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Description
Rating
Title
The
Langevin
equation
: with
applications
to
stochastic
problems
in
physics
,
chemistry
and
electrical
engineering
Creator
Coffey, William, 1948
Contributors
Kalmykov, Yu. P.
World Scientific (Firm)
DescriptionAbstract
This
volume
is
the
third
edition
of the
firstever
elementary
book
on the
Langevin
equation
method
for the
solution
of
problems
involving
the
translational
and
rotational
motion
of
particles
and
spins
in a
potential
highlighting
modern
applications
in
physics
,
chemistry
,
electrical
engineering
, and
so
on. In
order
to
improve
the
presentation
, to
accommodate
all
the
new
developments
, and to
appeal
to the
specialized
interests
of the
various
communities
involved
, the
book
has been
extensively
rewritten
and a
very
large
amount
of
new
material
has been
added
. This has been
done
in
order
to
present
a
comprehensive
overview
of the
subject
emphasizing
via
a
synergetic
approach
that
seemingly
unrelated
physical
problems
involving
random
noise
may
be
described
using
virtually
identical
mathematical
methods
in the
spirit
of the
founders
of the
subject
,
viz.
,
Einstein
,
Langevin
,
Smoluchowski
,
Kramers
, etc. The
book
has been
written
in
such
a
way
that
all
the
material
should be
accessible
both
to an
advanced
researcher
and a
beginning
graduate
student
.
It
draws
together
, in a
coherent
fashion
, a
variety
of
results
which
have
hitherto
been
available
only
in the
form
of
scattered
research
papers
and
review
articles
.
DescriptionTable Of Contents
ch
.
1
.
Historical
background
and
introductory
concepts
.
1.1
.
motion
.
1.2
.
Einstein's
explanation
of
movement
.
1.3
. The
Langevin
equation
.
1.4
.
Einstein's
method
.
1.5
.
Essential
concepts
in
statistical
mechanics
.
1.6
.
Probability
theory
.
1.7
.
Application
to the
Langevin
equation
.
1.8
.
Wiener
process
.
1.9
. The
FokkerPlanck
equation
.
1.10
.
Drift
and
diffusion
coefficients
.
1.11
.
Solution
of the
onedimensional
FokkerPlanck
equation
.
1.12
. The
Smoluchowski
equation
.
1.13
.
Escape
of
particles
over
potential
barriers
:
Kramers
'
theory
.
1.14
.
Applications
to the
theory
of
movement
in a
potential
.
1.15
.
Rotational
motion
:
application
to
dielectric
relaxation
.
1.16
.
Superparamagnetism
:
magnetic
aftereffect
.
1.17
.
Brown's
treatment
of
Néel
relaxation
.
1.18
.
Asymptotic
expressions
for the
Néel
relaxation
.
1.19
.
Ferrofluids
.
1.20
.
Depletion
effect
in a
biased
bistable
potential
.
1.21
.
Stochastic
resonance
.
1.22
.
Anomalous
diffusion

ch
.
2
.
Langevin
equations
and
methods
of
solution
.
2.1
.
Criticisms
of the
Langevin
equation
.
2.2
.
Doob's
interpretation
of the
Langevin
equation
.
2.3
.
Nonlinear
Langevin
equation
with a
multiplicative
noise
term
:
Itô
and
Stratonovich
rules
.
2.4
.
Derivation
of
differentialrecurrence
relations
from the
onedimensional
Langevin
equation
.
2.5
.
Nonlinear
Langevin
equation
in
several
dimensions
.
2.6
.
Average
of the
multiplicative
noise
term
in the
Langevin
equation
.
2.7
.
Methods
of
solution
of
differentialrecurrence
relations
arising
from the
nonlinear
Langevin
equation
.
2.8
.
Linear
response
theory
.
2.9
.
Integral
relaxation
theory
.
2.10
.
Linear
response
theory
for
systems
with
dynamics
governed
by
singlevariable
FokkerPlanck
equations
.
2.11
.
Smallest
nonvanishing
eigenvalue
:
continuedfraction
approach
.
2.12
.
Effective
relaxation
time
.
2.13
.
Evaluation
of the
dynamic
susceptibility
using
[symbol]
,
[symbol]
and
[symbol]
.
2.14
.
Nonlinear
transient
response
of a
particle

ch
.
3
.
motion
of a
free
particle
and a
harmonic
oscillator
.
3.1
.
Introduction
.
3.2
.
OrnsteinUhlenbeck
theory
of
motion
.
3.3
.
Stationary
solution
of the
Langevin
equation
: the
WienerKhinchin
theorem
.
3.4
.
Application
to
phase
diffusion
in
MRI
.
3.5
.
Rotational
motion
of a
fixedaxis
rotator
.
3.7
.
Torsional
oscillator
model
:
example
of the
use
of the
Wiener
integral

ch
.
4
.
Rotational
motion
about
a
fixed
axis
in
Nfold
cosine
potentials
.
4.1
.
Introduction
.
4.2
.
Langevin
equation
for
rotation
about
a
fixed
axis
.
4.3
.
Longitudinal
and
transverse
effective
relaxation
times
.
4.4
.
Polarizabilities
and
relaxation
times
of a
fixedaxis
rotator
with
two
equivalent
sites
.
4.5
.
Effect
of a
d.c
.
bias
field
on the
orientational
relaxation
of a
fixedaxis
rotator
with
two
equivalent
sites
. ;
8
ch
.
5
.
motion
in a
tilted
periodic
potential
:
application
to the
Josephson
tunneling
junction
.
5.1
.
Introduction
.
5.2
.
Langevin
equations
.
5.3
.
Josephson
junction
:
dynamic
model
.
5.4
.
Reduction
of the
averaged
Langevin
equation
for the
junction
to a
set
of
differentialrecurrence
relations
.
5.5
.
Currentvoltage
characteristics
.
5.6
.
Linear
response
to an
applied
alternating
current
.
5.7
.
Effective
eigenvalues
for the
Josephson
junction
.
5.8
.
Linear
impedance
.
5.9
.
Spectrum
of the
Josephson
radiation
.
5.10
.
Nonlinear
effects
in
d.c
. and
a.c
.
currentvoltage
characteristics
.
5.11
.
Concluding
remarks

ch
.
6
.
Translational
motion
in a
doublewell
potential
.
6.1
.
Introduction
.
6.2
.
Characteristic
times
of the
position
correlation
function
.
6.3
.
Converging
continued
fractions
for the
correlation
functions
.
6.4
.
Twomode
approximation
.
6.5
.
Stochastic
resonance
.
6.6
.
Concluding
remarks

ch
.
7
.
Noninertial
rotational
diffusion
in
axially
symmetric
external
potentials
:
applications
to
orientational
relaxation
of
molecules
in
fluids
and
liquid
crystals
.
7.1
.
Introduction
.
7.2
.
Rotational
diffusion
in a
potential
:
Langevin
equation
approach
.
7.3
.
rotation
in a
uniaxial
potential
.
7.4
.
rotation
in a
uniform
d.c
.
external
field
.
7.5
.
Nonlinear
transient
responses
in
dielectric
and
Kerreffect
relaxation
.
7.6
.
Nonlinear
dielectric
relaxation
of
polar
molecules
in a
strong
a.c
.
electric
field
:
steadystate
response
.
7.7
.
Concluding
remarks

ch
.
8
.
Anisotropic
noninertial
rotational
diffusion
in an
external
potential
:
application
to
linear
and
nonlinear
dielectric
relaxation
and the
dynamic
Kerr
effect
.
8.1
.
Introduction
.
8.2
.
Anisotropic
noninertial
rotational
diffusion
of an
asymmetric
top
in an
external
potential
.
8.3
.
Application
to
dielectric
relaxation
.
8.4
.
Kerreffect
relaxation
.
8.5
.
Concluding
remarks

ch
.
9
.
motion
of
classical
spins
:
application
to
magnetization
relaxation
in
superparamagnets
.
9.1
.
Introduction
.
9.2
.
Brown's
model
:
Langevin
equation
approach
.
9.3
.
Magnetization
relaxation
in
uniaxial
superparamagnets
.
9.4
.
Reversal
time
of the
magnetization
in
superparamagnets
with
nonaxially
symmetric
potentials
:
escaperate
theory
approach
.
9.5
.
Magnetization
relaxation
in
superparamagnets
with
nonaxially
symmetric
anisotropy
:
matrix
continuedfraction
approach
.
9.6
.
Nonlinear
a.c
.
stationary
response
of
superparamagnets
.
9.7
.
Concluding
remarks

ch
.
10
.
Inertial
effects
in
rotational
and
translational
motion
for a
single
degree
of
freedom
.
10.1
.
Introduction
.
10.2
.
Inertial
effects
in
nonlinear
dielectric
response
.
10.3
.
motion
of a
fixedaxis
rotator
in a
doublewell
potential
.
10.4
.
motion
of a
fixedaxis
rotator
in an
asymmetric
doublewell
potential
.
10.5
.
motion
in a
tilted
periodic
potential
.
10.6
.
Translational
motion
in a
doublewell
potential
.
10.7
.
Concluding
remarks
. ;
8
ch
.
11
.
Inertial
effects
in
rotational
diffusion
in
space
:
application
to
orientational
relaxation
in
molecular
liquids
and
ferrofluids
.
11.1
.
Introduction
.
11.2
.
Inertial
rotational
motion
of a
thin
rod
in
space
.
11.3
.
Rotational
motion
of a
symmetrical
top
.
11.4
.
Inertial
rotational
motion
of a
rigid
dipolar
rotator
in a
uniaxial
biased
potential
.
11.5
.
Itinerant
oscillator
model
of
rotational
motion
in
liquids
.
11.6
.
Application
of the
cage
model
to
ferrofluids

ch
.
12
.
Anomalous
diffusion
and
relaxation
.
12.1
.
Discrete
and
continuoustime
random
walks
.
12.2
.
Fractional
diffusion
equation
for the
continuoustime
random
walk
model
.
12.3
.
Solution
of
fractional
diffusion
equations
.
12.4
.
Characteristic
times
of
anomalous
diffusion
.
12.5
.
Inertial
effects
in
anomalous
relaxation
.
12.6
.
Barkai
and
Silbey's
fractional
kinetic
equation
.
12.7
.
Anomalous
diffusion
in a
periodic
potential
.
12.8
.
Fractional
Langevin
equation
.
12.9
.
Concluding
remarks
.
Publisher
World Scientific Pub. Co.
Subject
Langevin equations.
motion processes.
Identifier
9789814355674
(electronic
bk.)
;
9789814355667
;
http://www.worldscientific.com/worldscibooks/10.1142/8195#t=toc
Language
eng
Type
Electronic books.
FormatExtent
xxii, 827 p. : ill. (some col.)
Date
c2012
.
RelationIs Part Of
World Scientific series in contemporary chemical physics
v. 27
OCLC number
874497704
CONTENTdm number
363
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