Diffusion kernels

Figure 1. Three limiting mechanisms for atomic processes which mediate step fluctuations, a) Step-edge diffusion b) Evaporation-recondensation c) Terrace diffusion with diffusion kernel P(). By appropriate choice of P(J), this case can reduce to cases a) and b) (see text).

A behavior similar to Eq. (22) occurs for any short range diffusion kernel. For example, taking... 

The key physics of our model (see Eqs. (9) and (10)) is contained in the nonlocal diffusion kernels which occur after integrating over the atomic processes which produce step fluctuations. We have calculated these kernels for a variety of physically interesting cases (see Appendix C) and have related the parameters in those kernels to atomic energy barriers (see Appendix B). The model used here is close in spirit to the work of Pimpinelli et al. [13], who developed a scaling analysis based on diffusion ideas. The theory of Einstein and co-workers and Bales and Zangwill is based on an equihbrated gas of atoms on each terrace. The concentration of this gas of atoms obeys Laplace s equation just as our probability P does. To make complete contact between the two methods however, we would need to treat the effect of a gas of atoms on the diffusion probabilities we have studied. Actually there are two effects that could be included. (1) The effect of step roughness on P(J) - we checked this numerically and foimd it to be quite small and (2) The effect of atom interactions on the terrace - This leads to the tracer diffusion problem. It is known that in the presence of interactions, Laplace s equation still holds for the calculation of P(t), but there is a concentration... 

J. Baier, M.F. Richter, R.J. Cogdell, S. Oellerich, J. K5hler, Determination of the spectral diffusion kernel of a protein by single molecule spectroscopy. Phys. Rev. Lett. 100, 8108-1-4 (2008)... 

The organization of this chapter is as follows. Section 1.5.2 introduces the TLS model and discusses the dynamics of a single TLS. In Section 1.5.3 general results for the spectral diffusion kernel and the absorption line shape when the chromophore is coupled to an arbitrary number of TLSs are presented. Section 1.5.4 will discuss how the experimental observables (single molecule line shapes, spectral diffusion tra-... 

A related quantity, the conditional probability density (also called the spectral diffusion kernel) P co, /jojo) is defined by... 

P (o, t (Oo) is the probability density that the chromophore has transition frequency (o at time t given that it had frequency coo at time 0. While the functional form of this spectral diffusion kernel is quite complicated in general, at short times certain simplifications occur. In particular, if the positions of the TLSs occupy a regular lattice in three-dimensional space, all of the relaxation rates Kj are the same, all of the occupation probabilities pj are the same and equal to 1/2 (the high-temperature limit), and the perturbations vj are dipolar, then it was shown by Klauder and Anderson [29] and more recently by Zumofen and Klafter [30] that the spectral diffusion kernel is Lorentzian ... 

In the second type of experiment that measures single molecule spectral dynamics one performs repeated fluorescence excitation scans of the same molecule. In each scan the line shape is described as above, but now there is the possibility that the center frequency of the line will change from scan to scan because of slow fluctuations. Thus one can measure the center frequency as a function of time, producing what has been called a spectral diffusion trajectory. This trajectory can, in principle, be characterized completely by the spectral diffusion kernel of Eqs. (16) and (19), but of course it must be understood that only the slow Kj < 1 /t) TLSs contribute. In fact, the experimental trajectories are really too short to be analyzed with this spectral diffusion kernel. Instead, it is useful [11, 12] to consider three simpler characterizations of the spectral diffusion trajectories the frequency-frequency correlation function in Eq. (14), the distribution of frequencies from Eq. (15), and the distribution of spectral jumps from Eq. (21). For this application of the theoretical results, in all three of these formulas j should be replaced by s, the labels for the slow TLSs. 

The results of these measurements are given in Table I. The over-all accuracy, excluding possible systematic errors, is of the order of i 1% in the microscopic parameters, and T, While In L, Is of the order of 2%. The numbers were also used in a four-factor, diffusion kernel to epmpute -buckling values. The last two columns of the table compare the computed buckling values with the directly measured buckllngs previously reported. This comparison Indicates that the measured parameters can be used to predict the buckling with an accuracy of 3 to 4% in most cases. 

If a slowing-down model is specified, the neutron age to thermal energy may also be deduced from, the mea- sured d. The two-group diffusion kernel model gives ... 

Eig. 7. Drying of com kernels by Hquid diffusion. The dashed line is that predicted by theory based on constant diffusivity. The solid curve shows actual... 

The proposed specification of the kernel for m- and J-diffusion models is mathematically closed, physically clear and of quite general character. In particular, it takes into consideration that any collisions may be of arbitrary strength. The conventional m-diffusion model considers only strong collisions (0(a) = 1 /(27c)), while J-diffusion considers either strong (y = 0) or weak (y = 1) collisions. Of course, the particular type of kernel used in (1.6) restricts the problem somewhat, but it does allow us to consider kernels with arbitrary y < 1. 

Keilson-Storer kernel 17-19 Fourier transform 18 Gaussian distribution 18 impact theory 102. /-diffusion model 199 non-adiabatic relaxation 19-23 parameter T 22, 48 Q-branch band shape 116-22 Keilson-Storer model definition of kernel 201 general kinetic equation 118 one-dimensional 15 weak collision limit 108 kinetic equations 128 appendix 273-4 Markovian simplification 96 Kubo, spectral narrowing 152... 

F. Takahashi and V. R. Katta, Further studies of the reaction kernel structure and stabilization of jet diffusion flames, Proc. Combust. Inst. 30 383-390, 2005. 

Takahashi, R and Katta, V.R., Reaction kernel structure and stabilizing mechanisms of jet diffusion flames in microgravity, Proc. Combust. Inst., 29,2509, 2002. 

Takahashi, F and Katta, V.R., Chemical kinetic structure of the reaction kernel of methane jet diffusion flames, Combust. Sci. Technol., 155, 243, 2000. 

For the case of the spin-echo for diffusion weighting, the full kernel can be written as... 

The stimulated echo diffusion-relaxation experiment exhibits a kernel that is similar to that of the one with the pulsed field gradients ... 

Hence, a series of measurements with several Tcp values will provide a data set with variable decays due to both diffusion and relaxation. Numerical inversion can be applied to such data set to obtain the diffusion-relaxation correlation spectrum [44— 46]. However, this type of experiment is different from the 2D experiments, such as T,-T2. For example, the diffusion and relaxation effects are mixed and not separated as in the PFG-CPMG experiment Eq. (2.7.6). Furthermore, as the diffusion decay of CPMG is not a single exponential in a constant field gradient [41, 42], the above kernel is only an approximation. It is possible that the diffusion resolution may be compromised. 

Friction interactions, multiparticle collision dynamics, single-particle friction and diffusion, 114—118 Friction kernel, transition state trajectory, colored noise, 209... 

It was assumed that the nickel crystallites are rapidly enveloped in a skin of a copper-rich alloy, from which diffusion towards the center of each crystallite then takes place. If xx and x2 are the atomic fractions of copper in the two equilibrium phases and x is the atomic fraction of copper in the alloy film under consideration, then the crystallites in the annealed film may have a variety of forms. Solid solutions occur at either end of the composition range but the values of Xi and x2 at 200°C are <0.1 and 0.8. Hence, over much of the composition range (i.e., where x lies between X and xi), the Cu-Ni films should consist of crystallites with a kernel which is almost pure nickel (composition xi) enveloped in a skin of a copper-rich alloy (composition x2). Eventually, when x is only slightly larger than Xi, the alloy skin does not completely surround the nickel crystallites small patches of alloy (x2) and almost pure nickel ( ci) are both exposed. 

Hence, the decision to use a heated substrate with simultaneous evaporation of the component metals as an aid to homogenization requires consideration of whether or not it might have an adverse effect, i.e., causing preferential nucleation of one component, which interdiffusion may not be able to remedy. It was believed (60) that in preparing Pd-Rh alloys by simultaneous deposition on a substrate at 400°C, rhodium nucleated preferentially and that crystallites grew by the addition of palladium (and rhodium) atoms. The diffusion of palladium atoms into this kernel formed a phase with 88 =t 5% Rh (phase II). The outer shell of the crystallite, phase I, was in effect a solid solution deficient in rhodium compared with the overall film composition, and the Rh content of phase I therefore increased as the Rh flux was increased. 

Note that the particle diffusion term is ignored, just like particle dispersion due to SGS motions (this was found justified in a separate simulation). The shape of the sink term in the right-hand term of this equation is due to Von Smoluchowski (1917) while the local value of the agglomeration kernel /i0 is assumed to depend on the local 3-D shear rate according to a proposition due to Mumtaz et al. (1997). 

The second stage of steeping is the sulfur dioxide diffusion stage in which sulfur dioxide diffuses with the water into the corn kernel through the base end of the tip cap and moves through the cross and tube cells of the pericarp to the kernel crown and then slowly into the horny endosperm. The second stage is diffusion limited because of the specific path required for water going into the kernel. 

Due to the structure of the corn kernel (cutinized outer layer of the pericarp surrounding the corn kernel), the diffusion of water and chemicals inside the kernel is through a very specific pathway. Initial results with the use of enzymes during steeping (Figure 1) indicated that enzymes were not able to penetrate the kernels and break down the protein matrix surrounding starch particles. For enzymes to penetrate the corn kernel, it was necessary... 

However, use of the grid-cell kernel induces a deterministic error similar to numerical diffusion due to die piece-wise constant approximation. 

Analytic or semi-analytic many-body methods provide an independent estimate of ec( .>0- Before the Diffusion Monte Carlo work, the best calculation was probably that of Singwi, Sjblander, Tosi and Land (SSTL) [38] which was parametrized by Hedin and Lundqvist (HL) [39] and chosen as the = 0 limit of Moruzzi, Janak and Williams (MJW) [40]. Table I shows that HL agrees within 4 millihartrees with PW92. A more recent calculation along the same lines, but with a more sophisticated exchange-correlation kernel [42], agrees with PW92 to better than 1 millihartree. 

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