64
Ci. S. IJ.4’17+RWN, JR.
If A and R are two events, which are 4 J to be independent, then
ProbtAB} = probability that both A and B occur
= Prob{AJ Prob{B}
or
(7)
( A D ) = (AXB)
Also if A and B are mutually exclusive, i.e., AB = 0, then
(8)
( A @ 8 ) - ( A ) + (B)
Because of the binary nature of digital computers, binary representation is
a very natural way of codin8 an experiment. It is such an efficient way,
moreover, that ensembles of 25,000 realizations can bc generated on relatively small machines. This can give remarkably good statistics.
To describe the class of walks on random binuy tbldr, M n c the following variables:
if the velocity at point m in space and
n in time equals + 1,
if the velocity equals - 1,
if a particle which kft (0,O) in a
particular terlization arrives at point
m at time n.
[:
U ( m n ) =
X ( m , t i ) = 1
V(rr) = I
if a particle’s vslocity at time n = + 1
I, if it equals - 1.
It is important to realize that once V(m, n) is specified for all m, n in a
particular realization, then X(m, n), V ( n ) art immediately deducible, for
1
and
.%‘(in, n ) = V ( n - I)X(m - 1, n - 1) @ P”(n - l)X(m + 1, n - 1)
with X ( m , 0) = 0 except for m = 0, where X(O.0) = 1.
Statistical quantities of interest include
velocities separated by a distance (m, n) in space-time, and R(j; n ) the autocorrelation of a particle’s velocity at time n with its velocity at time n + j .
Afier some manipulation, we have
Ci. S. IJ.4’17+RWN, JR.
If A and R are two events, which are 4 J to be independent, then
ProbtAB} = probability that both A and B occur
= Prob{AJ Prob{B}
or
(7)
( A D ) = (AXB)
Also if A and B are mutually exclusive, i.e., AB = 0, then
(8)
( A @ 8 ) - ( A ) + (B)
Because of the binary nature of digital computers, binary representation is
a very natural way of codin8 an experiment. It is such an efficient way,
moreover, that ensembles of 25,000 realizations can bc generated on relatively small machines. This can give remarkably good statistics.
To describe the class of walks on random binuy tbldr, M n c the following variables:
if the velocity at point m in space and
n in time equals + 1,
if the velocity equals - 1,
if a particle which kft (0,O) in a
particular terlization arrives at point
m at time n.
[:
U ( m n ) =
X ( m , t i ) = 1
V(rr) = I
if a particle’s vslocity at time n = + 1
I, if it equals - 1.
It is important to realize that once V(m, n) is specified for all m, n in a
particular realization, then X(m, n), V ( n ) art immediately deducible, for
1
and
.%‘(in, n ) = V ( n - I)X(m - 1, n - 1) @ P”(n - l)X(m + 1, n - 1)
with X ( m , 0) = 0 except for m = 0, where X(O.0) = 1.
Statistical quantities of interest include
velocities separated by a distance (m, n) in space-time, and R(j; n ) the autocorrelation of a particle’s velocity at time n with its velocity at time n + j .
Afier some manipulation, we have