Dirac delta distribution

An alternative and illustrative derivation of the residence-time equations involves Dirac delta-distributions. Let us assume that Cin (f) = M (0)8(f)/Q or, equivalently, that a mass M (0) of i is injected into the reservoir at t=0, since... 

Wu, Yuan and Xu have studied nonuniform pellets in the hydrogenation of CO over Ni on alumina supports.39 Selectivity is directly related to the CO/H2 ratio inside the pellets. The active Ni particles were deposited as a narrow band in order to approximate a Dirac delta distribution. Experimental data followed theoretical predictions of the researchers regarding enhanced selectivity to... 

It is noted that, if the Dirac delta distribution is placed away from the external surface of the pellet (i.e.,suniformly distributed catalyst This is shown in Figure 9.7(a) [Yeung et al., 1994]. 

When the Dirac delta distribution is placed closer to the permeate side (i.e., a subsurface step distribution) of an CMR, the total conversion is actually lower than that with a uniform catalyst distribution (Figure 9.7). For a performance index other than the total conversion (such as product purity or product molar flow rate), the optimal distribution of the catalyst concentration can be rather complex even for reversible first-order reactions as displayed in Figure 9.8. 

This result is exactly identical to the equation given by Kim and Page [33] on the basis of another theory. Now, we might see that this is the density function of a modulated Gaussian distribution, where the modulating term has finite amplitude which runs over in time, while the Gaussian distribution sharpens toward to a Dirac delta distribution. This means that the particle will get closer and closer to the equilibrium point as f From last result, we can conclude that in the case of (3 -> 0 we get back to the well-known wave function of the simple oscillator. 

This is the charge density distribution for the point-like nucleus case (PNC), which we include for completeness and because of the importance of this model as a reference for any work with an extended model of the atomic nucleus (finite nucleus case, FNC). The charge density distribution can be given in terms of the Dirac delta distribution as... 

A uniform distribution of charge over the surface of a sphere of radius R can be represented as charge density distribution in terms of the Dirac delta distribution as follows ... 

We only briefly mention that a similar modification, i.e., a change from a Dirac delta distribution to an extended distribution, would be required for the spin-dependent electron-nucleus contact term, known as Fermi contact term, if the usual point-like nuclear magnetization distribution (the pointlike nuclear magnetic dipole approximation) is replaced by an extended nuclear magnetization distribution. 

The use of extended nuclear charge density distributions, instead of the simple point-like Dirac delta distribution, is almost a standard in present-... 

Taking into account that the integration over n. yields another Dirac delta distribution 0 — 0 ) we finally obtain... 

Imagine a distribution po(X) which we may take to be an initial macroscopic state of a stochastic differential equation system. This might be a smooth probability density such as a Gaussian, the indicator function for a small disk D in the phase space, or, in the extreme case a Dirac delta distribution (indicating that all initial conditions are clustered at a single point in phase space). The density evolves according to the partial differential equation... 

It is clear from Eq. (4.79) that the Langevin law can be obtained for infinitely narrow Dirac-delta distribution function x(P) = 8(P - P) (i.e. all particles have the same radius) and low energy barriers Po( << ksT so that exp(-Po AbP) 1. This means that nanoparticle radius is much smaller than the freezing one R Rf(T). The explicit expression reads ... 

The dependence of mean polarization (P3) on the applied electric field in shown in Fig. 4.38a for uniaxial material (Rochelle Salt) parameters, Dirac-delta distribution [T(7 ) = 8(7 — Rq) of particle sizes, fixed freezing radius Rf(T) at temperature 7 = 0 °C and different average nanoparticle radii Rq. Curves 1 for Rq < Rf correspond to Langevin law, while curves 5-7 for Rq > Rf indicate the hysteresis loop appearance. 

One should note the clever way in which the Dirac delta distribution makes its appearance on the right. In view of Eq. (7.5.21), adapted to the continuum case, we end up with the following differential equation for GJj- — r ) ... 

Technically, Gs is actually a Green function, because the right-hand side involves the Dirac delta distribution. 

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