Lawrence Berkeley National Laboratory
Recent Work
Title
THE FOKKER-PLANCK EQUATION FOR AN INVERSE-SQUARE FORCE
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https://escholarship.org/uc/item/2gk1s1v8
Authors
Rosenbluth, Marshall N.
MacDonald, William M.
Judd, David L.
Publication Date
1956-04-13
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UCRL
3375
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UNIVERSITY
OF
CALIFORNIA
Radiation
Laboratory
Berkeley,
California
Contract
No.
W-7405-eng-
48
UCR.L-3375
THE
FOKKER-PLANCK
EQUATION
FOR
AN
INVERSE-SQUARE
FORCE
Marshall
N.
Rosenbluth,.
William
M.
MacDonald,
and
David L.
Judd
April.l3,
1956
Printed
for
the
U.
S.
Atomic
Energy
Commission
••
-
Z-
UGRL-3375
THE
FOKKER-
PLANCK
EQUATION
FOR
AN
INVERSE-SQUARE
FORCE
Marshall
N.
Rosenbluth,
William
M.
MacDonald,
and
David
L.
Judd
Radiation
Laboratory
University
of
California
Berkeley,
Californi.a
April
13, 1956
ABSTRACT
The
contribution
to
the
Fokker-
Planck
eqttation.
for
the
distribution
function
for
gases,
due
to
particle-particle
interactions
in
which
the
fundamental
two-body
force
obeys
an
inverse
square
law,
is
investigated.
The
coefficients
in
the
equation,
(~)
(the
average
change
in
velocity
in
a
short
time)
and
f#
~),
are
obtained
using
the
collision
cross
sections
for
such
forces.
These
coefficients
are
obtained
in
terms
of
two
fundamental
integrals
which
are
dependent
on
the
distribution'function
itself.
The
transformation
of
the
equation
to
polar
c.oordtnates
in
a
case
of
axial
symmetry
is
carried
out.
By
expanding
the
distribution
function
in
Legendre
functions
of
the
angle
the
equation
is
cast
into
the
form
of
an
infinite
set
of
one
dimensional
coupled
nonlinear
integra-differential
equations.
If
the
distribution
function
is
approximated
by
a
finite
series,
the
reaultant
Fokker-
~lanck
equation
may
be
treated
numerically
using
a
compUting
machine.
Keeping
only
one
or
two
terms
in
the
series
corresponds
to
the
approximations
of
Chandrasekhar,
and
Cohen,
Spitzer
and
McRoutly,
respectively
.