BlNARY RANIXIM N A I KS: ADDENDUM
63
with
e-r'2TI ,{r/2T)
1,/2(t/2T j.4(1) = - - -
In the limit L -+ 'xi. this reduces to Goldstein's problem, and we recover the
telegraph equation.
The merit in Eq. (3) and its corollaries lies less in the possibility that it
might model turbulent diffusion, but more in the heuristic example it gives
by illustrating the extreme difficulties in obtaining a solution to an EulerLagrange problem and the inefficacy of ad koc type solutions. As regards
solutions of the Euler-Lagrangc problem where the random velocity fields
concerned are solutions to the Navicr-Stokes equations, there exists to my
knowledge only one analytical solution- that ofKamp6de Fhriet (1959) for
a quite restricted class o onedimchsional shearing motions, and one computer experiment using f numerically generated etatistically isotropic, but
nonstationary fields (Riley and Patterson, 1974).
2. FIN^ WALKS
2.1 Preliminary Mathematics
Binary random walks where a partick move either one place to the right
or one place to the left at each step can be treated mathematically in a very
straightforward way by using binary algebra. Let a given scalar function
take on values 1 or 0 (true or false, respectively, if one thinks of the function
as the occurrence of an event). Define multiplication identically with conventional multiplication and addition ps addition modulo 2, so that 1 @I 1 = 0.
Define the complement of a scalar A as
(4)
A ' = l $ A
(Note that A @ A = 0, A 0 A' = I, AA' = 0. Most proofs can be done by
cx haustion ; i.e., by enumerating all possibilities.)
If A is it random binary event, the ensemble average ( A ) is defined as
1 "
n-w n 1- I
( A ) = Irm -- C A ,
where ,4, is the value of A in the ith realization of the ensemble. By the strong
law of large numbers (Feller, 1957),
( 6 )
Prob{A = I ) = probability that A = t OT probability
that the event, occurs
= ( A )
63
with
e-r'2TI ,{r/2T)
1,/2(t/2T j.4(1) = - - -
In the limit L -+ 'xi. this reduces to Goldstein's problem, and we recover the
telegraph equation.
The merit in Eq. (3) and its corollaries lies less in the possibility that it
might model turbulent diffusion, but more in the heuristic example it gives
by illustrating the extreme difficulties in obtaining a solution to an EulerLagrange problem and the inefficacy of ad koc type solutions. As regards
solutions of the Euler-Lagrangc problem where the random velocity fields
concerned are solutions to the Navicr-Stokes equations, there exists to my
knowledge only one analytical solution- that ofKamp6de Fhriet (1959) for
a quite restricted class o onedimchsional shearing motions, and one computer experiment using f numerically generated etatistically isotropic, but
nonstationary fields (Riley and Patterson, 1974).
2. FIN^ WALKS
2.1 Preliminary Mathematics
Binary random walks where a partick move either one place to the right
or one place to the left at each step can be treated mathematically in a very
straightforward way by using binary algebra. Let a given scalar function
take on values 1 or 0 (true or false, respectively, if one thinks of the function
as the occurrence of an event). Define multiplication identically with conventional multiplication and addition ps addition modulo 2, so that 1 @I 1 = 0.
Define the complement of a scalar A as
(4)
A ' = l $ A
(Note that A @ A = 0, A 0 A' = I, AA' = 0. Most proofs can be done by
cx haustion ; i.e., by enumerating all possibilities.)
If A is it random binary event, the ensemble average ( A ) is defined as
1 "
n-w n 1- I
( A ) = Irm -- C A ,
where ,4, is the value of A in the ith realization of the ensemble. By the strong
law of large numbers (Feller, 1957),
( 6 )
Prob{A = I ) = probability that A = t OT probability
that the event, occurs
= ( A )