Random walker

Sample the initial distribution function P x,p,t = Q) by a set of random walkers. The ith random walker carries the information of its phase-space position (xi,pi), density-matrix component (nm)-, weight VF(xi, Pi, t), and phase a(xi, pi, t). Divide the whole time interval into N small enough pieces such that the desired accuracy of the Trotter scheme is guaranteed. 

In order to get a first impression on the performance of the QC Liouville approach, it is instructive to start with a simple one-mode spin-boson model, that is. Model IVa [205]. In what follows, the QCL calculations used the first-order Trotter scheme (61) with a time step 8r = 0.05 fs. If not noted otherwise, we have employed the momentum-jump approximation (59) and the initial number of random walkers employed was N = 50 000. 

In order to study the origin of the deviations observed, we first consider the statistical convergence of the QCL data. As a representative example. Fig. 14 shows the absolute error of the adiabatic population as a function of the number of iterations N—that is, the number of initially starting random walkers. The data clearly reveal the well-known 1/Vn convergence expected for Monte Carlo sampling. We also note the occurrence of the sign problem mentioned above. It manifests itself in the fact that the number of iterations increases almost exponentially with propagation time While at time t = 10 fs only 200 iterations are sufficient to obtain an accuracy of 2%, one needs N = 10 000 at t = 50 fs. 

Figure 15. Average number of random walkers generated for a single iteration as obtained for Model IVa [205], The full and short dashed lines correspond to the upper and lower electronic populations, respectively, while the long dashed line corresponds to the sum of the coherences of the electronic density matrix.

Due to the large-level density of the lower-lying adiabatic electronic state, the chances of a back transfer of the adiabatic population are quite small for a multidimensional molecular system. To a good approximation, one may therefore assume that subsequent to an electronic transition a random walker will stay on the lower adiabatic potential-energy surface [175]. This observation suggests a physically appealing computational scheme to calculate the time evolution of the system for longer times. First, the initial decay of the adiabatic population is calculated within the QCL approach up to a time to, when the... 

This general feature of the stochastic scheme may cause convergence problems. For example, consider a situation in which the molecular system is predominately in a single state, say pu. Although the expectation values of the population P2 = trp22 and the corresponding coherences are zero, there are the same number of random walkers in these states which need to cancel... 

The simplest of these models which permits a detailed discussion of the decay of correlations is a random walk model in which a set of random walkers whose positions are initially correlated is allowed to diffuse the motion of any single random walker being independent of any other member of the set. Let us assume that there are r particles in the set and motion occurs on a discrete lattice. The state of the system is, therefore, completely specified by the probabilities Pr(nlf n2,..., nr /), (tij = — 1, 0, 1, 2,. ..) in which Pr(n t) is the joint probability that particle 1 is at n1( particle 2 is at n2, etc., at time l. We will also use the notation Nj(t) for the random variable that is the position of random walker j at time t. Reduced probability distributions can be defined in terms of the Pr(n t) by summation. We will use the notation P nh, rth,..., ntj I) to denote the distribution of random walkers iu i2,..., i at time t. We define... 

One may ask the following question. Suppose the random walker starts out at site m at t = 0 how long does it take him to reach a given site R for the first time This first-passage time is, of course, different for the different realizations of his walk and is therefore a random quantity. Our purpose is to find its probability distribution, and in particular the average or mean first-passage time ]... 

Exercise 7.5 shows explicitly for a random walker on a primitive-cubic lattice that the mean values of the cos terms in Eq. 7.49 sum to zero and, therefore, that f = 1. Use Eq. 8.29 to demonstrate the same result. 

Here, we present an approach for the description of such anomalous transport processes that is based on the continuous-time random walk theory for a power-law waiting time distribution w(t) but which can be used to find the probability density function of the random walker in the presence of an external force field, or in phase space. This framework is fractional dynamics, and we show how the traditional kinetic equations can be generalized and solved within this approach. 

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]... 

If an external force field acts on the random walker, it has been shown [58, 59] that in the diffusion limit, this broad waiting time process is governed by the fractional Fokker-Planck equation (FFPE) [60]... 

Consider now a random walker in one dimension, with probability, R, of moving to the right, and, L, for moving to the left. At, l = 0, we place the walker at x = 0, as indicated in Figure 5.6. The walker can then jump, with the above probabilities, either to the left or to the right for each time-step. Every... 

FIGURE 5.6 One-dimensional random walker that can jump either to the left or to the right. 

Let us now consider a random walker in a three-dimensional cubic lattice. The atom will jump between sites of the normal lattice for a substitutional diffuser, and from interstitial to interstitial site for an interstitial diffuser. In the present case, the Einstein-Smoluchovskii equation for the diffusion coefficient in three dimensions which is a generalization of Equation 5.36, that is,... 

Using the EAM barriers, for typical parameters T = 320 K and Z = 401 the values for the return probabilities were peq = 1 — 2.4 x 10-7, p[Krp = 1.1 x 10 1, popp = 4.2 x 10 9, and the recombination probability prcc = 1.1 x 10 s. These values depend weakly on Z (e.g. the dependence of the mean square displacement, calculated later in this chapter from the return probabilities, is logarithmic (r2) a log (Z/Zo). This is a consequence of the fact that 2 is the marginal dimension for the return problem of the random walker. In higher-dimensional space the vacancy does not necessarily return and the return probability is asymptotically independent of the lattice size [43]. 

The probability for an one-dimensional, free and unconstrained random walker, starting from the origin, to reach the distance 2 after Ni steps of length l is well approximated (at large Ni) by... 

The definition of spectral dimension ds refers to the probability p(t) of a random walker returning to its origin after time t ... 

The description of these phenomena in complex media can be performed by means of fractal geometry, using the spectral dimension ds. To express the kinetic behavior in a fractal object, the diffusion on a microscopic scale of an exploration volume is analyzed [278]. A random walker (drug molecule), migrating within the fractal, will visit n (t) distinct sites in time t proportional to the number of random walk steps. According to the relation (2.9), n(t) is proportional to tdA2j so that diffusion is related to the spectral dimension. 

For ds > 2, a random walker has a finite escape probability-microscopic behavior conducive to re-randomize the distribution of reactants around a trap and deplete the supply of reactive pairs, and thus a stable macroscopic reactivity as attested by the classical rate constant [296,297]. The scale of the self-organization is microscopic and independent of time, such that n (t) oc t (is linear) and k = n (t) is a constant, so the reaction kinetics are classical. 

In the fractal porous medium, the diffusion is anomalous because the molecules are considerably hindered in their movements, cf. e.g., Andrade et al., 1997. For example, Knudsen diffusion depends on the size of the molecule and on the adsorption fractal dimension of the catalyst surface. One way to study the anomalous diffusion is the random walk approach (Coppens and Malek, 2003). The mean square displacement of the random walker (R2) is not proportional to the diffusion time t, but rather scales in an anomalous way ... 

Here the dimensionless time z=t/t is normalized by the characteristic relaxation time t, the time required for a charge carrier to move the distance equal to the size of one droplet, which is associated with the size of the unit cell in the lattice of the static site-percolation model. Similarly, we introduce the dimensionless time zs = ts/t where ts is the effective correlation time of the s-cluster, and the dimensionless time z = tm/t. The maximum correlation time t, is the effective correlation time corresponding to the maximal cluster sm. In terms of the random walker problem, it is the time required for a charge carrier to visit all the droplets of the maximum cluster sm. Thus, the macroscopic DCF may be obtained by the averaging procedure... 

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