Addendum’
RANDOM WALKS ON MARKOVIAN BINARY
VELOCITY FIELDS
G. S. PATTERSON, JR.*
Adiwic.td Srirdy Progrimi.
Nutiorid Center ,jbr Atniosphc~ric~ Rcseurclt. Boukder, Colorado 80302, U.S. A.
1. INTRODUCTION
Binary random walks in one or more dimensions have a long history of
being useful in giving insight into various physical phenomena-principally
as heuristic examples. The best known use of the random walk is in the
problems of molecular diffusion and Brownian motion (Einstein, 1905; von
Smoluchowski, 1916; Chandrasekhar, 1943), although similar walks had
been used by Rayleigh (1880) in a sound wave problem. In the simplest
one-dimensional case, the walking particle moves to the right or left a distance dx with equal probability, and arrives a time dr later. If a limiting
process is used whereby dx, dr + 0, but in such a way that (dx)’/dt -+ D, a
constant, then the probability distribution for the walking particles P(x, t )
satisfies the diffusion equation with diffusivity D.
(1)
I?P(x, t)/& = D ( ~ ’ P ( x ,
t ) / 3 X 2
While the diffusion equation is a satisfactory representation of molecular
diffusion, it is not completely satisfactory for the case of turbulent
diffusion- satisfactory perhaps for engineering approximations, but not
from a theoretical or experimental point of view. As opposed to molecular
diffusion, fluid point velocities are correlated over a significant time
compared to the time scale of the motion. They are also finite as compared
to molecular velocities which, in the simple random walk, are infinite in the
’ The work reported in this paper was referrod to in the review paper by Dr. Corrsinduring
the Symposium. The author prepared this Addendum at the request of the Editors of the
Proceedings.
On leave from Department of Engineering, Swarthmore College.
61
RANDOM WALKS ON MARKOVIAN BINARY
VELOCITY FIELDS
G. S. PATTERSON, JR.*
Adiwic.td Srirdy Progrimi.
Nutiorid Center ,jbr Atniosphc~ric~ Rcseurclt. Boukder, Colorado 80302, U.S. A.
1. INTRODUCTION
Binary random walks in one or more dimensions have a long history of
being useful in giving insight into various physical phenomena-principally
as heuristic examples. The best known use of the random walk is in the
problems of molecular diffusion and Brownian motion (Einstein, 1905; von
Smoluchowski, 1916; Chandrasekhar, 1943), although similar walks had
been used by Rayleigh (1880) in a sound wave problem. In the simplest
one-dimensional case, the walking particle moves to the right or left a distance dx with equal probability, and arrives a time dr later. If a limiting
process is used whereby dx, dr + 0, but in such a way that (dx)’/dt -+ D, a
constant, then the probability distribution for the walking particles P(x, t )
satisfies the diffusion equation with diffusivity D.
(1)
I?P(x, t)/& = D ( ~ ’ P ( x ,
t ) / 3 X 2
While the diffusion equation is a satisfactory representation of molecular
diffusion, it is not completely satisfactory for the case of turbulent
diffusion- satisfactory perhaps for engineering approximations, but not
from a theoretical or experimental point of view. As opposed to molecular
diffusion, fluid point velocities are correlated over a significant time
compared to the time scale of the motion. They are also finite as compared
to molecular velocities which, in the simple random walk, are infinite in the
’ The work reported in this paper was referrod to in the review paper by Dr. Corrsinduring
the Symposium. The author prepared this Addendum at the request of the Editors of the
Proceedings.
On leave from Department of Engineering, Swarthmore College.
61