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... 

These equations determine both moments, provided their initial values are known. Our aim was to solve the master equation with initial delta distribution (2.8) and we took x0 as initial value of the macroscopic part, see (3.2). It follows that the initial fluctuations vanish,... 

Exercise. Solve the diffusion equation in an inverted parabolic potential with initial delta distribution,... 

The quantity 8Ia is the linearized collision operator corresponding to (4.62). The subscript 0 in the second term means that the operator acts only on the function fa and not on the delta distribution. The contribution Aap in (5.14) is determined by /a/3, that is by the large-scale fluctuations. We consider only the approximation that Aa/ is equal to zero. 

If we only consider the static case w = 0we can distinguish two limiting cases First, if the typical range rjj of the screened Coulomb potential Vc is much smaller than the mean electron distance, the potential can be assumed to be a delta distribution and Ttc can be approximated by... 

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... 

The computed residence time distribution of the Sy elements is closely approximated by a delta distribution, and that is what is assumed here. Further, the reactant mole ratio in the Sy elements is taken to be the same as in the overall feed. 

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. 

Two special cases, the initial delta distribution and initial Boltzmann distribution are of interest. If... 

Now, if xCQ-ufoO) X (Q-u(ai°)) = 8(Q-u(aj0)) (Dirac s delta distribution) one gets (q u(ai°)) = 0, (q c(°) as one would formally expect. The difference is now apparent. The frame oq0 is related to the classical arrangement of sources and can be used to define frames helping describe the motion of the nuclear masses. But Q0 is defined with the nuclear framework. To use the nuclear coordinates for that task leads to problems as shown by Woolley and Sutcliffe [1 4 7 17]. 

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-... 

The four-dimensional delta distribution arises as a result of four-momentum conservation. If the momentum transfer g = p2 — P2 is small compared to mz, then... 

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... 

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