Dirac delta function models

Mathematically,/(l) can be determined from F t) or W t) by differentiation according to Equation (15.7). This is the easiest method when working in the time domain. It can also be determined as the response of a dynamic model to a unit impulse or Dirac delta function. The delta function is a convenient mathematical artifact that is usually defined as... 

When a is large (> 100) then the peak is symmetrical with a mean value of 1.20. In the extreme as a - then the simulated peak approaches the input function which in our model simulation is a Dirac delta function at 0 = 1.2. In the real situation it should approach the true MWD of the polymer being analysed. As a decreases the simulated peak is first broadened and then skewed. It is apparent that the peak maximum shifts in a manner expected. 

The family of curves represented by eqn. (46) is shown in Fig. 11 and the mean and variance of both the E(f) and E(0) RTDs are as indicated in Table 5. When N assumes the value of 0, the model represents a system with complete bypassing, whilst with N equal to unity, the model reduces to a single CSTR. As N continues to increase, the spread of the E 0) curves reduces and the curve maxima, which occur when 0 = 1 —(1/N), move towards the mean value of unity. When N tends to infinity, E(0) is a dirac delta function at 0 = 1, this being the RTD of an ideal PER. The maximum value of E(0), the time at which it occurs, or any other appropriate curve property, enables the parameter N to be chosen so that the model adequately describes an experimental RTD which has been expressed in terms of dimensionless time see, for example. Sect. 66 of ref. 26 for appropriate relationships. 

Another approach to approximating the scaling functions is to construct some model spectral densities. In turn, Eqs. (14)—(19) yield the approximate Fe and Fx. The choice of the Dirac delta function for T yields trivial relations... 

The derivation of the given whole field formulation, introducing the Dirac delta function (d/) into the surface tension force relation to maintain the discontinuous (singular) nature of this term, is to a certain extent based on physical intuition rather than first principles (i.e., in mathematical terms this approach is strictly not characterized as a continuum formulation on the differential form). Chandrasekhar [31] (pp 430-433) derived a similar model formulation and argued that to some extent the whole field momentum equation can be obtained by a formal mathematical procedure. However, the fact that the equation involves /-functions means that to interpret the equation correctly at a point of discontinuity, we must integrate the equation, across the interface, over an infinitesimal volume element including the discontinuity. 

Note that, since L has units (m/sf, the nonnegative function h ) would be dimensionless. With this model for A the realizability condition in Fq. (B.52) would always yield a nonzero upper bound on At when h ) is finite. physically, E is null in the limit of pure particle trajectory crossing where the true NDF is a sum of Dirac delta functions. On the other hand, when E reaches its maximum value, the NDF is Gaussian. Thus, since mixed advection is associated with random particle motion, the model in Fq. (B.56) also makes physical sense. Nonetheless, the potential for singular behavior in the update formula makes the treatment of mixed advection problematic. 

How can we model such a problem To do the analysis we need to introduce and become comfortable with two new functions the Dirac-Delta function and the UnitStep (or Heaviside) function. The Dirac-Delta function is infinitely intense and infinitesimally narrow—like a pulse of laser light. We can imagine it arising in the following way. We begin by considering a pulse that is quite broad, such as the function that is plotted here ... 

In Eq. 5, the surface tension force p is modeled using the continuum surface force model [5]. It is reformulated as an equivalent body force acting within a band of 2a at the interface using a smeared-out Dirac delta function D(iJ). The surface tension force can be expressed as... 

Although the band intensity in Eqn 24.1 is written with an infinitesimally narrow bandwidth (via the Dirac delta function), in practice the delta function is replaced by a more realistic band shape factor with a finite bandwidth. There are several contributions to the bandwidth including vibrational relaxation and dephasing, inhomogeneous broadening, and lifetime broadening. Since each of these separately is difficult to calculate exactly, the bandwidth is often represented by a phenomenological function based on an appropriate model. Bandwidth analyses are most... 

The model is applicable for any form of the energy distribution. The energy distribution is a result of structural heterogeneity or distribution of surface defects or other surface factors (for example, surface chemistry) that cause the variation in the adsorption energy. Two extremes we could expect for the energy distribution. One is the ideal surface where the distribution is the Dirac delta function, i.e. 

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