Random walk calculations

Viswanathan R, Raff L M and Thompson D L 1984 Monte Carlo random walk calculations of unimolecular dissociation of methane J. Chem. Phys. 81 3118-21... 

This expression is plotted in Figure 3.5. The similarity of Equation 3-36, which is a solution to the conservation of mass equations, to the Gaussian distribution obtained from random walk calculations (see Equation 3-14) is obvious. 

Random-walk calculations have been used to explain why, in the oxidation of fatty acids, C-7—C-10 methane protons are oxidized more than others in acetic acid media ... 

The connection of random-walk calculations based on concentration-gradient theory may be summoned as follows. As we have noted (Section 2.3), the Equation for diffusion-controlled rate constants based on Pick s law agrees with that derived from a random-walk model (Section 2.3). Pick s law is in fact a macroscopic consequence of the random-walk model of molecular-scale processes or, to put it the other way round, the random-walk model is an interpretation of Pick s experimental law [4]. 

This one-dimensional random walk calculation can then be generalized to three dimensions because macromolecular chains are not confined in only one direction. Assuming that the three directions of space are equiprobable—which is the case for flexible chains—the probability for the occurrence of n steps in a given direction of space has to be calculated with a number of steps equal to n/3 for each of the three directions of space x,y,z ... 

This result enables us to calculate the probability of any specified outcome for the one-dimensional random walk. We shall continue to develop this one-dimensional relationship somewhat further, since doing so will produce some useful results. 

With this probability expression, it is an easy matter to calculate the average dimensions of a coil. Because of the back-and-forth character of the x, y, and z components of the random walk, the average end-to-end distance is less meaningful than the average of r. The latter squares positive and negative components before averaging and gives a more realistic parameter to characterize the coil. To calculate r, we remember Eq. (1.11) and write... 

This idea that the heat was transfered by a random walk was used early on by Einstein [21] to calculate the thermal conductance of crystals, but, of course, he obtained numbers much lower than those measured in the experiment. As we now know, crystals at low enough T support well-defined quasiparticles—the phonons—which happen to carry heat at these temperatures. Ironically, Einstein never tried his model on the amorphous solids, where it would be applicable in the / fp/X I regime. 

Derivation of the Gaussian Distribution for a Random Chain in One Dimension.—We derive here the probability that the vector connecting the ends of a chain comprising n freely jointed bonds has a component x along an arbitrary direction chosen as the x-axis. As has been pointed out in the text of this chapter, the problem can be reduced to the calculation of the probability of a displacement of x in a random walk of n steps in one dimension, each step consisting of a displacement equal in magnitude to the root-mean-square projection l/y/Z of a bond on the a -axis. Then... 

Wang, F. Landau, D. P, An efficient, multiple range random walk algorithm to calculate the density of states, Phys. Rev. Lett. 2001, 86, 2050-2053... 

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