Delta function, definition

It is often important to be able to extend our present notion of conditional probability to the case where the conditioning event has probability zero. An example of such a situation arises when we observe a time function X and ask the question, given that the value of X at some instant is x, what is the probability that the value of X r seconds in the future will be in the interval [a,6] As long as the first order probability density of X does not have a Dirac delta function at point x, P X(t) = x = 0 and our present definition of conditional probability is inapplicable. (The reader should verify that the definition, Eq. (3-159), reduces to the indeterminate form in this case.)... 

Solution This solution illustrates a possible definition of the delta function as the limit of an ordinary function. Disturb the reactor with a rectangular tracer pulse of duration At and height A/t so that A units of tracer are injected. The input signal is Cm = 0, t < 0 = A/Af, 0 < t < At ... 

The vertical spring and mass is an example of a stable system and by definition this means that an arbitrary small external force does not cause the mass to depart far from the position of equilibrium. Correspondingly, the mass vibrates at small distances from the position of equilibrium. Stability of this system directly follows from Equation (3.102) as long as the mechanical sensitivity has a finite value, and it holds for any position of the mass. First, suppose that at the initial moment a small impulse of force is applied, delta function, then small vibrations arise and the mass returns to its original position due to attenuation. If the external force is small and constant then the mass after small oscillations occupies a new position of equilibrium, which only differs slightly from the original one. In both cases the elastic force of the spring is directed toward the equilibrium and this provides stability. Later we will discuss this subject in some detail. 

The solution of Eq. (78) can be obtained with the use of the Laplaee transform. However, it is first necessary to develop the expression for the Laplace transform of the delta function, as given on the right-hand side of Eq. (78). With the use of the definition of the Laplace transform [Eq. (43)] and f(t) = (t -t ), the desired result becomes... 

The kernel K(x,x — y) must satisfy this constraint for any integrable function (p. This is just the definition of the Dirac delta function K(x, x — y) = S(x — y). Note that, in this limit, the kernel function is equivalent to the filter function used in LES. As is well known in LES, filtering a function twice leads to different results unless the integral condition given above is satisfied. 

Since all tracer entered the system at the same time, t = 0, the response gives the distribution or range of residence times the tracer has spent in the system. Thus, by definition, eqn. (8) is the RTD of the tracer because the tracer behaves identically to the process fluid, it is also the system RTD. This was depicted previously in Fig. 3. Furthermore, eqn. (8) is general in that it shows that the inverse of a system transfer function is equal to the RTD of that system. To create a pulse of tracer which approximates to a dirac delta function may be difficult to achieve in practice, but the simplicity of the test and ease of interpreting results is a strong incentive for using impulse response testing methods. 

Alternatively, a Gaussian could have been used [491, 499]. In the limit A -+ 0, only when x is very close to x0 (i-e- x — x0 is about zero) does the delta function depart from zero and there it tends to infinity. From the definition of the delta function [e.g. eqn. (310)]... 

Substitution of the resulting equation into %"BB(k, co) and subsequent use of the definition of the delta function yields... 

The definition and properties of the delta function will reduce this equation to the solution of the PDE for the approximate u at each collocation point Xj... 

Distributions like those in Figure 10.4, for example, indicate that Yp or T differs from Yp(Z) or T(Z), respectively. If mixing were complete in the sense that all probability-density functions were delta functions and fluctuations vanished, then differences like T — T Z) would be zero. That this situation is not achieved in turbulent diffusion flames has been described qualitatively by the term unmixedness [7]. Although different quantitative definitions of unmixedness have been employed by different authors, in one way or another they all are measures of quantities such as Yp — Yp(Z) or T — T(Z). The unmixedness is readily calculable from P(Z), given any specific definition (see Bilger s contribution to [27]). 

A simple extension is the three-dimensional delta function (x — y), whose definition is... 

Three of the five terms in the final rearrangement contain operator strings of reduced length, and the first term contains only Kronecker delta functions. Note also that all the operator strings on the right-hand side of the final equality are normal-ordered by Merzbacher s definition. If we now evaluate the quan-... 

To put the definition of this property into direct correspondence with the definition of other atomic properties, as one for which the property density at r is determined by the effect of the field over the entire molecule, we express the perturbed density in terms of the first-order corrections to the state function. This is done in a succinct manner by using the concept of a transition density (Longuet-Higgins 1956). The operator whose expectation value yields the total electronic charge density at the position r may be expressed in terms of the Dirac delta function as... 

As a consequence of this definition, if f (x) is an arbitrary function which is well-defined at X = 0, then integration of /(x) with the delta function selects out the value of f (x) at the origin... 

In laminar flow, /f reduces to a delta function. In turbulent flow, /f can be modeled using PDF methods. Thus, by definition, there is no mass transfer between phases. 

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