Random walk definition

The amount of decrease of the resonance width may be simply estimated in the following way 50). Let the motion of the spins be characterized by a time tc, that is t is the average time a spin stays in a definite environment or the correlation time for the motion. This environment will cause a difference 5w in the precessional frequency of the spin which may be positive or negative from some average value to. During the time Tc the spin acquires a phase angle 60 = TcSu in addition to that acquired by the uniform precession at to. If we consider the motion to be a random walk process (51), after n such intervals during a time t the mean square phase acquired will be... 

It is a random walk over the integers n = 0,1,2,... with steps to the right alone, but at random times. The relation to chapter II becomes more clear by the following alternative definition. Every random set of events can be treated in terms of a stochastic process Y by defining Y(t) to be the number of events between some initial time t = 0 and t. Each sample function consists of unit steps and takes only integral values n = 0,1, 2,... (fig. 5). In general this Y is not Markovian, but if the events are independent (in the sense of II.2) there is a probability q(t) dt for a step to occur between t and t + dt, regardless of what happened before. If, moreover, q does not depend on time, Y is a Poisson process. 

Many years ago Polya [20] formulated the key problem of random walks on lattices does a particle always return to the starting point after long enough time If not, how its probability to leave for infinity depends on a particular kind of lattice His answer was a particle returns for sure, if it walks in one or two dimensions non-zero survival probability arises only for the f/iree-dimensional case. Similar result is coming from the Smoluchowski theory particle A will be definitely trapped by B, irrespectively on their mutual distance, if A walks on lattices with d = 1 or d = 2 but it survives for d = 3 (that is, in three dimensions there exist some regions which are never visited by Brownian particles). This illustrates importance in chemical kinetics of a new parameter d which role will be discussed below in detail. 

Several types of autocorrelation are often used for landscapes. In several important papers, Weinberger and Stadler consider both autocorrelation between adjacent points along a random walk in the landscape and autocorrelation between points a given Hamming distance apart independent of any walk [67,77,78,82,83], Both definitions yield similar information about the landscape and can be computed from one another for stationary landscapes. Other types of autocorrelation are based on neighborhoods defined by complex mutation operations such as crossover [45-49,85],... 

This model predicts that the sum of the exponents of the current decay before and after tT will be 2. As a tends to zero, the temporal distribution j/(2) broadens and the sharp knee seen in Fig. 8.25(b) becomes much less prominent, since the rate of decay of the current is similar both before and after tT. tT will depend on the ratio of the sample thickness to the average displacement of the carrier in the field direction during its random walk through the sample. Use cjf the formal definition of mobility, Equation (4.2), leads to the result that the mobility has a dependence on electric field and sample thickness, L, of the form ... 

Start with Eq. (2-14a), the definition of R, . For a random-walk chain, each step of the walk is independent of the others. Hence, the ensemble average can be brought inside the summations. Thus,... 

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... 

Walks. - The definitions of walks and random walks are given in a number of mathematical texts, e.g.9,11 103,190,191 j ere we hrielly repeat these definitions. A walk in a (molecular) graph G is an alternating sequence of vertices and edges of G, such that each edge e begins and ends with the vertices... 

Random Walks on a One-Dimensional Lattice. We consider [249, 251] stable random walks on a one-dimensional lattice. Here the particle moves at random, and the direction is defined by the direction of the previous step. Each step is only carried out to the nearest neighbor. The mathematical definition of stability demands that at any time and position on the lattice of the wandering particle, two previous coordinates and the direction of the previous step be known. To describe the random walk process, we consider two probabilities, p,i 1 and p, where pi is the probability to be at place j at step n from place j — 1 at the previous step, is the same probability but from place j + 1. 

When we use a time step of constant interval At (t = i At), the diffusion process can be expressed as a simple sequence of A% . This random walk model is convenient to express the normal Brownian motion [9,10]. The definition of Brownian motion,... 

Markov processes have no memory of earlier information. Newton equations describe deterministic Markovian processes by this definition, since knowledge of system state (all positions and momenta) at a given time is sufficient in order to determine it at any later time. The random walk problem discussed in Section 7.3 is an example of a stochastic Markov process. 

This definition of the spectral dimension agrees with that of Eq. (4.41) in the limit as n goes to infinity. The numerical simulation of the spectral dimension based on random-walk models requires enormously large lattices and long walklengths to obtain reliable results [59]. 

If this cyclohexane is cooler than 36.6 degrees, the attraction becomes stronger than the compensating value. Polymers in such a solvent become progressively smaller than the random-walk size they have a D that is larger than 2. Such polymers attract one another in solution as well as attracting themselves. Thus it is difficult to isolate them and measure their D definitively. Such solvents are called poor solvents. 

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