Random walk cubic lattice

With the Monte Carlo method, the sample is taken to be a cubic lattice consisting of 70 x 70 x 70 sites with intersite distance of 0.6 nm. By applying a periodic boundary condition, an effective sample size up to 8000 sites (equivalent to 4.8-p.m long) can be generated in the field direction (37,39). Carrier transport is simulated by a random walk in the test system under the action of a bias field. The simulation results successfully explain many of the experimental findings, notably the field and temperature dependence of hole mobilities (37,39). 

Figure 1 Two examples of random walks 10,000 steps on a cubic lattice. 

In Figure 1, a is plotted vs. ax for a face-centered lattice (a close-packed lattice) and for a simple cubic lattice (a loose-packed lattice). We notice that (1) the dependence of a on ax can be regarded as being practically the same for both lattices and that, (2) tx undergoes a rapid change around x = xc, which is the point at which a = 0 (Fig. 1). However, p/a does not attain the value it would have for the case of the unrestricted random walk model at x = xc, since at this point, p/a > 1 (Fig. 2), while for unrestricted chain pja = 1. Moreover, the dependence of p/a on ax is not the same for the two lattices while a as a function of ax is practically independent of the lattice. 

Blokland (14) recently also considered the stress-strain behaviour of structured networks. His approach is schematically illustrated in Fig. 29. Consider a cubical lattice on which the chain configurations are laid out in a partially obstructed random walk. Of the N steps of each chain there will be on the average m steps which participate in a bundle structure... 

Suppose a random walk occurs on a primitive cubic lattice and successive jumps are uncorrelated. Show explicitly that f = 1 in Eq. 7.49. Base your argument on a detailed consideration of the values that the cos Oi,i+j terms assume. 

In the random walk description, the energy transfer rate constant is related to the number of hops made by the exciton without revisiting any site in its lifetime. The nearest-neighbor random walk in an isotropic medium may be approximated by a random walk on a simple cubic lattice [145]... 

Consider a restricted random walk on a 3D cubic lattice. Let us assume That a walker is not allowed step back (but can go forward, turn up, down,... 

For a chain modeled by an ideal A -step random walk on a cubic lattice there are 6 different states with a fixed position of one end (see Fig. 9.30). It is impossible to sample all of these states for large N. Therefore, the... 

The random number generator produces random numbers in the interval between 0 and 1. For a random walk on a cubic lattice, there are six possible directions. If the first random number is less than 1/6, the first chosen to be direeted to the right. If it is between 1/6 and 1/3,... 

Estimate the A-dependence of the success rate of a simple Monte Carlo simulation of a self-avoiding walk with 100 steps. Assume that random walks are generated on a cubic lattice. Each step of the walk is not allowed to step back (they can only go forward, up, down, turn left, or right with equal probability of 1/5). Walks that intersect themselves are discarded. 

First, consider the simplest case in which the sensitizer-activator interaction is treated as equivalent to the sensitizer-sensitizer interaction so the exciton becomes trapped only when it happens to hop onto an activator site. Also let us use as an example a simple cubic lattice of sensitizers and assume an electric dipole-dipole interaction as the mechanism causing the energy transfer. The hopping time is represented by th and the probability of host fluorescence per time of one step is a. The fraction of lattice sites which are traps is Cj and the probability of luminescence emission from a trapped exciton per time of one step is (3. The probability of host luminescence at the nth step in the random walk is... 

Table III.5 gives a comparison of these two representations of the data for (n) for nearest-neighbor random walks on finite, cubic lattices with a centrosymmetric trap and subject to periodic boundary conditions. A similar analysis [14] shows that for d = 3, tetrahedral lattices (v — 4)...

It is in the dependence of (n) on the (average) valency (u) that the results here stand in contrast to the analytic and numerical results obtained for lattices subject to periodic boundary conditions. From studies on periodic lattices, n) should decrease systematically with increase in the uniform valency v. This result pertains as well to random walks on ci = 3 dimensional periodic lattices of unit cells and can also be demonstrated analytically and numerically for walks on higher-dimensional [d < 8) cubic lattices [15,16]. In these problems, v = 2d and hence the higher the dimensionality of the space, the greater the number of pathways to a centrally located deep trap in a periodic array of (cubic) cells the decrease in (n) is found to be quite dramatic with increase in d, and hence v. However, an increase in v will also result in a greater number of pathways that allow the random walker to move away from the trap. For periodic lattices, this latter option positions the random walker closer to the trap in an adjacent cell. For finite lattices, moving away from the trap does not position the walker closer to a trap in an adjacent unit cell it positions the walker closer to the finite boundary of the lattice from whence it must (eventually) work itself back. It is evident, therefore, why the v dependence for periodic lattices is modified when one studies the same class of nearest-neighbor random-walk problems on finite lattices. 

In order to describe the collapse of a long-chain polymer in a poor solvent, Flory developed a nice and simple theory in terms of entropy and enthalpy of a solution of the polymer in water [14]. In order to obtain these two competing thermodynamic functions, he employed a lattice model which can be justified by the much larger size of the polymer than the solvent molecules. The polymer chains are represented as random walks on a lattice, each site being occupied either by one chain monomer or by a solvent molecule, as shown in Figure 15.8. The fraction of sites occupied by monomers of the polymer can be denoted as 0, which is related to the concentration c, i.e., the number of monomers per cm by 0 = ca, where is the volume of the unit cell in the cubic lattice. Though the lattice model is rather abstract, the essential features of the problem are largely preserved here. This theory provides a convenient framework to describe solutions of all concentrations. 

Shaffer s bond fluctuation model (Shaffer, J. S., 1994. Effects of chain topology on polymer dynamics—Bulk melts, J. Chem. Phys., 101 4205 13). Polymers are grown as random walks on a simple cubic lattice, subjected to the excluded volume, chain connectivity, and chain uncrossability constraints described in the text. 

We have compared our analytical results with numerical simulations performed by Dayantis et al. [30], and also comment on the relation to earlier simulations done by Baumgartner and Muthukumar [28]. Dayantis et al. carried out simulations of free chains (random-flight walks) confined to cubes of various linear dimensions 6 — 20, in units of the lattice constant. These chains can intersect freely and lie on a cubic lattice. They introduced random obstacles with concentrations r = 0,0.1,0.2 and 0.3. The length of the chains vary between 18 — 98 steps. They also simulated self-avoiding chains that we will not discuss here. They measured the quenched entropy, the end-to-end distance, and also the radius of gyration which is a closely related quantity. Unfortunately, these... 

Abstract. The square and cubic lattice percolation problem and the selfavoiding random walk model were simulated by Monte Carlo method in order to obtain new understanding of the fractal properties of branched and hnear polymer molecules. The central point of this work refers to the comparison between the cluster properties as they emerge from the percolation problem on one hand and the random walk properties on the other hand. It is shown that in both models there is a drastic difference between two and three dimensional systems. In three dimensions it is possible to find a regime where the properties converge towards simple non-avoided random walk, while in two dimensions the topological reasons prevent a smooth transition of the properties pertaining to avoided and non-avoided random walks. 

The main topics in lattice theories, which are relevant for the polymer subject are avoided random walk, lattice percolation [3] and lattice spin models. In this work we shall put the emphasis on the numerical investigation of the systems in the framework of lattice percolation methodologies and avoided random walks on square and cubic lattices. 

Figure 2. Fractal dimension of clusters on cubic lattice. See also the caption of Fig. 1. Note that in three dimensional case there are three values of fractal dimensions in close vicinity lattice animal limit d = 2.08, marked by the o symbol), self-avoiding random walk (d = 1.7) and ordinary random walk (d = 2) represented by the arrows at the left vertical axis.

Figure 6. Log-log plot of the size versus mass relation of the self-avoiding random walk on cubic lattice. The step length is 20 3 lattice points. The vedue of the slope of the straight line which can be drawn through the points leads to the value of the fractal dimension d = 1.96.

The X component of the position after the A-step random walk on the three-dimensional (3D) cubic lattice has a zero mean and a variance of Afe"/3. When A 1, the probability density PJir) for the x component approaches that of a normal distribution with the same mean and variance. Thus,... 

Figure 1.17. Step motion in a three-dimensional random walk on a cubic lattice.

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