C is
unity. regular
subvolumes,
smal1-displacement expansion
there
for
the
volume,
n(p)
u
of
particles
constant
intersection
where
such
expansion
R(t)
where
of
=
1 -
p V" Q(u
)
1
+ o(p )
,
2
is
the
unit
vector
in
the
is
the
area
of
projection
(2.20)
direction
of
p
P
and
Q(u)
onto
a plane
R(t)
The is Urn t->0
=
normal
1 -
V
integration a sharply P(p,t)
=
to
u.
dp
P(p,t)p
may be
peaked 6(p)
the
of
the
subvolume
Hence
taken
function
Q(u
over as
)
(2.21)
+ o
all t->-0 .
2
p
since In
fact,
P(p,t)
24
For a spherical
an
isotropic
subvolume,
R(t)
(Here
again
the
If to
the
a formula
constant
of
where
C
is
n
[Cn-l/n
c
n V"
isoperimetric rapidly
for
of
is
convex,
its
Q
•=
its
A" Q 1
the
A ,
which
inequality,
isotropic
the
form
average
area
V
over
then,
- 1
Q
angle.)
according area
Q is
in
n-1
of
the
in
n
unit
(2.22)
indicates, that
solid
A,
n C
sphere
volume
factor
(2.22)
mean p r o j e c t i o n
surface
o_r
subvolume
spherical
An density
1
Q
1
an
to
the
V"
distribution
becomes.
to
again
Consequently, by
- 1
result
refers
Cauchy,
ratio
A
the
~ 1 -
bar
displacement
R(t)
due
n-sphere. may be
to
the
decreases
replaced generalized least
subvolumes. n-dimensional
Gaussian
displacement
25
—1
r—
P(p,t)
leads
-n/2
h (t)
2TT
'
exp
2
2h
p2 — (t)_
(2.23)
2
to
n C < ( )> p
.
t
n c
n
and
)
(2.24)
. n-i
therefore,
R ( t
For so
h ( t
diffusion
(with
)
1 - "A
=
diffusion
Hil)
(2.25)
constant
D),
/2Dt
h(t)
that
Dt 7T
Note
that
that
of
Q = \ ,
the
Lax as
and
distribution, implies
term
Mengert
required For
This
second
for
differs
(their
eq.
is
process
a factor
(1.11)
mathematical
a free-flight P(p,t)
by
with
2//TT
from
their
surfaces). with
Gaussian with
Maxwellian h(t)
nJ
kT m
velocity
26
kT
V >| 2lTm
c
In
fact,
going
for
results
R
When
an
lead
(t)
either
symmetry,
we
arbi trary
=
t • V"
f(v)
or
find
that
experiment
involved
of
block.
It
Furth's
results
speed
values
of
of
has
dv
1
the
Z V
Fu'rth,
system been may
f(v)
process,
• v •Q(u ) v
subvolume
RrAt)
determination
one-dimensional
free-flight
the
fore-
to
1 -
The
t
- 1
of
pedestrians
quantity
used
be
spherical
by to
(2.27)
earlier
the
thus
2
possesses
of
out
(2.26)
o(t )
Q t
mentioned
pointed
+
in
this
1 - R(t) moving
for
along
Chandrasekhar estimate
chapter,
the
a
a city
that mean
pedestrians. The
above
relations
t.
Since
the
hold
for
autocovariance
small
non-negative
coefficient
27 '
(t) = n
R 111
that
the
is
symmetric
about
t = 0,
(2.26)
implies
,
v.
first
derivative
exhibits
a finite
geometry
of
the
of
$ (t)
discontinuity
subvolume
derivative
therefore
In
Chapter
4 we
at
which
particles
and
contains
shall
see
how
enter
for
n
and
at
free-flight
t = 0
arbitrary a term
for f(v)
.
The
proportional
fluctuations leave
arbitrary
in
V give
rise
process
with
Gaussian
with
second
to
)
at
to
the
asymptotic high
28
frequencies. 1948)
to
An
the
application
small-time
Watson's
lemma
(Watson,
expansion
R(t)
results
of
z
1 -
-
C r
C t'a
in
S(iw)
z
(iw)"
1
(a+l)(ico) - a -
I
>
(2 S(w)
= 2Cr
It
is
obey
seen an
that,
at
= 2 Re
(a+l)
high
sin
three-halves
Fasset;
Lax
Mengert),
lead
an
to
sionality
inverse and
geometry
Long-time of
the
square
remainder
aT
0 1
power
law
"
.
1
(van
free-flight
law.
The only
The
behaviour. can
be
(2
diffusion
while
enters
function
^
frequencies,
inverse and
S(ico)
Vliet
the
long-time
obtained
and
processes
dependence through
spectra
from
on
dimen-
constant
properties an
C
29
expansion verse
of
the
powers
of
R(t)
displacement t.
=
To
P(p =
this
probability
e n d we
0,t)
dp
density
in
in-
write
P(p,t)
fifp)
P(p=0,t)
and e x p a n d
the
term
in
square
P(p,t)
brackets.
Let
_ L l L t L P(p=0,t)
s
Then
p(p,t)
=
i
+
f
1
9P t
—
00
(2.30)
=
It
follows
1 + t"'
that
b (p) x
+ t~
z
b (p) 2
+
30
R(t)
=
P(p = 0 , t )
•V
1
+ Bx t ~
+ B 1"
1
2
+
2
(2.31)
where
B.
a n d we
The
have
is
P(p,t)
independent The
the
following
R (t)
=
d
For
the
R
f f
, of
B.. but
=
depend
on
the
first
fi(p)
long-time
n / 2
f ( v - 0 )
V
the
shape
term
on
of
the
the
RHS
of
subvolume (2.31)'
geometry.
n-dimensional
(47rDt)-
=
dp n ( p )
the
n-dimensional
(t)
dp b . ( p )
1
used
coefficients
a n d on
V'
E
diffusion
leads
to
result,
V
1
+
\diff
free-flight
• t"
process
1 1
•V 1
+ B
*
t _ 1
process
1
>
f
f
•
(2.32)
+
we
t"
1
find
+
• • •
(2.33)
31
Free-flight rapidly
remainder
than The
a simple
case
free
near-zero
flight,
these hence
The
gas
ideal Maxwell
where
m
K
a modified
is
is
In
the
the
those
large
are
the
rest
more
(2.31)
are
which
have
still
to
of
particles
affords
t.
In
which
clas s
dimensions
distribution
with
general
has
(2.34)
2
c
is
the
speed
of
light
«
1) ,
function. limit
=
(x
>>
1),
2ukT V.
while,
in
the
extreme
have
f(v=0).
i 3/2 (0)
the
K (x)
classical
f
be
x e 2
mass,
Bessel
in
values
factor
three
1
decay
particles
for
velocity
to
processes.
P(p=0,t)
the
in
4TTC
seen
displacement
subvolume
f ( v - O ) •-
2
net
velocity;
relativistic
0
of
only
small
the
are
diffusion
appearance
a very
inside of
for
explanation:
undergone found
those
functions
J
relativistic
limit
(x
and
32
rm =
TT
cn
0
kT
2TT
3/2 f
Thus, tion
as
one
decays
would
expect,
much m o r e
the
,
(0)
clas s
relativistic
rapidly
than
observe
that
its
remainder
classical
func-
counter-
part. Finally, rapidly
for
sionality
we
higher
dimensionality.
implies
more
means
of
R(t)
decreases
Since
higher
escape,
this
more
dimen-
too
is
not
s u r p r i s i ng .
Decaying system
are
T~ )
which
then
the
1
If
particles.
subject
to
operates
remainder
a decay
process
independently function
R(t,x)
=
e~
the
of
particles (with the
of
decay
transport
our constant process
becomes
t
/
T
R ( t , T = ~)
.
(2
_t /T•
Here
e
decay
for
'
is a time
the t.
probability
that
a particle
does
not
33
The the
effect
replacement
looks part
like of
of
S
and
Fasset.
demonstrated However,
domain where decay
Section
2.6
for
most
more
diffusion
form
our
.
is
Although
this
the e v a l u a t i o n
of
of
the
laborious,
processes
we a r e p r i m a r i l y
of
)
_ 1
function
by
(2.35)
calculations
in
allows
(see,
as
van
interested
real
Vliet the
us
to
however,
5.3.3) .
The the
number
can
be d e r i v e d
of
stationary
where
for
density
S(iw + T
considerably
the simple
Fluctuations
cov
by
alteration,
can become
been
ignore
S(ico)
a simple
has
time
on t h e s p e c t r a l
N(t
dN/dt.
correlation particles from
the
in
function
for
the
the subvolume
following
rate
at
varies
relation
for
which
with
time
covariance
processes,
0
+ t),N(t )
of
dt
0
N = dN/dt
variance
in
N and
and
:
cov
= 0 .
defining
N(t
0
+ t),N(t )
(2
0
Normalizing with
the
34
$; (t) N
=
T
(varN)
cov N ( t
- 1
+ t),N(t )
0
0
yields
*.(t) N
For $• N
may a l s o
the
transit
of The
the
a particle
distribution
of
p(£)
of
The of
has
length
I +dl.
the
of
the
definition
sively
depends above
I,
of
Chapter
6.
.
We h a v e
seen
the
times and
one
convex
subvolume
to
$> N
provides
the
traverse in
turn
the
random
gives
the
probability with
statistics
depends
the
path
from
subvolume.
on
This
on
the
secant
the
that
subvolume
is
a purely
geometrical
(which
depends,
however,
that
intersection
discussion
to
"randomness").
in
on
which
a particle's
subvolume
and
(2
reference
requires
latter
intersection
property
without
velocities
.
.
process
transit
the
in
* (t) N
T
distribution
distribution
density
A
otained
time
time
-
a free-flight be
J
=
the
It
discussed
autocovariance
volume
suspects
is
ft.
In
a connection
view
of of
between
on exten-
N(t) the ft
and
35
the
random
4 and in
6 and
secant leads
geometrical
density to
p(£).
a very
probability.
This
simple,
is
but
explored useful,
in
new
Chapters theorem
Chapter
3
FREE-FLIGHT PROCESSES
3.1
Introduction A free-flight
tion
p = vt
ticles (and
move w i t h
therefore
determined of
for
the
by
the
process
the
velocity
present
fluctuations are
chapter in
considered
single-speed speed),
the
velocity frequency
case
means
that
the
the
and
by
the
rela-
par-
remainder
autocovariance)
distribution
dv
is
Section (in
treatment
f(v)
devoted
free-flight in
by
This and
number
defined
function
is geometry
i.e.
R(t)
The
velocity
particle
subvolume,
been
displacement.
constant
the
has
distributions.
to
an
systems.
3.2.
which is
ft(vt)
all
After
domain.
36
to 3.3
(3.1)
investigation Temporal
move
some m o r e is
of
number
correlations
a discussion
particles
extended Section
.
at
of the
the same
general
concerned
with
the
36a
Frequently one-dimensional system, domain aries
this
case.
case
enclosed
subvolume drical
is
of
infinite these
may
the
subvolume
is
3.2
Temporal The
The
a delta
ation
is
velocity
other
length.
(in
compound
interest
this
chapter:
into
distributions,
the a
cylin-
being
simple
whenever the
Cartesian
of
Particle
be
defined which
distribution
non-spherical
of
the
velocity
factors
and
shape.
particles
of
bound-
two-
e.g.
from
of
"slab"
plane
dimension
Aside
the
Number
Gas
will
point
function
are
a
the
two,
on
physical is
Similarly,
the
decomposed
system
velocity
L.
than
cases
Single-Speed
non-interacting
to
of
focussed
parallel
one
Correlations
This
speed.
be
be
subvolume
whenever
distribution
distribution)
the
infinite
a distance
larger
mathematically,
3.2.1
two
much
domain
displacement
when
results
will
a three-dimensional
results
by
case
attention
In
between
separated
dimensional
our
v. we
When
the
restrict
f(v)
as all
a collection move is
then
subvolume our
at
the
same
proportional
under
treatment
of
to
considerisotropic
36b
.1-n
f (v)
n C
n
where and to
c C
the
= ir
n
in
tion
may
f(v)
For
of
drift"
have
for
to
any the
an
to
average
that
apply,
(3.
denotes of
over
the ft
case
the
restriction
the
direction
intersection
volume
is
inde-
of
course
velocity
whatever,
of
R(t)
for
example,
a beam o f
dimensionality
the
simplify
dependence
calculation
to
n
subvolumes
angular
i.e.
c)
effect
is
In
n-spheres
.
distribu-
since
only
Consequently to
particles
the
the
"pure
moving
with
c. In
speed
gas
now
any
case,
takes
the
the
R(t)
Thus
The
spherical
process,
speed
.
direction.
enters
results
speed,
distributions
(3.1)
alone.
pendent
particle
/rCy+l)
isotropic
average ft
is
S(v -
n
it
is
normalized metrical below.
sufficient volume
problem.
of
to
remainder
following
=
for
particularly
ft(ct)
determine
intersection,
Several
function
specific
the
simple
single form,
.
(3.
the
directi
which
is
examples
on-averaged
a purely are
geo-
discussed
37
If
n-Spheres. hypersphere L = 2a), lized
of
radius
then
fi
incomplete
n
the subvolume
a (for
n = 1 :
c a n be e x p r e s s e d beta
2 C n (p)
function
n-1
'n
1
in
-y
an
line
n-dimensional
segment
terms
(Appendix
dy
n
is
of
of
the
,
(P
length
norma-
A),
:
i 2a)
p/2a
(3.4)
n+1
(We
use t h e
Q
vanishes.
n
notation:
1
A series
(p)
l
4a
I(a,b;x)
= I
expansion
2 C «
;
r
.
n-1 n
*
Y
£ k=0
X is
_
r
2
(a,b)).
For
p >
2a,
'n+ll
r n+1
(3.5) 2k+l k
(2k+l)
[2aJ
(p
, 2TTIT1
.
other
expansions
Ri(t)
=
R (t)
-
f
(v
=0)
these
n-dimensional
1
u
-
2
equations
• t"
to
R(t)
of
velocity,
i.e.
those
n
when
+
1
• V
bution
(3.20)
Maxwellian
(cf.
those
t
>>
T
h
(u
we
the
evaluating
the
form
- 1
obtain
(j) e
1 +
The
one-dimensional
-xcox
(icox) - 2
.
changing
and
1
+
- 1
autocovariance
1 -{ i+e
:
(3
exp(-(p/9) 1 +8 2
where
6 E WX^
dimensional
and
subdomain
(1 - 6 ) c o s cp -
20 s i n ) .
is
similar
again in
form
the
rectangular
The
one-dimensional
logarithmic
(3.57c)
behaviour
velocity
as
Chapter
4
FLUCTUATIONS IN NCr) AND A THEOREM IN GEOMETRICAL PROBABILITY
4.1
I n t r o d u c t i on The
be
obtained
Section
autocorrelation
from
2.6).
entering
the
for
particle
ft(p).
In
that
number,
the
N(t)
may
also
be
t i mes ,
derived the to
the
volume Hence the
extent
of
(called the
time
N E dN/dt
basic
the
volume
(Section the
of
4.2)
we s h a l l
durations
of
time which
density
distribution
and 78
of
individual
is
x = £/v
,
of
its
path
with
I)
and
v
its
W(T) the
distribution
of
subvolume.
time
length
show
autocovariance
the
of
quantity
intersection
of
transit
(see
coefficient
terms
the
may-
geometrical
autocovariance
intersection
a secant
transit
velocity
the
in
for
differentiation
flight,
traverse
A particle's
the
is
chapter free
by
the ,
N
in
i.e.
require
of
* (t)
present
particles
is
that
calculation
for
particles
N(t)
We r e c a l l
how,
transi t
of
function
is
may be
where
I
the
sub-
speed.
obtained
random s e c a n t
from
density
p(l).
79
It secant this
length
the
between
tion
of
this
function shall
of
Obviously
then,
in
the
two
the
point
of
N,
chapter
deriving Later
relation view
the
some
a physical
of
quantity
without
of
be
elucidaview
by
correlation
in
between
relation
the
point
on,
in
should
is
the
of
connection
intersection
this
4.3).
of
view
N and of
of
from
methods
N (Section
of
volume
purpose
re-establish
number
Chapter
p(£)
6,
we
and Q
reference
to
from particle
fluctuations. In
spectrum p(£).
geometrical
and
mathematical
distribution
basic
connection
the
probability
the
A major
comparing
the
autocovariances
p(£)
expected.
that
becomes
approach.
between
the
follows
to
Two
Section the
4 . 4 we
relate
characteristic
examples
are
the
number
functions
discussed
in
of
fluctuation W(T)
Section
and
4.5.
• 4.2
The
Correlation
Time
Function
particles
occur
at
after in
N(t)
and
Transit
Distributions Fluctuations
exit
of
the
surface
a random
accord
in
with
N(t) of
transit
are
V
at
time
a Poisson
due
to
random x
.
process
the
arrival
times,
Let with
the
and
arrival
intensity
of
their times (mean
80
rate
of
random
arrivals) variable
equal N(t)
N(t)
to
v .
in
the
I
-
T h e n we c a n r e p r e s e n t form
h t - t.
h ( t - t
arriving
at
the
The independent with
, x
is
random
transit
the
time times
the
density
following
cov
W
to
N of
and h a v i n g
transit
arriving
particles
for
distributed A
(4.1)
1
contribution t
and i d e n t i c a l l y
probability
provides
)
, T .
1
]_ = — 0 0
where
the
("0
.
random
Campbell's
a
particle
time
x
.
are
variables theorem
then
relation:
N(t ) , N(t 0
0
+ t)
(4.2)
dx
v
ds
W.(T)
A
h(s , x ) h ( s + t , x )
o
For of
a pair
time
by
of
particles oppositely
the t r a n s i t
time
in
free
flight,
directed x ,
i.e.
h(s,x)
impulses,
consists
separated
in
81
h(s,x)
Utilizing
this
cov
function
N(t ) 0
The
, N(t + 0
and, except
is
indicates
impulses.
the t r a n s i t
order
to
distribution
,
- v W A
(4.3)
reason
2, for
of
is the
factor
2 delta
term
particles
result
2 is
when
that
functions. (4.3)
correlation
entering
in
sta-
on t h e RHS o f
behaviour
contribution
time
the usual
time-dependent this
mentioned
the t r a n s i t
factor
expect of
6(t)
since
the
is positive
the subvolume they
exit
is
at a
later.
convenient
for
N
yields
of
the second
a negative
In
]
The
that
f o l l o w e d - by
W .(x)
the
t f= 0
to
+ T)
on t h e RHS h a s b e e n
combination
One w o u l d
x
2v
For
contribution
secant
for
a linear
negative.
time
t)
independent
tistics
h(s,x)
(4.2)
It
is
6(s
in
term
2.5.
Poisson
-
first
Section
for
= 6(s)
to
which
particles
and t h e
define
first
refers which
derive
to
W.(x) A
in
velocity another
terms
distribution
transit
the d i s t r i b u t i o n
are i n s i d e
of
time of
the subvolume.
the it
random is
density,
transit Thus
times W (x)dx
82
is
the p r o b a b i l i t y
has
transit
time
in
W Cx) arriving
we t a k e
number
of T - c l a s s
T-class
known
t o be i n s i d e
V,
T , r + dx .
particles)
• vW ( x ) d x ,
a particle,
may be r e l a t e d
ticles
T
that
to
as f o l l o w s :
W (x)dT
particles.
A
( ) T
(which
vW
refers
for non-interacting
t o be p r o p o r t i o n a l
particles
since
W
inside
(x)dT
par-
t o t h e mean
V, which
is
to
is
the rate
equal
to
of a r r i v a l
of
Hence
W (T) R
-
T
W
a
( T )
(4.4)
and
-1 E (x)
where E.Cx
0 1
dx
E (x ) A a
A
)
(4.5)
EJ-CT" ) 1
A
T W (T) , a
and s i m i l a r l y
for
. Dividing
particles
i n V,
and u s i n g «
coefficient
( 4 . 3 ) by
for N :
vE
(4.4),
(x),
t h e mean n u m b e r
we d e f i n e
a
of
correlation
83
= cov[N(t )
,
0
N(t
+
0
t)]
N
(4.6)
=
It secant through the be
I
volume
the
density
for
procedure tion in
of
the
be
to
of
6(t)
-
is
as
secant
corresponds
is
W (T)
the
uniform
Let
the
by
path
a point
point
and
inside
direction
distributions.
of
the
path
Then
the
resulting
denoted
by
the
choice
position
and
isotropic
approach
random
a particle's
to
function
.
defined
Let
intersection
length
x
follows:
uniquely
length.
w (|t|)
|t
determine
independent
secant
remainder
)
a direction.
the
initial
1
defined
and
from
magnitude is
I
subvolume
subvolume selected
E ( T "
remains
density the
2
of
to
with
p^iZ) uniform
The
the
probability
.
This
distribu-
displacement particle
sub-
density
number
c o r r e l a t i ons. Denoting for
the
transit
the
time
speed density,
density
by
f(v)
we
then
have
84
W-j- ( T
dv
}
f(v)
d£ p U )
6
r
I
" V
T
(4.7)
dv
If particles,
desired, W (x)
,
f(v)
the may
v
p (vx) I
transit be
time
obtained
density
from
for
a r r i v i ng
(4.4)
and
(4.7).
not
easily
A
However, without
the use
result of
W (x)
in
terms .
of
P (^)
Instead,
W (x)
by
p
in
(£)
terms
of
a related
, which
is
defined
V is
defined
by
section
with
a plane
IT t h r o u g h
and
the
point
pendent may
direction
be
of
ti.
Then
expressed
be
derived
with
distributions. as
u
and
results
n are The
density,
a particle's
some c h o s e n
P (£) y
secant
follows:
a direction
intersection
uniform
random
as
through
to
may
seen
A
J_
directly
is
I
by
its
the
selected transit
and
normal
direction
from
time
path
inter-
origin if
denoted
inde-
density
85
f *W ( T )
=
dv
_ 1
f(v)
dl
v
p (£)
6
y
A
0
i)
r
0
(4.8)
=
where that
is
v/
particle's
W CT) 1 tion
speed
of
on
of
P {1)
1969;
thus
been
results
are
Bertrand with this
was
an
fluctuations
on
{I)).
fact
arriving
p
of
N
on
dependence The
the
of
determina-
\1 an i n t e r e s t i n g Bertrand's of
interest
secant by
in
which (Coleman,
densities
Coleman.
for Further
6.
concerned
with
two
dimensions.
dimensionality.
true
problem
paradox),
renewed
published
Chapter
also
the
.
Explicit
been
used
that
the
(cf.
arbitrary is
have
through
subject
in
+ dv
the
itself
have
derived
deals
of
W (x) A
the
(VT)
a n d we
v , v
enters
in
p
2
probability
1965,1969).
geometries
stated,
in
(or
several
wise
the
probability
Kingman,
Kingman
is
is
T
v
mean s p e e d
lies
p,(£) J-
recently
f(v)
dependence V
geometrical has
the
f(v)dv
The geometry
dv
for
our
work.
Unless
other-
86
4.3
A New T h e o r e m We come
geometrical the
(normalized)
of
which that
the
the
volume
order
(4.6)
of
random
between
secant
intersection p
and
the
density
ft
two p
and
I
.
ft
we s h a l l
transit
and
combining
is
this
x
of
_ ~
with
between
with
make
(4.6)
2
2
of
N(t)
the d i s c u s s i o n observing
the
with
.
that
remainder
F o r we h a v e
d R(t) dt
(4.4),
N(t)
worth
density.
i (t)
view
it
a link
time
coefficient
process
proceeding
quantities
provides
in
the
connection
relate
derivative
the
or,
to
the
the autocovariance
Before metrical
now t o
Probability
relation
links of
Geometrical
quantities,
In use
in
and ( 4 . 9 )
geo-
(4.9)
function
$ (*) N
of
=
with and
R(|t|)
>
yields
(t
> 0)
(4
87
W ( )
A
Let gas
(speed
t
K L ]
= c).
In
that
r
2
the i s o t r o p i c
case,
= c
T
P
according
]
single-speed
to
(4.7),
(4.11)
.(CT)
(3.3),
R(t)
Consequently,
P
Equation
/(t) dt
d 2
us now c o n s i d e r
W ( )
and , by
«
according
_
1
(4.12)
only
and i s
made
in deriving
restriction apparently
P
x
is
quite
( P ) -
=
to
fi(ct)
(4.10),
j^r
(
a relation independent $„
t o convex new, r e s u l t
we m u s t
and
between
> 0)
subvolumes.
.
geometrical
of the physical
4>. , e x c e p t
in that
P
have
(4.12)
quantities
assumptions
f o r the
geometrical
It
is
a very
simple,
branch
of
geometrical
but
88
probability
which
convex
domains.
sented
in
is
A direct
Chapter The
ft
,
concerned
and
the
calculation
random
mathematical
paths
derivation
provided
usefulness
by
(4.12),
of
the
would
link
appear
of
random
secant
densities,
generally
a fairly
simple
matter
to
dependent
volume
culty but
in
average. on
the
care
use
The
than
of
is
is
1
hand,
approach
properties second
given
theorem
in
.
for
transit
equations
(4.7)
densities (1965),
(4.4), p
and
p
arbitrary
for
obtains ,
more
An e x a m p l e
time
(4.8),
one
In
in
it
is
major
diffi-
laborious,
the
direction
first
principles,
insight
and
a number
easily is
di r e c t i on-
by
the
of
making
proof
of
dimensionality
6.
the
and
from
handled
comparing
two
of
addition,
interesting
also
pre-
lie
the
sometimes
ft
ft
Chapter
The
greater
.
to
since
requires
are of
the
p^l)
Another
Using
is
between
down
.
evaluation
via
p
write
ft(p)
then
of
usually
involving
Crofton's which
P (l)
derivation
the
the
intersection
straightforward,
other
problems
of
deriving
usually
through
6.
principal
p
with
first
result
may
be
derived
by
densities
W
the
speed
case.
between
the
single
a relation
demonstrated
and
by
,
secant
Kingman
89
P
From
this
one
may
U)
y
E
(£ )
its
E-j- ( - ^ " )
(4.14)
and
p U) y
Again,
E (£ ) x
(4.13)
1
dl r
=
a
y
U)
deduce,
E
where
cc lp
U )
l
a
consequences
are
discussed
the more
similarly
relation
for
(4.13)
extensively
and
in
Chapter
6. For relation secant
completeness
between
length,
the
we
moments
which
is
the
E
(r )
E
U)
moments
of
easily
present
transit deduced
the
time from
following
and (4.7)
those
of
:
a
1 _
provided
also
a
X
exist.
< v
-a>
(4.15)
90
Finally, of
(4.6)
making
with use
let
that
of
the
us
compare
obtained
by
the
delta
differentiating
s m a l l - t i m e expansion
:
* (t) N
1 -
^
function
of
ft(p)
term
$ (t)
,
N
,
|t|
for
where
we
recall
|t|
-->P - P,
,
flight
process,
dr
3.2.4.
mainder
function
diffusion
(This
in
densities the of
if
effect
may be
the
motion
of
is
for
the
The
may be
displacement
density
process,
which
h
X
observe velocity
is
the
here
2 /
t
N
_
2kT
Langevin
that
in
for
the
case
At
of
-
doing
2
of
the
.)
as
more
/2Dt most
free-flight
probability process
limiting
general
and forms
Brownian
with
1 + e
-Xt
(5.4)
coefficient.
h (t) the
h(t)
h(t)
holds
friction
general
autocorrelation
directly
considered
(5.1)
in
re-
for
the
free-,
discussed
displacement
for
by
the
of
Maxwellian
p
,
d r
for
reason terms
v
(3.33)
the in
velocity
been
substitution
Motion.
for
over
particles,
one-dimensional
has
equation
of
replacing
the
3.2.4
process
with,
the
drift
taken
we make
both
For
again,
Brownian
diffusion
Here
the
motion
(5.1),
t .
dr
Section
motion
modify
dr
Once
diffusive
a drift to
correspondence
algebra
the
- * - - > • p, = v,
with
Section
of
on
is
related
particles
by
[We to
might the
105
d h 2
2
=
2
or ft h (t) 2
where
=
2
refers
Vi
Again,
to the
dt'
(t - t « )
one
component
remainder
flight
(Section
3.2.2)
h (t)
as
by
2
A =
0
given
limit
continuum
of
of
motion
At
for
the
1
this
remainder it
does
function for
to
has
several
As
times
A
from
of
Figure
values
in
5.1
free where
A
in
less
plotted of
case
which
the
decays
particles
been
increases the
for .
of
is
mobility rapidly flight.
Maxwellian
motion
we
if
pass
use is
the
to
h (t)
gives
2
1/mA
t
and
,
the
diffusion
than
We i l l u s t r a t e
this
Brownian
motion
a one-dimensional
a
when
interval
for
free
through
flight
reached
u =
we
which
2
free
Since
a time
the
for
(kT/m)t
interest.
with
velocity.]
Brownian
replace
displacement
function
in
to
of
functions
to
diffusion,
increases
for
behaviour
(5.4).
all
mean-square
since
(5.4)
situations
Brownian >>
apply
Vi(t')>
(5.15a)
^ -
2" £og 2
(5.15b)
(5.15c)
113
Evidently
only
to
S
and
2
larity
of
process. time in
was the
The
the
found
as
to -»• 0
same k i n d
in
Chapter
one-dimensional
of
for
this
$ (t)
,
n
.
Si
of
3 for
the
similarity is
as
logarithmic
Maxwellian
which
varies
singu-
spectral
density
free-flight
lies
in
proportional
the
long-
to
t"
1
cases. The
easily
finite
reason
asymptote
both
is
3
exhibits
2
which
function
S
from
I
n
from which
c
high-frequency
the
(a)
v
'
we
asymptote
of
S
n
follows
expansion
K (a)
= sr— 2a
n •
1
-
( n - 1) 8 a 2
2
deduce
i \
2na
(n
2
- 1) 40 2
(toa /D 2
2
»
.1)
.
(5.16)
In
each
case
in
Chapter
2.
S
n
«
to~ ' 3/
2
,
a s was
already
pointed
out
114
Further in
6 the
article
an e x t e n s i v e
5.3.
by
van
Time
A recent of
region
of
shall
extend
some
of
starting
position.
positronium
an
Section
are
absorbing
theory,
we
In
case
that
fusion
no
previous
which
In
These
may
be
also
found
contains
ledge
only
of
time,
and
this
in
solids,
which
not
with
to
has
may
escape
to
allow
arise to
in
which
a
Here
include
shall
the
from
case
for
random
the we
we
the
of
phenomenon
apply
the
leave
the
region
return,
the
surface
the
we
Gaussian shall
moment g e n e r a t i n g be
obtained
from
of
passage"
probability
simple
However,
language
a "first
displacement
the
treats
diffusion.
results we
(1970)
for
conditions
(In
dealing
section.
Particles
our
5.4.
assumed
longer
Popov
undergoing
particles
the
by
addition
boundary.
are
Diffusing
mean t i m e
Popov's
diffusion
Since sideration
the
particles
particles.
in
Fassett
of
publication
decaying
results
and
spectra
Notes
determining
given
of
Vliet
diffusion
Distribution
Introductory
problem
on
bibliography.
Escape
5.3.1
results
form
function
con-
acts
as
probability problem.)
density
require
the
under
for
used
in
explicit for
the
difthe knowescape
Laplace-transformed
11 5
diffusion
equation,
condition. with
the
For
the
subscript
taking
remainder zero
finite
lifetime,, i.e.
5.3.2
Popov's
particle, will the
be
initially
0
particle particles
is moves P
0
inside
the
the
i
functions
particles
with
in-
that
a
non-decaying
inside
a time
t
a region later.
V
This
, is
function
=
r
boundary
particles.
r
at
dr
conditional
from is
at
V
remainder
±
chapter
to
probability
, r )
absorption
Results
which
Ro(t
the
this
non-decaying
the
is
of
of
refer
Consider
somewhere
P
will
Relevant
conditional
where
account
to Green
P (r
, t ;
0
in
time
function
equation
r- =
D V
±
probability
r
2
P
0
of
(5.17)
r )
t
density
that
.
diffusing
the
For
diffusion
the
116
When R
( t , r )
0
q
0
is
becomes
escaped escape
V
from
V
time.
CT , r )
,
qo(x
in
a time
that
interval
the c o n d i t i o n a l
given
, r)
by an a b s o r b i n g
the p r o b a b i l i t y
Then
is
bounded
surface
the p a r t i c l e
t
.
Let
escape
T
time
has n o t
denote
density,
by
dx = R (x , r )
-
0
R (x+dx,r) 0
(5.18) =
which
is
escape
If time
go(s
then
it
Ro(T , r)dx
the probability
x , x + dx . for
-
follows
that
the p a r t i c l e
we now d e f i n e
t h e moment
as t h e L a p l a c e
, r)
that
dt
g
q ( t , r) 0
D V g 2
0
= sg
0
e
_
escapes
in
generating
transform
satisfies
0
,
of
s
t
q
0
function
>
,
the d i f f e r e n t i a l
(5.19)
equation
(5.20)
\
with
boundary
surface
of
V
condition
V
transformed
is
the
to
Bessel's
go(s
with
from
v =
given
by
-
The
moments
by
Popov,
1
a_ r
and
K = of
the
script
the
are
TQ(r
easily
is
I
v
I
used
of
r
radius
in
the
a
,
(5.20)
with
solution
(KIT)
V_
escape
'
(5.21)
(ica)
time
The
may be
mean a n d
obtained
variance,
determined:
1
-
r. 2 1 - i
(5.22a)
-1
2 n ( n + 2)
)
diffusion
zero
for
equation
differentiation.
±
where
1
/s/D
T.(r )
var
=
n-sphere
, r)
n/2
(5.21)
, r)
0
.
When may be
g (s
117
(5.22b)
2
times to
is
denote
r
d
=
a /D 2
,•
non-decaying
and
the
sub-
particles.
118
5.3.3
Decaying
Particles
We s h a l l have
to
be made t o
when
the d i f f u s i n g
now i n v e s t i g a t e the treatment particle
what
of
modification
the preceding
has a n o n - z e r o
will
section-
probability
of
decayi ng. Let dt
be
ydt,
the p r o b a b i l i t y with
V
we s h a l l
mean c r o s s i n g
by
decay
not
is
that
a particle,
t , t
+ dt
considered initially
tion.
relation
These
a time
y •
By
"escape"
the boundary
of
rate
as e s c a p e . at
q ( t , r )
is
a finite
V,
i.e.
will
= e"
holds
relations
R(t + dt , r)
Clearly
in
r
in
V .
Then V,
interval from
Thus
the
removal
probability
escapes
in
is
q(t , r)dt
A similar
decay
constant
of
for
Y
t
q
0
( t , r ) d t
the c o n d i t i o n a l
are c o n s i s t e n t
probability not escape
not
integrate
that at
to
In
remainder
- ydt]
unity
the p a r t i c l e
a l l .
(5.23)
func-
with
+ q ( t , r ) d t = R ( t , r ) - [ l
does
.
fact,
will
.
since decay
(5.24)
there inside
119
P (r)
dt
e
is
the escape
probability
For the
e
to
, r)
go(s , r)
function
g
e
0
does
escape
of
escape
time
one c a n d e f i n e to
one c a n
q
define
as
.
£
corresponding
decay.
density
= q(x , r ) / P ( r )
,
(5.25)
=.g (s = y , r)
the presence
which
normalized)
q (x
ating
in
a particle
(properly
Analogous
q ( t , r)
(5.26)
t h e moment .
e
This
gener-
results
i n
9 (s e
The
mean e s c a p e
, r)
time
= g ( s + Y » r)/g .(Y » ) 0
in
the presence
r
0
of
decay
•
(5.27)
is
120
x(r )
=
±
dx
q (x
log
ds
Similarly,
the
log
variance
of
±
When sphere,
g
the
is
0
given
differentiations
T(r ) ±
and
one
=
27
g
by
(s , r . )
0
(Y
-
log
under
±
time
g
0
is
(5.29)
r )
(Y ,
±
consideration
(5.21).
readily
(5.28)
r )
,
escape
= ^
region
g
) • V
s = 0
Of
var x (r )
, r
e
Performing
is the
an
n-
indicated
obtains
e a F (e a ) n
-
er.
i
F
n
(e r . ) i
(5.30a)
121
var
where
T
(r.)
e = /yTD
F
n
,
1
-
6
(ea)
-
I
(x)
n.
G
(x)
=
I
B. 2
(x)
/
EL 2
(x)
2
G (x)
= XF
n
is
n
(x)
2
n
dx
clear
that
these
expressions
complicated
than
their
counterparts
ticles, above
equations
when
is
(5.22),
y "*• 0
.
the
one-
In
and
which
are
of
modified
spherical
be w r i t t e n
in
terms
the
of
Fi(x)
are
more
=
3
is
the
Langevin
=
considerably
for
Bessel familiar
tanhx
coth
function].
x
-
x
_ 1
more
non-decaying
recovered
and
F3(x)
2
three-dimensional
a ratio
[F
(5.30b)
x
and
d F
It
(sr.)
n
from
the
cases,
functions
par-
F
n
which
hyperbolic
can
functions:
1 22
For sphere
a particle
t h e mean e s c a p e
x ( r . = 0)
starting
time
ea
=
at the centre
simplifies
Let
us i n t r o d u c e
and
the d i f f u s i o n
ea =
/T./T„
.
mean
d I escape
time
time
x
d
f "
from
this
limit
7(0)
smaller
than
(5.22a)
by a f a c t o r
is
the l o n g - l i f e t i m e
(5.32)
o f d i m e n s i o n and
value /x^/x
d
(2n) .
contribution
t o t h e mean e s c a p e
which
quickly,
time
i n the absence
probability
of decay
of decaying
The the centre
i n the presence
variance
while
would
x
- 1
This
major
escape
the
T
interpreted:
particles
»
1
£ d
independent
of order
-
( x »