Delta Dirac

The Boltzmann constant is ks and T the absolute temperature. — is the Dirac delta function. Below we assume for convenience (equation (5)) that the delta function is narrow, but not infinitely narrow. The random force has a zero mean and no correlation in time. For simplicity we further set the friction to be a scalar which is independent of time or coordinates. 

The application in [24] is to celestial mechanics, in which the reduced problem for consists of the Keplerian motion of planets around the sun and in which the impulses account for interplanetary interactions. Application to MD is explored in [14]. It is not easy to find a reduced problem that can be integrated analytically however. The choice /f = 0 is always possible and this yields the simple but effective leapfrog/Stormer/Verlet method, whose use according to [22] dates back to at least 1793 [5]. This connection should allay fears concerning the quality of an approximation using Dirac delta functions. 

Dirac delta function S Elution volume, exclusion Vo... 

Dirac delta function, an ideal pulse change ... 

Inthiscasethenonzerovalueofthefunctionhasbeenpushedtothepositiveextremumof this interval on t. If we integrate the product of the dye mass m[Pg.184]

Fig, 7,2 Spin-glass overlap probability Pspin-giaes(5) versus overlap q (equation 7,57) 6 x) is the Dirac-Delta function. 

Fig. 3-3. Some Important Probability Density Functions and Their Corresponding Distribution Functions. Arrows are used to indicate Dirac delta functions with the height of the arrow indicating the area under the delta function.

Xq) denotes a unit Dirac delta function located at x x0,... 

The next example will illustrate the technique of calculating moments when the probability density function contains Dirac delta functions. The mean of the Poisson distribution, Eq. (3-29), is given by... 

In most cases of interest, this n + m order derivative can be written as an ordinary n + mth order derivative and some Dirac delta functions. Situations do exist in which this is not true, but they do not seem to have any physical significanpe and we shall ignore them. In any event, all difficulties of this nature could be avoided by replacing integrals involving probability density functions by their corresponding Lebesque-Stieltjes integrals. 

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.)... 

The statistical matrix may be written in the system of functions in which the coordinate x is diagonal. In one dimension, the eigenfunction of is the Dirac delta function. The expansion of (x) in terms of it is... 

The right side of this is the Dirac delta function.3... 

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... 

The Fourier representation of the Dirac delta function leads then to the result... 

The integral over k may be expressed in terms of the Dirac delta function through equation (C.6) in Appendix C, so that we have... 

This relation may be obtained by the same derivation as that leading to equation (B.28), using the integral representation (C.7) for the three-dimensional Dirac delta function. 

The follotving properties of the Dirac delta funetion can be demonstrated by multiplying both sides of each expression hy /(x) and observing that, on integration, eaeh side gives the same result... 

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