7. c‘ohimh;c;ocs LIMITS cw FOKMAL RANDOM WALKS
The simple diffusion equation [Eq. (l4)] is a particular continuous limit of
the simplest formal random walk process. a connection apparently made
first in essence by Rayleigh (1894) in 8 different context.
The siniple wave equation is a different continuous limit of the random
walk process. In a sense this is implied by the more general work of Goldstein (1951), bur seems to have been explicitly stated first in unpublished
reports by Michelson (1954). and by navies et al. (1954).
Motivated by the wish to subsume in a single model the highly correlated
’‘ small time *’ dispersion effect of turbulent motion with the uncorrelated
“large time” effect, Goldstein (1951) dcviscd a random walk limiting
procedure which led to the so-called “telegraph equation.” A similar hyperbolic model [the diffusion equation, Cq. (14), is parabolic] had been
proposed :ilso be Lyapin (1948. 1950; see also the paper of Monin, 1955).
Without the details of the general difference equation for the random walk,
a sense of the three different limits alluded to above can be given briefly.
We consider the elementary, onedimensional walk on a regular lattice.
Each step length is SZ and has duration 61. Suppose that the probability
that two successive steps are in the same direction is p; the probability of
a reversal must. of course, be 1 - p.
The difliision cqzration limit follows from choosing
(34)
p = f ; (SZ)*ldf 4 2 0 as 6Z-+O.
where D is a nonzero and noninfinite constant, essentially the diffusivity.
Thcn the diffcrence equation (which is not shown here) becomes
( 3 5 )
PI = DP,, .
The NYWP rqtrution hJJ? follows from choosing
(30)
p = 1:
iiZJ6i -+ V w 6Z 4 0 ,
wlicre I/ is ;I tionzero and noninfinitc constant, essentially the wave speed.
Thzn
( 3 7 )
f’,l = VZP,, .
The 1 ~ 4 y r ~ 1 p t t
ryitcrr iorr linril follows from choosing
as SZ -0,
I
1) -+ I with ( 1 - p)/dr 3 1/2T
and 62161 -+ V
( 3 8 )
The simple diffusion equation [Eq. (l4)] is a particular continuous limit of
the simplest formal random walk process. a connection apparently made
first in essence by Rayleigh (1894) in 8 different context.
The siniple wave equation is a different continuous limit of the random
walk process. In a sense this is implied by the more general work of Goldstein (1951), bur seems to have been explicitly stated first in unpublished
reports by Michelson (1954). and by navies et al. (1954).
Motivated by the wish to subsume in a single model the highly correlated
’‘ small time *’ dispersion effect of turbulent motion with the uncorrelated
“large time” effect, Goldstein (1951) dcviscd a random walk limiting
procedure which led to the so-called “telegraph equation.” A similar hyperbolic model [the diffusion equation, Cq. (14), is parabolic] had been
proposed :ilso be Lyapin (1948. 1950; see also the paper of Monin, 1955).
Without the details of the general difference equation for the random walk,
a sense of the three different limits alluded to above can be given briefly.
We consider the elementary, onedimensional walk on a regular lattice.
Each step length is SZ and has duration 61. Suppose that the probability
that two successive steps are in the same direction is p; the probability of
a reversal must. of course, be 1 - p.
The difliision cqzration limit follows from choosing
(34)
p = f ; (SZ)*ldf 4 2 0 as 6Z-+O.
where D is a nonzero and noninfinite constant, essentially the diffusivity.
Thcn the diffcrence equation (which is not shown here) becomes
( 3 5 )
PI = DP,, .
The NYWP rqtrution hJJ? follows from choosing
(30)
p = 1:
iiZJ6i -+ V w 6Z 4 0 ,
wlicre I/ is ;I tionzero and noninfinitc constant, essentially the wave speed.
Thzn
( 3 7 )
f’,l = VZP,, .
The 1 ~ 4 y r ~ 1 p t t
ryitcrr iorr linril follows from choosing
as SZ -0,
I
1) -+ I with ( 1 - p)/dr 3 1/2T
and 62161 -+ V
( 3 8 )