x decline faster than any algebraic function (something that you need to remember from calculus), the product goes to 0. However, if x = 0, the exponential term is always equal to 1, but the reciprocal square root still goes to infinity as t approaches 0. What do we conclude from this discussion? When v = 0, as t! 0 px, t) approaches a function 6(x) with these properties oo

In physics, a function with these properties is called the Dirac delta function, named after Paul Dirac (see Connections); in applied mathematics it is called a generalized function, so named by Sir James Lighthill (Lighthill 1958), although at the time he was just M. J. Lighthill (also see Connections). We will use generalized functions considerably when we deal with stochastic population theory in Chapters 7 and 8.

For the time being, this takes care of the infinite domain. What about a bounded region? That is, suppose that the range that the walker can take is 0 < x < L. Once again, we need to apply two spatial conditions. We might have one at x = 0 and one at x = L, or both at one of the boundaries. For example, if the walker disappears (is absorbed) at x = 0 or x = L, we would have the boundary condition ¿>(0, t) = pL, t) = 0 because there is no chance of finding the walker at those points. Suppose, on the other hand, we knew that the walker was always constrained to stay within (0, L). We then have the condition that J^px , t)dx = 1 and if we differentiate this equation with respect to time, we obtain J^^x , t) = 0. We now use Eq. (2.51) to conclude that tf [—vp + ((j2/2)pjdx = 0, which we integrate to obtain

0 0

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