Jump process

Figure 5. Arrhenius-plot of three relaxation times involved in total ordering process. ( ) Xi, ( ) X2, atomic jump processes within domains (o) xj, change in domain size.

Atomic jump processes studied by order-order relaxation experiments, Acta Mater. 44 1573 (1996). 

It should be emphasized here that usual tracer diffusion experiments in LI2 ordered alloys due to diffusion of majority atoms mainly over their own sublattice do not give any of the strongly desired information about ordering kinetics. The study of order-order relaxations in contrast, yields a selective information Just about those atomic Jump processes which are related to ordering phenomena. 

In molecular doped polymers the variance of the disorder potential that follows from a plot of In p versus T 2 is typically 0.1 eV, comprising contributions from the interaction of a charge carrier with induced as well as with permanent dipoles [64-66]. In molecules that suffer a major structural relaxation after removal or addition of an electron, the polaron contribution to the activation energy has to be taken into account in addition to the (temperature-dependent) disorder effect. In the weak-field limit it gives rise to an extra Boltzmann factor in the expression for p(T). More generally, Marcus-type rates may have to be invoked for the elementary jump process [67]. 

Extraframework cations are needed in anionic zeolites for charge balance, and for several zeolite topologies their locations are well investigated [281, 282]. Different cations have been investigated by solid state NMR in the past with different NMR properties and different project targets. We restrict this section to a tutorial example on sodium cation motion in sodalite and cancrinite structures [283-285], 23Na has a nuclear electric quadrupole moment, and quadrupolar interaction is useful to investigate jump processes, especially when they are well defined. 

Local mixing is best defined in terms of stochastic models. However, this condition is meant to mle out models based on jump processes where the scalar variables jump large distances in composition space for arbitrarily small df. It also rales out interactions between points in composition space and global statistics such as the mean. 

As will be shown for the CD model, early mixing models used stochastic jump processes to describe turbulent scalar mixing. However, since the mixing model is supposed to mimic molecular diffusion, which is continuous in space and time, jumping in composition space is inherently unphysical. The flame-sheet example (Norris and Pope 1991 Norris and Pope 1995) provides the best illustration of what can go wrong with non-local mixing models. For this example, a one-step reaction is described in terms of a reaction-progress variable Y and the mixture fraction p, and the reaction rate is localized near the stoichiometric point. In Fig. 6.3, the reaction zone is the box below the flame-sheet lines in the upper left-hand corner. In physical space, the points with p = 0 are initially assumed to be separated from the points with p = 1 by a thin flame sheet centered at... 

Pulsed field gradient (PFG)-NMR experiments have been employed in the groups of Zawodzinski and Kreuer to measure the self-diffusivity of water in the membrane as a function of the water content. From QENS, the typical time and length scales of the molecular motions can be evaluated. It was observed that water mobility increases with water content up to almost bulk-like values above T 10, where the water content A = nn o/ nsojH is defined as the ratio of the number of moles of water molecules per moles of acid head groups (-SO3H). In Perrin et al., QENS data for hydrated Nation were analyzed with a Gaussian model for localized translational diffusion. Typical sizes of confining domains and diffusion coefficients, as well as characteristic times for the elementary jump processes, were obtained as functions of A the results were discussed with respect to membrane structure and sorption characteristics. ... 

Proof for Continuous-Time Jump Processes These stochastic processes are ruled by master equations of Pauli type ... 

These results are commonly interpreted in terms of distributions of elemental local jump processes [116]. For instance, a superposition of Debye elemental processes with a Gaussian distribution of energy barriers g(E) ... 

The description of the chain dynamics in terms of the Rouse model is not only limited by local stiffness effects but also by local dissipative relaxation processes like jumps over the barrier in the rotational potential. Thus, in order to extend the range of description, a combination of the modified Rouse model with a simple description of the rotational jump processes is asked for. Allegra et al. [213,214] introduced an internal viscosity as a force which arises due to a transient departure from configurational equilibrium, that relaxes by reorientational jumps. Thereby, the rotational relaxation processes are described by one single relaxation rate Tj. From an expression for the difference in free energy due to small excursions from equilibrium an explicit expression for the internal viscosity force in terms of a memory function is derived. The internal viscosity force acting on the k-th backbone atom becomes ... 

Orbit is reached by optimization of the energy density functional through inter-orbit jumping. This process, which is illustrated in Fig. 7 by means of a sequence of arrows, is discussed in detail below. The inter-orbit jumping process is repeated until one finally reaches orbit This is the orbit where, by definition, one finds the exact ground state wavefunction that satisfies the Schrodinger equation = El Pl. For this reason, we call this the... 

In this section, we consider the description of Brownian motion by Markov diffusion processes that are the solutions of corresponding stochastic differential equations (SDEs). This section contains self-contained discussions of each of several possible interpretations of a system of nonlinear SDEs, and the relationships between different interpretations. Because most of the subtleties of this subject are generic to models with coordinate-dependent diffusivities, with or without constraints, this analysis may be more broadly useful as a review of the use of nonlinear SDEs to describe Brownian motion. Because each of the various possible interpretations of an SDE may be defined as the limit of a discrete jump process, this subject also provides a useful starting point for the discussion of numerical simulation algorithms, which are considered in the following section. 

Fig. 12. Mossbauer line shape of 57Fe in a constant eqQ field and a random hyperfine field assumed to be a two-state-jump process for different values of the jump rate W. (a) The random hyperfine field along the eqQ axis. (b) Perpendicular to the eqQ axis. (Calculations of Tjon and Blume.)...

We shall meet more general Fokker-Planck equations the special form (1.1) is also called Smoluchowski equation , generalized diffusion equation , or second Kolmogorov equation . The first term on the right-hand side has been called transport term , convection term , or drift term the second one diffusion term or fluctuation term . Of course, these names should not prejudge their physical interpretation. Some authors distinguish between Fokker-Planck equations and master equations, reserving the latter name to the jump processes considered hitherto. 

One sees that the Ito equation reproduces the correct equation (V.1.9) for the average and the Stratonovich equation does not. The reason is that in this process the jump probability from n to n — 1 is proportional to the number n of available nuclei before the jump, just as was assumed by Ito in (4.11). The same thing holds for chemical reactions, emission of photons, etc. But the higher moments give difficulties. At any rate these jump processes are better treated by means of the master equation - to be solved by means of the expansion developed in the next chapter. 

The relationship between the macroscopic isotropic diffusivity, D, and microscopic jump processes can be evaluated in three dimensions. The equivalence of Eqs. 7.31 and 7.35 means that... 

With the cylindrical cryptands, each macrocycle may bind one cation so that both mono- and dinuclear cryptates may be formed. Although the 12-membered (N202) macrocycles of ligand 5 are too small to bind two cations within each of the macrocycles, variable temperature 13C-NMR measurements have revealed intramolecular cation exchange between identical sites at the top and bottom of this cryptand, for Ca2+, Sr2+, and Ba2+. Cation jump between the two sites is fast with respect to intermolecular cation exchange, modeling the elementary jump processes of cations between binding sites in membrane channels (91). 

Polymers, because of their long chemical structure cohere as solids even when discrete section of the chain are undergoing Brownian motions moving by diffusional jump processes from place to place. This is the main difference between elastic solids and polymers [1-7]. 

Here DR and >, are the diffusion coefficients for the isotropic overall and free internal motions, respectively. Equation 31 assumes a diffusional process for the methyl group. If a jumping process between three equivalent positions separated by 120° is considered,47 the last term becomes C/(6DK + Z), ). Parameters A, B, and C are geometric constants similar to those in Eq. 27, but here the angle is that formed between the methyl C—H vectors and the axis of rotation. Assuming tetrahedral angles, for free internal motion ( >, ) ), 1/7 ,(CH3) is decreased to one-ninth of the value expected for a rigidly attached CH carbon. For slow internal rotation (D, — ) ), l/r1(CH3) becomes one-third of the value of a methine carbon in the same molecule, as predicted by Eq. 16. 

The description of the internal motion of the epoxypropyl ring of 23 is strongly model-dependent. This motion can be satisfactorily approximated either by free rotation about the C-5—C-6 bond or by a jumping process between two stable conformations. Discrimination between these two models from the relaxation data was not possible owing to a fortuitous similarity in the activation energies ( 17 kJ/mol) of the internal and overall diffusional motions.13 Inspection of molecular models indicates, however, that the rotation of the epoxypropyl ring is not sufficiently constrained to justify restricted rotation about the C-5—C-6 bond. 

Contents 1. Introduction 104 2. Theory 106 2.1. Specifically for spin-1 nuclei 112 2.2. Two-axis jump processes 114 2.3. 2-by-2-site jump 114 2.4. 2-by-3-site jump 115 3. Numerical Simulations 117 3.1. Spin-1 nuclei 118 3.2. Half-integer quadrupolar nuclei 118 4. Results and Discussion 119 4.1. Spin-1 nuclei 119 4.2. Dynamic effects in 14N MAS spectra by SQ or DQ coherences 123 4.3. Multi-axis jump processes 124 4.4. Half-integer quadrupolar nuclei 129 5. Conclusions 134 Acknowledgements 135 References 135... 

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