Jump length probability

We may determine from /(x, t) both the jump length probability density function... 

The deviations from Gaussian behaviour were successfully interpreted as due to the existence of a distribution of finite jump lengths underlying the sublinear diffusion of the proton motion [9,149,154]. A most probable jump distance of A was found for PI main-chain hydrogens. With the model... 

The dependence on r describes the relative probability of various jump lengths, while the dependence on X determines the overall transition prob-... 

In the continuous-time random walk model, a random walker is pictured to execute jumps at time steps chosen from the waiting time pdf w(t). In the isotropic and homogeneous (that is, force-free) case, the distance covered in a single jump event can be drawn from the jump length pdf X x). Then, the probability t) (x, t) of just having arrived at position x is given through [49]... 

Figure 6 The distribution of jump lengths of embedded atoms in Cu(00 1), measured for (a) 1461 In hops at 320 K and (b) 887 Pd hops at 335 K. Plotted is the probability for a jump of an In/Pd atom from its starting position to any site at a given distance from the starting position. Probabilities have been normalized so that the probabilities for the entire lattice add up to one. To illustrate the distinct nature of the diffusion behavior, the Gaussian jump length distribution that would be expected for the case of simple hopping is also plotted in (a).

In Section 3 we derive that for the vacancy-mediated diffusion mechanism, one expects the shape of the jump length distribution to be that of a modified Bessel function of order zero. Both distributions can be fit very well with the modified Bessel function, again confirming the vacancy-mediated diffusion mechanism for both cases. The only free parameter used in the fits is the probability prec for vacancies to recombine at steps, between subsequent encounters with the same embedded atom [33]. This probability is directly related to the average terrace width and variations in this number can be ascribed to the proximity of steps. The effect of steps will be discussed in more detail in Section 4. 

Using this, the tracer atom is described as if it forms a pair with the vacancy on one of the bonds adjacent to its original site, it walks on the bond lattice, and at the end of the walk (which happens after each move with probability prec) it is released with equal probability at either end of the last visited bond. Results for the probabilities of the different jump lengths (beginning-to-end vectors of these trajectories) are shown in Fig. 9. Note, that the model calculations in Fig. 9 contain no adjustable parameters. 

Figure 9 The probabilities of the jump lengths of the tracer atom for T = 320 K and l = 401 lattice spacings. Filled circles correspond to experimental values (measured at this temperature and terrace size), open circles are from the model described in the text, and the solid curve is the continuum solution described in Section 3.3. The distribution depends only on the magnitude of the jump lengths with no directional dependence. (Each dataset is normalized separately such that the probabilities corresponding to a subset of the jump vectors, 1 < r < 6, add up to unity. These are the probabilities that are determined with good accuracy in the experiment.)...

Equation 21 is the microscopic equivalent of the Einstein equation (3) and implies that succeeding jumps are uncorrelated. In general, however, backward jumps will take place with a higher probability (the correlation effect [2]), so that eq 21 provides an upper limit of the diffusivity. Alternatively, with known values for D and t, eq 21 allows an estimate of the lower limit of the mean-square jump length [67, 68]. 

Here is the probability that a molecule can make a jump in the right direction given the jump length is dp and the velocity is... 

In some experiments (118) the spatial probability distribution of the initially localized particle was measured at the early stages of evolution. At long times the evolution is universal, controlled by the diffusion equation and the shape of the distribution is Gaussian. At the early stage, however, the shape of the distribution depends on the particle jump length between successive trappings. 

It is possible to derive the time-dependent probability distribution allowing any jump length (113). The location of the particle on the lattice is denoted by two indices /, m, such that the location of the particle at time t = 0 is at / = m = 0. The probability distribution w, ,(0 for the particle to be at the /, m site at time t is determined by the following master equations ... 

Particles, such as molecules, atoms, or ions, and individuals, such as cells or animals, move in space driven by various forces or cues. In particular, particles or individuals can move randomly, undergo velocity jump processes or spatial jump processes [333], The steps of the random walk can be independent or correlated, unbiased or biased. The probability density function (PDF) for the jump length can decay rapidly or exhibit a heavy tail. Similarly, the PDF for the waiting time between successive jumps can decay rapidly or exhibit a heavy tail. We will discuss these various possibilities in detail in Chap. 3. Below we provide an introduction to three transport processes standard diffusion, tfansport with inertia, and anomalous diffusion. 

CTRWs display subdiffusive behavior if the variance of the jump length PDF remains finite, but the waiting time PDF is heavy-tailed, such that the mean waiting time T is infinite. An example is a waiting time PDF derived from a Mittag-Leffler function for the survival probability, F(t) = Ey(-t ) with 0 < y < 1 [381]. The... 

Recent experimental work [221] for another invasive thistle, C. acanthoides, was used to test our theoretical predictions for the invasion rate. Rosettes of C. acanthoides were introduced into uninvaded plots in Maryland (USA), where each rosette was considered as a founder individual for new invasive thistle populations. The cumulative probability distribution for jump lengths W(r) was measured for different years and different treatments, named Ox, lx, and 2x clippings. The relation between W(r) and w r) is given by W(r) = 2n /q r w r )dr or... 

In this model the polymer is made up of N beads connected by Af — 1 bonds, each having constant length b (see Fig. 4.16). In a small time interval At, each bead makes the following jump with probability w At. (i) For the internal beads (i.e., beads 2,3,..., N -1)... 

The diffusive movement of atoms is assumed to have a range of values, and the probability that a jump of length between j and j + dj can occur is... 

In other words, if we assume that the counting function N(t) has statistically independent increments (Eq. (3-237)), and has the property that the probability of a single jump occurring in a small interval of length h is approximately nh but the probability of more than one jump is zero to within terms of order h, (Eq. (3-238)), then it can be shown 51 that its probability density functions must be given by Eq. (3-231). It is the existence of theorems of this type that accounts for the great... 

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