Random walk square lattice

In a random walk on a square lattice the chain can cross itself. 

Direct evaluation of p(l, m) is not immediate. For example, in the restricted random walk on a square lattice in which squares are eliminated, the transition p 2, 2) is smaller than the transition p 3, 2), However, in an unrestricted random walk on a square lattice with no self reversals, p(2,2)=p(3,2) = i. 

Random walks on square lattices with two or more dimensions are somewhat more complicated than in one dimension, but not essentially more difficult. One easily finds, for instance, that the mean square distance after r steps is again proportional to r. However, in several dimensions it is also possible to formulate the excluded volume problem, which is the random walk with the additional stipulation that no lattice point can be occupied more than once. This model is used as a simplified description of a polymer each carbon atom can have any position in space, given only the fixed length of the links and the fact that no two carbon atoms can overlap. This problem has been the subject of extensive approximate, numerical, and asymptotic studies. They indicate that the mean square distance between the end points of a polymer of r links is proportional to r6/5 for large r. A fully satisfactory solution of the problem, however, has not been found. The difficulty is that the model is essentially non-Markovian the probability distribution of the position of the next carbon atom depends not only on the previous one or two, but on all previous positions. It can formally be treated as a Markov process by adding an infinity of variables to take the whole history into account, but that does not help in solving the problem. 

Exercise. Find the mean square distance after r steps of a random walk on a two-dimensional square lattice when U-turns are forbidden. (This is not the excluded volume problem, because each site can be visited many times, provided more than two steps intervene.)... 

Exercise. For the random walk on a two-dimensional square lattice, either with discrete or continuous time, show that every lattice point is reached with probability 1, but on the average after an infinite time. In three dimensions, however, the probability of reaching a given site is less than unity there is a positive probability for disappearing into infinity. 

From a formal point of view it describes random walks on a square lattice with the non-linear transition probabilities. All possible kinds of transitions and their probabilities are given in Fig. 2.12. 

We assume that the time resolution of the STM-measurements is sufficient to discriminate the effect of the entire random walk of one individual vacancy from that of the next, which was already shown to be the case in Section 2.2. The starting situation is shown in Fig. 13. A tracer atom is embedded in the origin of a square lattice with a vacancy sitting next to it at (1, 0). We assume that the only diffusion barrier that is modified by the tracer atom is that for vacancy-tracer exchange. 

It should also be remarked that, in attempts to find the actual configurations of macromolecules, lattice models have played important roles (73). The main interest here is the investigation of self-avoiding walks on a given lattice as a model of a real chain. One tries to find, for example, the mean square length of the random walk as a... 

Equation (1) may be derived using a variety of microscopic models of the relaxation process. In the derivation of Eq. (1), Debye [1] used the theory of the Brownian motion developed by Einstein and Smoluchowski. Einstein s theory of Brownian motion [2] is based on the notion of a discrete time walk. The walk may be described in simple schematic terms as follows. Consider a two-dimensional lattice then, in discrete time steps of length At, the random walker is assumed to jump to one of its nearest-neighbor sites, displayed, for example [7], on a square lattice with lattice constant Ax, the direction being random. Such a process, which is local both in space and time, can be modeled [7] in the one-dimensional analogue by the master equation... 

A two-dimensional random walk on a square lattice. The direction of each step is randomly chosen from four possible diagonals. ... 

Consider a restricted random walk on a square lattice. Let us assume that a walker is not allowed to step back (but can go forward, turn right, or turn left with equal probability). Calculate the mean-square end-to-end distance for such a restricted n-step random walk. What is the characteristic ratio Coo for this walk The lattice constant is equal to /. 

Consider a chain represented by an unrestricted random walk of N steps on a square lattice, as sketched in Fig. 9.37. Topological constraints are represented by obstacles placed in the middle of each cell. The chain is not allowed to cross any of these obstacles. The primitive path (thick line in Fig. 9.37) is then defined as the shortest walk on the lattice with the same end... 

Consider a random-walk chain of no monomer units in a medium 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 surroundings be represented by a permanently connected rigid lattice of uncrossable lines enveloping the chain contour. We assume that the effect of this obstacle lattice on the conformations of the chain is specified simply by a distance scale, the mesh size d, as follows. Pieces of the chain which have a mean-square end-to-end distance (r2) much smaller than d2 can explore all conformations with the same probability as free chains of the same length. For longer 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... 

Figure 4 Chains of N = 11 steps (bonds), that is, 12 monomers (solid circles) on a square lattice (open circles). Immediate chain reversals are not allowed therefore the maximum directions v available are 4 for the first step and 3 for the later steps, (a) An ideal chain (random walk) starting from the origin (1) the chain intersects itself and the last step (dashed line) goes on the third one. (b) A self-avoiding walk (SAW) is not allowed to self-intersect. The SAWs are a subgroup of the ideal chains, (c) A SAW with a finite interaction e between nonbonded nearest-neighbor monomers. The total energy of the chain is 6e.

Figure 1. (a) A random walk in d dimensions with 2 as the variable along the contour of the polymer i.e. giving the location of the monomers, (b) Directed polymer on a square lattice. A polymer as of (a) can be drawn in d -I-1 dimensions. This is like a path of a quantum particle in nonrelativistic quantum mechanics, (c) A situation where both the transverse space (r) and z are continuous, (d) The directed polymers on a hierarchical lattice. Three generations are shown for 4 bonds, (e) A general motif of 26 bonds. 

One may consider a lattice (square lattice in the Fig. lb) with the polymer as a random walk on the lattice with a bias in the diagonal z-direction, never taking a step in the —z direction. The length of the polymer is then the number of steps on... 

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