Distribution of jump distance

In Sect. 3, the Noyes approach to analysing reaction rates based on the molecular pair approach is discussed [5]. Both this and the diffusion equation analysis are identical under conditions where the diffusion equation is valid and when the appropriate recombination reaction rate for a molecular pair is based on the diffusion equation. Some comments by Naqvi et al. [38] and Stevens [455] have obscured this identity. The diffusion equation is a valid approximation to molecular motion when the details of motion in a cage are no longer of importance. This time is typically a few picoseconds in a mobile liquid. When extrapolating the diffusion equation back to such times, it should be recalled that the diffusion is a continuum form of random walk [271]. While random walks can be described with both a distribution of jump frequencies and distances, nevertheless, the diffusion equation would not describe a random walk satisfactorily over times less than about five jump periods (typically 10 ps in mobile liquids). Even with a distribution of jump distances and frequencies, the random walk model of molecular motion does not represent such motion adequately well as these times (nor will the telegrapher s or Fokker-Planck equation be much better). It is therefore inappropriate to compare either the diffusion equation or random walk analysis with that of the molecular pair over such times. Finally, because of the inherent complexity of molecular motion, it is doubtful whether it can be described adequately in terms of average jump distances and frequencies. These jump characteristics are only operational terms for very complex quantities which derive from the detailed molecular motion of the liquid. For this very reason, the identification of the diffusion coefficient with a specific jump formula (e.g. D = has been avoided. 

The type of diffusion discussed here may be termed normal" or Gaussian diffusion. It arises simply from the statistics of a process with two possible outcomes, which is attempted a very large number of times. In Section 2.1.2.7, the statistical basis of diffusion is enlarged to include random walks in continuous rather than discrete time, and also situations where different distributions of jump distances occur. 

In recent years, more complex types of transport processes have been investigated and, from the point of view of solid state science, considerable interest is attached to the study of transport in disordered materials. In glasses, for example, a distribution of jump distances and activation energies are expected for ionic transport. In crystalline materials, the best ionic conductors are those that exhibit considerable disorder of the mobile ion sublattice. At interfaces, minority carrier diffusion and discharge (for example electrons and holes) will take place in a random environment of mobile ions. In polycrystalline materials the lattice structure and transport processes are expected to be strongly perturbed near a grain boundary. 

Disorder was introduced into this system by postulating a distribution of waiting times. A complementary extension of the theory may be made by considering a distribution of jump distances. It may be shown that, as a consequence of the central limit theorem, provided the single-step probability density function has a finite second moment, Gaussian diffusion is guaranteed. If this condition is not satisfied, however, then Eq. (105) must be replaced by... 

Figure 6 The distribution of jump lengths of embedded atoms in Cu(00 1), measured for (a) 1461 In hops at 320 K and (b) 887 Pd hops at 335 K. Plotted is the probability for a jump of an In/Pd atom from its starting position to any site at a given distance from the starting position. Probabilities have been normalized so that the probabilities for the entire lattice add up to one. To illustrate the distinct nature of the diffusion behavior, the Gaussian jump length distribution that would be expected for the case of simple hopping is also plotted in (a).

Regarding the distribution of jump lengths, we know that the settlers did not always cover the same distance, and the distribution w x) should include the possibility of different jump lengths. This can be done by fitting the observed data to a continuous distribution. In [64], the jump distances covered by settlers were estimated from individual records obtained from the migrations.org project database, available at http //www.migrations. org. The authors collected 400 individual... 

To get an idea of the probable displacement distribution of the atoms, the jump distances are squared, and the mean of these squares, over the n steps, is computed. Thus, for a single walk ... 

The deviations from Gaussian behaviour were successfully interpreted as due to the existence of a distribution of finite jump lengths underlying the sublinear diffusion of the proton motion [9,149,154]. A most probable jump distance of A was found for PI main-chain hydrogens. With the model... 

Fig.4.19 Tseif(Q) obtained for a all the protons in PVE empty MD simulations,/ /// NSE, /=0.55) and b the main chain (filled circle, /=0.66) and the side group hydrogens (empty circle, /=0.51), both from the MDS. Dotted lines are expected Q-dependence from the Gaussian approximation in each case. Solid lines are description in terms of the anomalous jump diffusion model. Insets Chemical formula of PVE (a) and distribution functions obtained for the jump distances (b)...

Hall and Ross (HR) [14] here, the jump distance is not fixed, but one has a jump length distribution of the form... 

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