Random walking particles

The emergence of slow kinetics with its typical slowly decaying memory effects is tightly connected to a scale-free waiting time pdf that is, the temporal occurrence of the motion events performed by the random walking particle is broadly distributed such that no characteristic waiting time exists. It has been demonstrated that it is the assumption of the power-law form for the waiting time pdf which leads to the explanation of the kinetics of a broad diversity of systems such as the examples quoted above. 

The Mean Square Distance Traveled in a Time fby a Random-Walking Particle... 

APPENDIX 4.1. THE MEAN SQUARE DISTANCE TRAVELED BY A RANDOM-WALKING PARTICLE... 

Mean distance covered by a particle during a specified time period Any specified length. Free motion length for a random walking particle Molar mass (of species S)... 

Dimensionless time. Duration of free motion for a random walking" particle... 

We can now show the relation between the random walk analogy (Fig. 9.1) and the diffusion coefficient. If we released a cluster of random walk particles with total mass Q, at x = 0 and t = 0, they would describe a normal distribution about x = 0 for t > 0. The number of particles, M, occupying a distance AI along the line is given by... 

The first thing to recognize is that eddy diffusion is also fundamentally a statistical problem that can be best understood on the basis of the random walk. Particle diameter is of particular importance here, because, unlike the infinitesimal step lengths in the diffusional random walk, the particle diameter characterizes the step length in developing an expression for eddy diffusion. 

For particles executing a random walk, like albumin molecules in buffered water, the calculations above suggest that individual steps in the random walk occur very quickly, over a short time interval. As a consequence, during a typical observation time, each particle takes many steps on the axis of Figure 3.2. The probability that a random walking particle took a total of k steps to the right after a sequence of n steps in the random walk is provided by the binomial distribution ... 

As demonstrated in previous sections, Eq. 2.27 is out of the direct use due to absence of the data on the activity coefficients y in HASP. Instead of Eq. 2.27, another equation that contains DC is more helpful, namely the Einstein-Smoluchowski relation (Eq. 2.28) for mean square distance of a freely randomly walking particle during time t ... 

VIII. MEAN FIRST PASSAGE TIME OF A RANDOM-WALK PARTICLE AND RELATED PHYSICAL PROBLEMS... 

Let a random walk particle start from site 0 at t = 0, and consider the probability P t) that this particle returns to its starting point at time t. We shall show that the shape of Pit) depends on the dimensionality d of the space covered by the walker. We assume that the space is homogeneous, that is, that after a given time t the walker has the same probability of being at any point within a volume V(/). We can thus write... 

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