Analysis of Tridiagonal Recurrence Relations in Continuum Approximation
Transition from difference to differential equation allows solving tridiagonal recurrence relations, which appear, among other things, in analysis of the rotation of an overdamped Brownian particle subjected to a periodic force. Replacement of the discrete integers in the Fourier series by continuum...
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Радіоастрономічний інститут НАН України
2001
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| Цитувати: | Analysis of Tridiagonal Recurrence Relations in Continuum Approximation / F.G. Bass, M. Gitterman // Радиофизика и радиоастрономия. — 2001. — Т. 6, № 1. — С. 71-78. — Бібліогр.: 11 назв. — англ. |
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irk-123456789-1222262017-07-01T03:03:21Z Analysis of Tridiagonal Recurrence Relations in Continuum Approximation Bass, F.G. Gitterman, M. Transition from difference to differential equation allows solving tridiagonal recurrence relations, which appear, among other things, in analysis of the rotation of an overdamped Brownian particle subjected to a periodic force. Replacement of the discrete integers in the Fourier series by continuum is justified for large numbers, i. e. for small angles. For the simplest case of the sinusoidal force, our solution, indeed, coincides with one obtained by expanding the sin in the original Fokker-Planck equation (The Ornstein-Uhlenbeck limit). However, for slightly more complicate potential the expansion for small angles does not transform the appropriate Fokker-Planck equation into the soluble. At the same time, the method suggested allows solving the problem for all periodic potentials which have finite number of terms in their Fourier series such as sinm(θ ) or cosm (θ). Even and odd functions require slightly different analysis, and are considered separately. Переход от разностного к дифференциальному уравнению позволяет решить тридиагональные рекуррентные соотношения, которые возникают, в частности, при анализе вращения броуновской частицы с трением при наличии периодической силы. Замена дискретных индексов в разложениях Фурье непрерывными оправдан для больших номеров, т. е. для малых углов. В простейшем случае синусоидальной силы наше решение действительно совпадает с решением, полученным путем разложения синуса в первоначальном уравнении Фоккера-Планка (предел Орнштейна-Уленбека). Однако уже в случае несколько более сложного потенциала разложение при малых углах не делает соответствующее уравнение Фоккера-Планка разрешимым. В то же время предлагаемый метод позволяет решить задачу для всех периодических потенциалов, для которых ряды Фурье содержат конечное число слагаемых типа sinm(θ ) или cosm (θ). Четные либо нечетные функции требуют несколько различного подхода и рассматриваются отдельно. Перехід від різницевого до диференціального рівняння дозволяє вирішити тридіагональні рекурентні співвідношення, які виникають, зокрема, при аналізі обертання броунівської частинки з тертям у присутності періодичної сили. Заміна дискретних індексів у розкладанні Фур’є неперервними виправдана для великих номерів, тобто для малих кутів. У найпростішому випадку синусоїдальної сили наше рішення співпадає із рішенням, отриманим шляхом розкладання синуса у початковому рівнянні Фоккера-Планка (границя Орнштейна-Уленбека). Однак уже у випадку дещо складнішого потенціалу розкладання при малих кутах не робить відповідне рівняння Фоккера-Планка вирішуваним. Водночас запропонований метод дозволяє вирішити задачу для всіх періодичних потенціалів, для яких ряди Фур’є містять кінцеву кількість доданків, типу sinm(θ ) або cosm (θ). Парні чи непарні функції вимагають дещо іншого підходу і розглядаються окремо. 2001 Article Analysis of Tridiagonal Recurrence Relations in Continuum Approximation / F.G. Bass, M. Gitterman // Радиофизика и радиоастрономия. — 2001. — Т. 6, № 1. — С. 71-78. — Бібліогр.: 11 назв. — англ. 1027-9636 http://dspace.nbuv.gov.ua/handle/123456789/122226 en Радиофизика и радиоастрономия Радіоастрономічний інститут НАН України |
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English |
| description |
Transition from difference to differential equation allows solving tridiagonal recurrence relations, which appear, among other things, in analysis of the rotation of an overdamped Brownian particle subjected to a periodic force. Replacement of the discrete integers in the Fourier series by continuum is justified for large numbers, i. e. for small angles. For the simplest case of the sinusoidal force, our solution, indeed, coincides with one obtained by expanding the sin in the original Fokker-Planck equation (The Ornstein-Uhlenbeck limit). However, for slightly more complicate potential the expansion for small angles does not transform the appropriate Fokker-Planck equation into the soluble. At the same time, the method suggested allows solving the problem for all periodic potentials which have finite number of terms in their Fourier series such as sinm(θ ) or cosm (θ). Even and odd functions require slightly different analysis, and are considered separately. |
| format |
Article |
| author |
Bass, F.G. Gitterman, M. |
| spellingShingle |
Bass, F.G. Gitterman, M. Analysis of Tridiagonal Recurrence Relations in Continuum Approximation Радиофизика и радиоастрономия |
| author_facet |
Bass, F.G. Gitterman, M. |
| author_sort |
Bass, F.G. |
| title |
Analysis of Tridiagonal Recurrence Relations in Continuum Approximation |
| title_short |
Analysis of Tridiagonal Recurrence Relations in Continuum Approximation |
| title_full |
Analysis of Tridiagonal Recurrence Relations in Continuum Approximation |
| title_fullStr |
Analysis of Tridiagonal Recurrence Relations in Continuum Approximation |
| title_full_unstemmed |
Analysis of Tridiagonal Recurrence Relations in Continuum Approximation |
| title_sort |
analysis of tridiagonal recurrence relations in continuum approximation |
| publisher |
Радіоастрономічний інститут НАН України |
| publishDate |
2001 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/122226 |
| citation_txt |
Analysis of Tridiagonal Recurrence Relations in Continuum Approximation / F.G. Bass, M. Gitterman // Радиофизика и радиоастрономия. — 2001. — Т. 6, № 1. — С. 71-78. — Бібліогр.: 11 назв. — англ. |
| series |
Радиофизика и радиоастрономия |
| work_keys_str_mv |
AT bassfg analysisoftridiagonalrecurrencerelationsincontinuumapproximation AT gittermanm analysisoftridiagonalrecurrencerelationsincontinuumapproximation |
| first_indexed |
2025-07-08T21:21:29Z |
| last_indexed |
2025-07-08T21:21:29Z |
| _version_ |
1837115312976166912 |
| fulltext |
Ðàäèîôèçèêà è ðàäèîàñòðîíîìèÿ, 2001, ò. 6, ¹1, ñòð. ??-??
© F. Bass, M. Gitterman, 2001
Analysis of Tridiagonal Recurrence Relations
in Continuum Approximation
F. Bass, M. Gitterman
Department of Physics, Bar Ilan University,
Ramat Gan, 52900 Israel
Received on April 11, 2001
Transition from difference to differential equation allows solving tridiagonal recurrence relations,
which appear, among other things, in analysis of the rotation of an overdamped Brownian particle
subjected to a periodic force. Replacement of the discrete integers in the Fourier series by continuum
is justified for large numbers, i. e. for small angles. For the simplest case of the sinusoidal force, our
solution, indeed, coincides with one obtained by expanding the sin in the original Fokker-Planck equation
(The Ornstein-Uhlenbeck limit). However, for slightly more complicate potential the expansion for small
angles does not transform the appropriate Fokker-Planck equation into the soluble. At the same time,
the method suggested allows solving the problem for all periodic potentials which have finite number of
terms in their Fourier series such as sin ( )m θ or cos ( ).m θ Even and odd functions require slightly different
analysis, and are considered separately.
The current radio physics is essentially nonlin-
ear science. This is caused by using high power
generators being concerned with their arrange-
ment as well as with the effect of the radiation
produced on various objects. Nonlinear processes
occur in the natural conditions too.
Now, one can speak with assurance about the
occurence in Kharkov of the lead of nonlinear
radio physics which found the world recognition.
The first from the papers in which the complicat-
ed nonlinear problem was solved, and which can
be called the basic one, was the theory of mag-
netron built by S. Ya. Braude. He succeeded in
solving the system of nonlinear equations describ-
ing the electron motion in magnetron. This work
as is known was approved by L. D. Landau.
In present paper the new method is proposed for
solving the nonlinear problems describing the radio
physical phenomena including the behavior of the
nonlinear circuit with strong resistivity, the motion of
an electron in nonlinear resistive medium, et cetera.
The present problem is related to the same
field as the pioneer work of S. Ya. Braude.
1. Introduction
Many ordinary and partial differential equations
used in a practice after suitable expansion reduce
to tridiagonal recurrence relation of the form
1
d
d
n
n n n n n n
c
Q c Q c Q c
t
− +
− += + + (1)
Some examples are listed in the Risken mono-
graph [1] (master equation with nearest-neighbor
coupling, the one � dimensional Shroedinger equa-
tion with an unharmonic potential, the Fokker �
Plank equation for lasers, and for the Brownian
particle moving in a periodic potential). The un-
known quantities nc in Eq. (1) might be scalars or
column vectors. We restrict our analysis to the
case of scalars; the extension to vectors should
present no problems.
Overdamped Brownian motion in a periodic
potential is a typical example leading to Eq. (1). Its
equation of motion has the following form:
F. Bass, M. Gitterman
2
Ðàäèîôèçèêà è ðàäèîàñòðîíîìèÿ, 2001, ò. 6, ¹1
d
( ) ( ),
d
g a f t
t
θ
+ θ = + (2)
where ( )g θ and a are the periodic and con-
stant forces acting on a particle, in addition
1 1( ) ( ) 2 ( ).f t f t D t t< >= δ −
The Fokker-Planck equation for the distribu-
tion function ( , ),P tθ which corresponds to the
Langevin equation (2), has the following form:
( )
2
2
d
( ) .
d
P P
g a P D
t
∂ ∂= θ − + ∂θ ∂θ
(3)
If Eq. (3) has coefficients periodic in θ, then
its solution is also periodic in θ, and, therefore,
may be expanded in the Fourier series
( , ) ( )exp( ).
n
n
n
P t c t in
=∞
=−∞
θ = θ∑ (4)
Substituting the expansion of the periodic func-
tion ( )g θ in the Fourier series
( ) exp( )
n
k
n
g d ik
=∞
=−∞
θ = θ∑ (5)
and (4) into (3), one obtains after simple transfor-
mation
2d
( ) .
d
k
n
n k n k
k
c
Dn ina c in d c
t
=∞
−
=−∞
= − − − ∑ (6)
If the sum in Eq. (6) contains only a finite
number of terms or if this sum converges so
rapidly that one can restrict our analysis to a
finite number of terms, then Eq. (6) takes the
form of scalar (or vector) tridiagonal recurrence
relations (1) [1].
The equation of motion of an overdamped
pendulum subject to a constant and random torque
is the simplest example of an equation of the form
(2) (with ( ) sing θ = θ ):
d
sin ( ).
d
b a f t
t
θ
+ θ = + (7)
Eq. (7) describes many different phenomena,
such as motion of fluxons in superconductors [2],
motion of ions in superionic conductors [1] and
biological channels [3], charge density waves [4],
phase locking in electric circuits [3], mode locking
in ring laser gyroscopes [5], and the Josephson
junction [6].
The Fokker-Planck equation corresponding to
Eq. (7) has the form (3) (with ( ) sing θ = θ ):
2
2
(sin ) ,
P P
a P D
t
∂ ∂ ∂= θ − +
∂ ∂θ ∂θ
(8)
or the form (6) with the coefficients
1 1
1
2
d d
i−= − = and 0kd = for 1k ≠ :
2
1 1
d
( ) ( ).
d 2
n
n n n
c bn
Dn ina c c c
t
+ −= − − − − (9)
The usual way to solve the tridiagonal recur-
rence relations of type (1) is by matrix continued
fractions (see [1] and references therein). The
aim of this note is to propose another method for
solving Eqs. (1), (9), namely, the transition from
a difference to a differential equation. Of course,
this method is not new, and it has been used
successfully, for example, in continuum approx-
imation of small oscillations near the equilibrium
positions for chains of equidistant particles with
nearest-neighbor interactions. However, this
method has never been applied to the equations
of a kind (9), which would be the aim of our
considerations.
Replacing the integers n in Eq. (9) by the
continuous variable, one can make the Taylor
expansion which gives
( )
3
1 1 3
2
2 .
3
n n
n n
c c
c c O
n n
+ −
∂ ∂
− ≅ + ∂ ∂
(10)
Analysis of tridiagonal recurrence relations in continuum approximation
3
Ðàäèîôèçèêà è ðàäèîàñòðîíîìèÿ, 2001, ò. 6, ¹1
The error due to retaining only the first term in
the series expansion in (10) is of order of
13
3 2
2 1
2 ,
3 3
n nc c
nn n
−∂ ∂ ≈ ∂∂
which strongly decreas-
es with n.
Substituting (10) into (9), one gets
2[ ] .n n
n
c c
bn Dn ina c
t n
∂ ∂
+ = − −
∂ ∂
(11)
Let us now turn to the general Eq. (6), assuming
that summation in this equation is restricted to some
max .k < ∞ Then, for max,n k> one can expand n kc −
in Eq. (6),
22
2
...,
2
n n
n k n
c ck
c c k
n n
−
∂ ∂
≈ − + +
∂ ∂
which
reduces Eq. (6) to the following form
max
max
2
k
n
k n
k k
c
Dn ina in d c
t =−
∂
= − − − −
∂
∑
max max
max max
2
2
2
( ) ( ) .
2
k k
n n
k k
k k k k
c cin
in kd k d
n n=− =−
∂ ∂
+
∂ ∂∑ ∑ (12)
Three combination of the Fourier coeffi-
cients kd defined by (5), enter Eq. (12),
max
max
0 ,
k
k
k k
K d
=−
= ∑
max
max
1 ,
k
k
k k
K kd
=−
= ∑ and
max
max
2
2 .
k
k
k k
K k d
=−
= ∑ The calculations are slightly
different for the odd and even function ( ).g θ
For odd functions ( ),g θ k kd d− = − so that
0 0K = and
max
1
0
2 ,
k
k
k
K kd
=
= ∑ whereas for even
functions ( ),g θ
max
0
0
2
k
k
k
K d
=
= ∑ and 1 0.K =
Therefore, for odd ( ),g θ one can neglect the
last term in Eq. (12), and rewrite it as
2
1 .n n
n
c c
Dn ina c inK
t n
∂ ∂ = − − − ∂ ∂
(13)
Whereas for the even function ( )g θ
2
2
0 2 2
.
2
n n
n
c cin
Dn ina inK c K
t n
∂ ∂ = − − − + ∂ ∂
(14)
Analogous to Eq. (10), one concludes that the
relative error due to neglecting the next term in
the expansion in Eq. (14) is of the order
4 2
1
4 2 2
1 1 1
( ) .
24 2 12
n nc c
n n n
−∂ ∂
≈
∂ ∂
We consider in Sections 2 and 3 the simple
cases of purely deterministic and steady-state
overdamped pendulum, and compare our result
with the exact solutions, leaving to Section 4 the
analysis of (11). The general analysis of even
and odd functions ( )f θ described by Eqs. (13)
and (14) is performed in Sections 5 and 6. Final-
ly, some discussion and conclusions complete the
analysis.
2. Non-biased pendulum
Let us start with the simplest case, without
deterministic (a) or random ( ( ))f t forces in
Eq. (7). Then, Eq. (7) takes the simple form,
d
sin 0,
d
b
t
θ
+ θ = which allows the exact solution
0tan tan exp( ),
2 2
bt
θθ
= − (15)
where 0( 0) .tθ = = θ
Eq. (8) with 0a D= = transforms after in-
serting sinG P= θ to a first order partial differ-
ential equation with constant coefficients and
characteristic equations of the form
F. Bass, M. Gitterman
4
Ðàäèîôèçèêà è ðàäèîàñòðîíîìèÿ, 2001, ò. 6, ¹1
d d d
.
1 sin 0
t G
b
θ
= =
θ
(16)
The latter equations show that t and θ enter
the solution only in the combination
log tan ,
2
bt
θ +
so that the solution of Eq. (8)
with a = D = 0 has the following form:
log tan
2
,
sin
f bt
P
θ + =
θ
(17)
where ( )f z is an arbitrary function which is found
from the initial conditions.
If 0( , 0) ( ),P tθ = = δ θ − θ then Eq. (17) gives
0
log tan
2
( ) ,
sin
f
θ
δ θ − θ =
θ
(18)
which means that
1 1
0( ) sin 2tan (exp ) 2tan exp .
2
f z z z− − θ = δ − θ
(19)
Substituting now (19) at value
log tan
2
z bt
θ = =
into (17), one finds the coef-
ficients nc in the Fourier series (4),
1sin 2 tan tan exp( )
2
d exp( )
sinn
bt
c in
− θ
= θ − θ ×
θ∫
1
02tan tan exp( ) .
2
bt− θ δ − θ
(20)
Using the well-known relation,
[ ]
1
0
d
( ) ( ) ( )
d
−ψ δ ψ θ −θ = θ = θ δ θ−θ θ
% %
with 0( )ψ θ = θ% one can perform integration in
Eq. (20) which finally gives
0
1 0
sin
( , )
sin 2 tan tan exp( )
2
P t
bt−
θ
θ = ×
θ −
2 0
2 0
1 tan exp( )
2
1 tan exp( 2 )
2
bt
bt
θ + − ×
θ + −
1 0exp 2tan tan exp( ) .
2n
in bt− θ θ − −
∑ (21)
The sum over n in the exponent of the latter
formula defines the delta-function which gives the
solution (15) while the pre-sum normalization fac-
tor equals to unity both for 0t = and .t = ∞ Hence,
using the more complicated calculation, we ob-
tained the same solution, which follows immedi-
ately from the equation of motion, as it should be
since our calculation is exact.
Let us now solve the approximate Eq. (11)
and, then, compare the result obtained with the
exact one found both from the equation of motion
and from the Fokker-Planck equation.
The first-order partial differential Eq. (11) with
constant coefficients has the characteristic equa-
tions of the form
dd d
.
1 0
nct n
bn
= − = (22)
The latter equation is similar to Eq. (16), and
its solution can be found analogously to Eqs. (16)-
(21), which gives
Analysis of tridiagonal recurrence relations in continuum approximation
5
Ðàäèîôèçèêà è ðàäèîàñòðîíîìèÿ, 2001, ò. 6, ¹1
0( , ) exp[ ( exp( ))]
n
P t in btθ = θ − θ −∑ (23)
with the deterministic solution 0 exp( )btθ = θ −
which is the limiting case of the exact solution
(15) for small θ.
3. Steady-state case
In the steady-state limit, 0,nc
t
∂
=
∂
Eq. (9) re-
duces to an ordinary equation in finite differences
2
1 1( ) ( ) 0,
2n n n
bn
Dn ina c c c+ −− − − − = (24)
which can be solved rigorously. The continued-
fraction method was used for the solution of
Eq. (24) by Cresser et al. [7], while Ivanchenko
and Zilberman [8] noticed that Eq. (24) bears a
resemblance to the recurrence relation for the
Bessel functions of the imaginary argument [9],
1 12 ( ) [ ( ) ( )] 0.I z z I z I zν ν+ ν−ν + − = Comparing the
latter equation with Eq. (24), one concludes that
(up to a normalized factor),
.n ia
n
D
b
c I
D+
≈
(25)
For the following discussion, we need the as-
ymptotic form of Eq. (25) for large numbers
.n ia D+ The latter can be obtained from the
integral representation of this function [9],
0
1
( ) exp( cos )cos( )dI z z
π
ν = θ νθ θ −
π ∫
0
sin( )
exp( cosh )d .z t t t
∞νπ − − ν
π ∫ (26)
For large ν one can neglect the second inte-
gral in (26), and the main contribution to the first
integral comes from small θ since for large θ
the kernel of this integral oscillates rapidly. In
line with this, one can extend the range of in-
tegration to infinity and expand the argument
2cos( ) 1 2.θ ≈ − θ Then,
2exp( )
( ) exp cos( )d
2 2
z z
I z
∞
ν
−∞
θ= − νθ θ = π
∫
2exp( )
exp
22
z
zz
ν−
π
(27)
and, according to Eq. (25)
2
exp .
2n
a
D n i
D
c
b
+ ≈ −
(28)
Let us solve now Eq. (24) by the transition
from a difference to a differential equation,
which gives
( ) .n
n
c
b Dn ia c
t
∂
= − +
∂
(29)
As is easy to see, the solution of Eq. (29)
coincides with the asymptotic (for large n) form
(28) of exact solution.
After checking the applicability of our approx-
imation by comparing with the known exact solu-
tions for the field-free and steady-state cases, we
proceed in the next Section to the analysis of the
general Eq. (11), for which the exact solution is
unknown.
4. Overdamped pendulum
The calculations to be described here are sim-
ilar to those performed in Section 2. Consider
Eq. (11), where a and b might be arbitrary func-
tions of t
2( ) ( ) .n n
n
c c
b t n Dn ina t c
t n
∂ ∂ + = − − ∂ ∂
(30)
F. Bass, M. Gitterman
6
Ðàäèîôèçèêà è ðàäèîàñòðîíîìèÿ, 2001, ò. 6, ¹1
The characteristic equations associated with
Eq. (30) have the following form
2
dd d
.
1 ( ) ( )
nct n
nb t Dn ina t
= = −
+
(31)
The first equation in (31) defines the first con-
stant of integration
1
0
exp ( )d
t
C n b
= − τ τ
∫ (32)
while from the second equation, one gets
2
0
exp ( )d
t
nc C n b
= − τ τ ×
∫
2
1
0 0
exp d exp 2 ( )
t z
DC z b
− τ −
∫ ∫
1
0 0
d ( )exp ( )d .
t z
iC za z b
τ τ
∫ ∫ (33)
Taking Eq. (33) into account, according to the
theory of partial differential equations, the second
constant of integration
0
0
exp exp ( )d
t
nc in b
= − − τ τ θ ×
∫
2
0 0 0
exp exp 2 ( )d d exp 2 ( )d
t t z
Dn b z b
− − τ τ τ τ −
∫ ∫ ∫
0 0 0
exp ( )d ( )exp ( )d .
t t z
in b za z b
− τ τ τ τ
∫ ∫ ∫
(34)
For the special case of time-independent a
and b, Eq. (34) reduces to
[ ]0exp exp( )nc in bt= − θ − ×
[ ] [ ]
2
exp 1 exp( 2 ) 1 exp( ) .
2
Dn ina
bt bt
b b
− + − − − −
(35)
It is a matter of direct verification to confirm
that Eqs. (34) and (35) are solutions of Eq. (30)
with ( ),a a t= ( )b b t= and const,a = const,b =
respectively.
Substituting Eq. (35) into Eq. (4), one obtains
the full solution of the Fokker �Planck equation (8)
corresponding to the Langevin equation (7), while
Eq. (34) relates to the Langevin equation (7) with
the time-dependent coefficient. Recall that all these
solutions have been obtained on the assumption
that one can replace the difference in n tridiagonal
recurrence relations by differential equations. Now
we are in a position to understand the final results
of this assumption. It turns out that our final result
(35) coincides with the solution of the Ornstein-
Uhlenbeck process (sin( )θ → θ in the original
equation) which can be found in the book by Gar-
diner [10] for a = 0. Therefore, our approximation
describes the limiting case of small angles, as we
have already seen in the field-free case described
in Section 2.
Provided the distribution function ( , )P tθ is
known, one can find all the correlation functions.
For our case, which reduces to the Ornstein-Uhlen-
beck approximation, the correlation functions are:
1( ) ( )t t< θ θ > and 1cos[ ( )]cos[ ( )] ,t t< θ θ > which
have been calculated in [10] and [7], respectively.
5. Odd potential
In two previous sections, we considered the
specific form of the periodic function ( ) sin ,g θ = θ
which is important for many applications. The
general case of the odd function ( )g θ is described
by Eq. (13) which looks exactly like Eq. (30) upon
replacing b by 1ibK and a by 0.a K+ Of course,
although these equations look similar, the accura-
cy of our approximation strongly depends on the
form of ( )g θ since one assumes max.n k>
As an example, let us consider the special
case of 3( ) sin ( ),g θ = θ which, in contrast to
Analysis of tridiagonal recurrence relations in continuum approximation
7
Ðàäèîôèçèêà è ðàäèîàñòðîíîìèÿ, 2001, ò. 6, ¹1
( ) sin( ),g θ = θ does not reduce to Ornstein-
Uhlenbeck equation, and cannot be solved by
expansion in θ. Since this function is odd, 0 0.K =
Using the well-known relation [9]
[ ]3 1
sin ( ) sin(3 ) 3sin
4
θ = − θ + θ (36)
one concludes that 3 3
1
8
d d
i−= − = − and
1 1
3
,
8
d d
i−= − = so that 1
1
.K
i
= Since 1 1,iK = the
final Eqs. (34) and (35) obtained in the previous
Section for ( ) sin( )g θ = θ and n > 1 also apply for
3( ) sin ( )g θ = θ for n > 3.
6. Even potential
The case of an even potential is described by
Eq. (14), which can be rewritten, after the sepa-
ration of variables ( , ) exp( ) ( )n nc n t t q n= −λ as
2
02
2
2 d
( ) 0.
d n
i q
i a nK Dn q
K nn
λ + + + + =
(37)
The solution of Eq. (37) for the steady-state
case, 0,λ = presents no problem since Eq. (37)
reduces to the differential equation for the Airy
function [11].
In the general case 0,λ ≠ one has to per-
form substitution
d
( ) ln[ ( )]
d
r n q n
n
= which trans-
forms Eq. (37) into the Ricatti equation for ( )r n
of the form
2 2
0
d
( ) ,
d 2
Kr
r i a nK Dn
n i n
λ + = + + +
(38)
after which one obtains the required solution
of Eq. (37),
( )d
( ) exp ( )d .
d
q
r n r n n
n
− = −∫ (39)
In general, the solutions of Eqs. (38) and (39)
cannot be obtained by quadratures, and one has to
use the approximate or numeral methods.
For 2( ) cos ,g θ = θ 2 2 02 1 4,d d d= − = = so
that in this case, 0 1 2K = and 2 2.K = To find
the approximate solution of Fokker-Planck equa-
tion (3) for arbitrary even periodic function ( )g θ
with a finite numbers of terms kd in its Fourier
series (5) (till some maxk ), one has to find these
,kd and calculate two numbers
max
max
0 ,
k
k
k k
K d
=−
= ∑
and
max
max
2
2
k
k
k k
K k d
=−
= ∑ in Eq. (37). The latter equa-
tion is justified for max.n k>
7. Conclusions
We have used the transition from difference to
differential equations as an approximate method of
solving tridiagonal recurrence equations. This well-
known method of replacing a discrete integer vari-
able n by a continuous variable has a relative error
of order of
2,n−
i. e. this approximation is justified
for large n, and the error is decreased with n.
According to expansion (4) in the Fourier series,
only small θ make an essential contribution to this
series for large n, compared with rapidly oscillating
large terms. It is no wonder, therefore, that for the
overdamped pendulum without periodic and ran-
dom forces (Section 2), and in the presence of
these forces in the steady-state (Section 3) and in
the general time-dependent case (Section 4), our
method coincides with the exact solution in the limit
of small θ. The periodic force acting on the pendu-
lum has a simple form ( ) sin ,g bθ = θ and after
replacing sinθ by θ in the Fokker-Planck equation,
the latter takes the Ornstein-Uhlenbeck form which
allows an exact solution. However, for a slightly
different form of a periodic force, say
2 1( ) sin ( ),mg −θ = θ the expansion for small θ has
the form 2 1 2 1sin ( ) ,m m− −θ ≈ θ and the appropriate
Fokker-Planck equation cannot be solved. Howev-
er, our method can be applied for arbitrary function
( ),g θ leading to the first-order partial differential
Eq.(13) for odd functions ( )g θ and to the second-
order partial differential Eq. (14) for even functions
( ).g θ The coefficients in these equations contain
F. Bass, M. Gitterman
8
Ðàäèîôèçèêà è ðàäèîàñòðîíîìèÿ, 2001, ò. 6, ¹1
a few simple combinations of the coefficients in
the Fourier expansions of ( )g θ under the assump-
tion that these expansions contain a finite number
maxk of terms, and the equations can easily be
solved, as we illustrated by a few simple examples
in Sections 5 and 6. Strictly speaking, our proce-
dure is applicable for max,n k> and it has a relative
error of order
2.n−
There are different ways to
improve the accuracy of our method. One way is
to restrict this procedure to the values of n that
are too small, min,n n≥ where the accuracy is high
2
min( ),n−≈ and the coefficients 1... nc c with minn n<
will be found from the appropriate tridiagonal re-
currence relations with the previously found
min
.nc
References
1. H. Risken. The Fokker-Planck Equation; Methods of
Solution and Applications, Second Edition. Berlin,
Springer-Verlag. 1996, Chapter 9-11.
2. B. Shapiro, I. Dayan, M. Gitterman, and G. H. Weiss.
Phys. Rev. 1992, B 46, p. 8416.
3. A. J. Viterbi. Principles of Coherent Communications.
New York, McGraw-Hill, 1966.
4. G. Gruner, A. Zavadowski, and P. M. Chaikin. Phys.
Rev. Letters. 1981, 46, p. 511.
5. W. Shleich, C. S. Cha, and J. D. Cresser. Phys. Rev.
1984, A 29, pp. 230.
6. A. Barone and G.Paterno. Physics and Applications
of Josephson Effect. New York, Willey, 1982.
7. J. D. Cresser, D. Hammonds, W. H. Louisell, P. Meystre,
and H. Risken. Phys. Rev. A. 1982, 25, p. 2226.
8. Yu. M. Ivanchenko and L. A. Zilberman. Zh. Eksp.
Teor. Fiz. 1968, 55, pp. 2395; Sov. Phys.-JETP. 1969,
28, pp. 1272.
9. I. S. Gradstein and I. M. Ryzhik. Table of Integrals,
Series and Products. Boston, Academic Press, 1994.
10. C. W. Gardiner. Handbook of Stochastic Methods.
Berlin, Springer-Verlag, 1997.
11. M. Abramowitz, I. A. Stegun. Handbook of
Mathematical Functions. New York, Dover, 1972.
Àíàëèç òðèäèàãîíàëüíûõ
ðåêóððåíòíûõ ñîîòíîøåíèé
â êîíòèíóàëüíîì ïðèáëèæåíèè
Ô. Áàññ, Ì. Ãèòòåðìàí
Ïåðåõîä îò ðàçíîñòíîãî ê äèôôåðåíöè-
àëüíîìó óðàâíåíèþ ïîçâîëÿåò ðåøèòü òðè-
äèàãîíàëüíûå ðåêóððåíòíûå ñîîòíîøåíèÿ,
êîòîðûå âîçíèêàþò, â ÷àñòíîñòè, ïðè àíàëè-
çå âðàùåíèÿ áðîóíîâñêîé ÷àñòèöû ñ òðåíè-
åì ïðè íàëè÷èè ïåðèîäè÷åñêîé ñèëû. Çàìå-
íà äèñêðåòíûõ èíäåêñîâ â ðàçëîæåíèÿõ Ôó-
ðüå íåïðåðûâíûìè îïðàâäàí äëÿ áîëüøèõ
íîìåðîâ, ò. å. äëÿ ìàëûõ óãëîâ. Â ïðîñòåé-
øåì ñëó÷àå ñèíóñîèäàëüíîé ñèëû íàøå ðå-
øåíèå äåéñòâèòåëüíî ñîâïàäàåò ñ ðåøåíè-
åì, ïîëó÷åííûì ïóòåì ðàçëîæåíèÿ ñèíóñà â
ïåðâîíà÷àëüíîì óðàâíåíèè Ôîêêåðà-Ïëàíêà
(ïðåäåë Îðíøòåéíà-Óëåíáåêà). Îäíàêî óæå
â ñëó÷àå íåñêîëüêî áîëåå ñëîæíîãî ïîòåí-
öèàëà ðàçëîæåíèå ïðè ìàëûõ óãëàõ íå äåëà-
åò ñîîòâåòñòâóþùåå óðàâíåíèå Ôîêêåðà-
Ïëàíêà ðàçðåøèìûì. Â òî æå âðåìÿ ïðåäëà-
ãàåìûé ìåòîä ïîçâîëÿåò ðåøèòü çàäà÷ó äëÿ
âñåõ ïåðèîäè÷åñêèõ ïîòåíöèàëîâ, äëÿ êîòîðûõ
ðÿäû Ôóðüå ñîäåðæàò êîíå÷íîå ÷èñëî ñëàãà-
åìûõ òèïà sin ( )m θ èëè cos ( ).m θ ×åòíûå ëèáî
íå÷åòíûå ôóíêöèè òðåáóþò íåñêîëüêî ðàçëè÷-
íîãî ïîäõîäà è ðàññìàòðèâàþòñÿ îòäåëüíî.
Àíàë³ç òðèä³àãîíàëüíèõ ðåêóðåíòíèõ
ñï³ââ³äíîøåíü ó êîíòèíóàëüíîìó
íàáëèæåíí³
Ô. Áàññ, Ì. óòòåðìàí
Ïåðåõ³ä â³ä ð³çíèöåâîãî äî äèôåðåíö³àëüíî-
ãî ð³âíÿííÿ äîçâîëÿº âèð³øèòè òðèä³àãîíàëüí³
ðåêóðåíòí³ ñï³ââ³äíîøåííÿ, ÿê³ âèíèêàþòü, çîê-
ðåìà, ïðè àíàë³ç³ îáåðòàííÿ áðîóí³âñüêî¿ ÷àñ-
òèíêè ç òåðòÿì ó ïðèñóòíîñò³ ïåð³îäè÷íî¿ ñèëè.
Çàì³íà äèñêðåòíèõ ³íäåêñ³â ó ðîçêëàäàíí³ Ôóð�º
íåïåðåðâíèìè âèïðàâäàíà äëÿ âåëèêèõ íîìåð³â,
òîáòî äëÿ ìàëèõ êóò³â. Ó íàéïðîñò³øîìó âè-
ïàäêó ñèíóñî¿äàëüíî¿ ñèëè íàøå ð³øåííÿ ñï³âïà-
äຠ³ç ð³øåííÿì, îòðèìàíèì øëÿõîì ðîçêëàäàí-
íÿ ñèíóñà ó ïî÷àòêîâîìó ð³âíÿíí³ Ôîêêåðà-
Ïëàíêà (ãðàíèöÿ Îðíøòåéíà-Óëåíáåêà). Îäíàê
óæå ó âèïàäêó äåùî ñêëàäí³øîãî ïîòåíö³àëó
ðîçêëàäàííÿ ïðè ìàëèõ êóòàõ íå ðîáèòü â³äïî-
â³äíå ð³âíÿííÿ Ôîêêåðà-Ïëàíêà âèð³øóâàíèì.
Âîäíî÷àñ çàïðîïîíîâàíèé ìåòîä äîçâîëÿº âèð-
³øèòè çàäà÷ó äëÿ âñ³õ ïåð³îäè÷íèõ ïîòåíö³àë³â,
äëÿ ÿêèõ ðÿäè Ôóð�º ì³ñòÿòü ê³íöåâó ê³ëüê³ñòü
äîäàíê³â, òèïó sin ( )m θ àáî cos ( ).m θ Ïàðí³ ÷è
íåïàðí³ ôóíêö³¿ âèìàãàþòü äåùî ³íøîãî ï³äõîäó
³ ðîçãëÿäàþòüñÿ îêðåìî.
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