribution obeys a desired diffusion equation. The drift velocity vector and diffusivity tensor are defined here as statistical properties of a stochastic process, which are proportional to the first and second moments of random changes in coordinates over a short time period, respectively. In Section VILA, we describe Brownian motion as a random walk of the soft generalized coordinates, and in Section VII.B as a constrained random walk of the Cartesian bead positions. 

In a one-dimensional random walk with excursion from the origin, the mth moment of the jump length distribution can be derived from... 

The effective polarizability of surface atoms can be determined with different methods. In Section 2.2.4(a) a method was described on a measurement of the field evaporation rate as a function of field of kink site atoms and adsorbed atoms. The polarizability is derived from the coefficient of F2 term in the rate vs. field curve. From the rate measurements, polarizabilities of kink site W atoms and W adatoms on the W (110) surface are determined to be 4.6 0.6 and 6.8 1.0 A3, respectively. The dipole moment and polarizability of an adatom can also be measured from a field dependence of random walk diffusion under the influence of a chemical potential gradient, usually referred as a directional walk, produced by the applied electric field gradient, as reported by Tsong et a/.150,198,203 This study is a good example of random walk under the influence of a chemical potential gradient and will therefore be discussed in some detail. 

Consider for a moment a rod-shaped particle of unit length. The orientation of the rod, u, can be specified by a unit vector u directed along its axis with spherical polar coordinates, D - id, random walk along the surface of the unit sphere. Debye [16] in 1929 developed a model for the reorientation process based on the assumption that collisions are so fiiequent that a particle can rotate throu only a very small angle before having another reorienting collision (i.e., small step diffusion). Debye began with the diffusion equation... 

Golayf ° and Giddings, respectively, described a modification of the rate theory for capillary columns (hollow tube with inner wall coated with liquid phase) and the random walk, non-equilibrium theory. The former derived an equation to describe the efficiency of an open tubular column, while the random walk theory describes chromatographic separation in terms of statistical moments. The non-equilibrium theory involves a rigorous mathematical treatment to account for incomplete equilibrium between the two phases. ... 

Consider a random-walk chain of no monomer units in a n dium which is densely filled with the contours of other chains. For the moment take the ends of the test chain to be fixed. Let its surrounding be r resented by a permanently connected rigid lattice of uncrossable lines enveloping the chain contour. We assume that the effect of this obstade lattice on the conformations of the chain is specified simply by a distance scale, the mesh so d, as follows. Pieces of the diain which have a mean quare end-to-end distance (r ) much smaller than (F can explore aU conformations with the same probability as free chains of the same length. For loiter pieces, the presence of the obstacles (and the fact that the pieces are connected in a definite sequence between the fixed end points of the... 

In the simple random walk the steps are statistically independent of one another. The simplest generalization of this model is to make each step dependent on the preceding step in such a way that the second moment is... 

The first is the long rests or fractal time random walk where the mean waiting time diverges however, the second moment of the jump length distribution remains finite. The fractal time random walk always leads to... 

For an unbiased symmetric random walk, P(S) = P(—S), the second term on the right vanishes and taking the time-continuous limit of small r one obtains a diffusion equation with diffusion coefficient determined by the variance of the jumps D = 5%)/(2r). In d dimensions the result is D = (<52)/(2g t). In the random walk context, the dispersion in Eq. (2.13) is giving the second moment of the position of a random walker which started at r = 0 ... 

The distribution for the distance from the origin that results from a random walk comprising N steps is the A-fold convolution of the probability density with itself. This is easily obtained in the space of the Fourier variable u or Laplace variable s, as being the characteristic or moment generating function for a single step raised to the power N ... 

It is a vector-matrix equation in s space and an integral equation in time. The unknown is the vector of functions This equation is called continuous time random walk (CTRW) and was used in phenomenological modeling of transport [17]. Equation 13.1 is closed and can be solved provided that A. /x) is known. Our contribution is to show how detailed microscopic dynamics is used to compute or its moments (see below). 

In the two-state random walk model, where the velocity of the particle is either /+ or we can conveniently calculate the first and second moments of the coordinate W by introducing the generating function G(k, T),... 

Consider now the process of diffusion in a thin layer of liquid, into which a substance (particles) with concentration Co is introduced at the initial moment t = 0, at the point x = 0. The substance diffuses in the liquid. In view of the analogy between diffusion and random walk of particles, it is assumed that in time t, the particle makes n displacements, where the number n is proportional to t ... 

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