40
STAXI.EY CORRSIN
spatial domains of size SZ have equal velocities at the same t , while q is the
probability that a point in space retains the same velocity over successive
time intervals St, then Patterson sets the following limiting process:
q 4 I with ( I - q)/& 4 1/2T as 62 + 0.
I
p 4 1 with (1 - p)/dZ 4 1/2L
(47)
and SZ/Gt-+ V
Clearly the limits are like Goldstein's, but the process is Eulerian. L and T
have been defined as Eulerian length and time scales.
Patterson's generalized telcgraph equation for the probability density
function of the particle walks which can occur on an ensemble of these
random fields is
1
(48)
fl, + - - p + -
2T
4LT
where
(49a)
a kind of" telegraph operator" on P,
(49b)
p(:, t ) E P,, + I /T(I + v n - *)P, - v*pZ, ,
A(r) = (4T/t) exp( -t/2T)II(r/2T),
and
I I is the modified Bessel function of first order. For L = a, this reduces to
the (ioldstcin case p = 0. as it should. In the Patterson equation, the line
: = V r is singular, as is the analogous line in the telegraph case. Equation
(48) has not yet been solved, although Patterson has developed a simpler
form by Laplace transformation.
9. A RANDOM WALK WIW LARGE STEPS
The wavelike character of a simple random walk at "small time" or with
"large mean free path" (Lea, step length not very much smaller than a
characteristic length of the mean conmtration field), is dramatically
demonstrated by the binary velocity caac. To make the problem more
physical," the initial concentration field is taken as