Approximate solution for Fokker-Planck equation
In this paper, an approximate solution to a specific class of the Fokker-Planck equation is proposed. The solution is based on the relationship between the Schrödinger type equation with a partially confining and symmetrical potential. To estimate the accuracy of the solution, a function error obtai...
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nasplib_isofts_kiev_ua-123456789-1552722025-02-09T16:34:56Z Approximate solution for Fokker-Planck equation Наближений розв’язок рiвняння Фоккера-Планка Drigo Filho, E. Araujo, M.T. In this paper, an approximate solution to a specific class of the Fokker-Planck equation is proposed. The solution is based on the relationship between the Schrödinger type equation with a partially confining and symmetrical potential. To estimate the accuracy of the solution, a function error obtained from the original Fokker-Planck equation is suggested. Two examples, a truncated harmonic potential and non-harmonic polynomial, are analyzed using the proposed method. For the truncated harmonic potential, the system behavior as a function of temperature is also discussed. У цiй статтi запропоновано наближений розв’язок спецiального класу рiвняння Фоккера-Планка. Розв’язок базується на зв’язку з рiвнянням типу Шредингера з частково обмеженим i симетричним потенцiалом. Щоб оцiнити точнiсть розв’язку, запропоновано функцiю похибок, яка отримана з оригiнального рiвняння Фоккера-Планка. Використовуючи запропонований метод, проаналiзовано два приклади, а саме, утятий гармонiчний потенцiал i негармонiчний полiном. Окрiм цього, для утятого гармонiчного потенцiалу обговорено поведiнку системи в залежностi вiд температури. The authors acknowledge the financial support by the Brazilian agency CNPq (Proj. ESN No. 233776/2014-1 and Proj. PDE No. 232865/2014-0) and the financial support of the Spanish MINECO (Project MTM2014-57129-C2-1-P) and Junta de Castilla y León (UIC 011). 2015 Article Approximate solution for Fokker-Planck equation / M.T. Araujo, E. Drigo Filho // Condensed Matter Physics. — 2015. — Т. 18, № 4. — С. 43003: 1–12. — Бібліогр.: 19 назв. — англ. 1607-324X DOI:10.5488/CMP.18.43003 arXiv:1512.07787 PACS: 05.10.Gg, 02.30.Mv https://nasplib.isofts.kiev.ua/handle/123456789/155272 en Condensed Matter Physics application/pdf Інститут фізики конденсованих систем НАН України |
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In this paper, an approximate solution to a specific class of the Fokker-Planck equation is proposed. The solution is based on the relationship between the Schrödinger type equation with a partially confining and symmetrical potential. To estimate the accuracy of the solution, a function error obtained from the original Fokker-Planck equation is suggested. Two examples, a truncated harmonic potential and non-harmonic polynomial, are analyzed using the proposed method. For the truncated harmonic potential, the system behavior as a function of temperature is also discussed. |
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Approximate solution for Fokker-Planck equation |
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Approximate solution for Fokker-Planck equation |
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Approximate solution for Fokker-Planck equation |
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Approximate solution for Fokker-Planck equation |
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Approximate solution for Fokker-Planck equation |
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approximate solution for fokker-planck equation |
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Approximate solution for Fokker-Planck equation / M.T. Araujo, E. Drigo Filho // Condensed Matter Physics. — 2015. — Т. 18, № 4. — С. 43003: 1–12. — Бібліогр.: 19 назв. — англ. |
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Condensed Matter Physics |
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Condensed Matter Physics, 2015, Vol. 18, No 4, 43003: 1–12
DOI: 10.5488/CMP.18.43003
http://www.icmp.lviv.ua/journal
Approximate solution for Fokker-Planck equation
M.T. Araujo1,2, E. Drigo Filho1,3
1 Instituto de Biociências, Letras e Ciências Exatas, UNESP - Univesidade Estadual Paulista,
2265 Cristóvão Colombo St., 15054-000, São José do Rio Preto/SP, Brazil
2 Institut für Physik, Universität Augsburg , Universitätsstr. 1, D-86135, Augsburg, Germany
3 Departamento de Física Teórica, Atómica y Óptica, and IMUVA (Instituto de Matemáticas),
Universidad de Valladolid, 47011 Valladolid, Spain
Received June 1, 2015, in final form October 6, 2015
In this paper, an approximate solution to a specific class of the Fokker-Planck equation is proposed. The solution
is based on the relationship between the Schrödinger type equation with a partially confining and symmetrical
potential. To estimate the accuracy of the solution, a function error obtained from the original Fokker-Planck
equation is suggested. Two examples, a truncated harmonic potential and non-harmonic polynomial, are ana-
lyzed using the proposed method. For the truncated harmonic potential, the system behavior as a function of
temperature is also discussed.
Key words: Fokker-Planck equation, Schrödinger equation, approximate solution
PACS: 05.10.Gg, 02.30.Mv
1. Introduction
Differential equations are used to model different natural phenomena such as diffusion that are of
great importance in many physical, chemical and biological processes [1].
In general, diffusive processes can be treated via the Fokker-Planck equation [2–6]. This equation is
obtained from the Langevin equation and gives the probability of finding a given particle in a state x at
time t . The usual representation of the Fokker-Planck equation is given as follows:
∂P (x, t )
∂t
=− ∂
∂x
[
f (x)P (x, t )
]+Q
∂2P (x, t )
∂x2 , (1.1)
where t is the time variable, and x is the variable characteristic of the system (which may be identified,
for example, as velocity).Q is the diffusion coefficient and P (x, t ) is the probability distribution. The f (x)
function is known as the external force that acts on the system, although this designation is only suitable
when x represents velocity. This function can be identified as the derivative of a potential function V (x):
f (x) =−∂V (x)/∂x.
Different methods for treating the Fokker-Planck equation have been suggested as, for example, its as-
sociation with a Schrödinger type equation [2, 7]. However, there are only a few cases where the equation
(1.1) has an exact analytical solution.
In a recent study [8], an approximate solution for partially confining potentials was proposed, where
part of the solution is written in terms of functions that arise from the solution of the Schrödinger type
equation for bound states and part is composed of functions originating from the free particle. This so-
lution method is discussed in reference [9] for the particular case of the Rosen-Morse potential. This
potential has an exact solution to the Schrödinger equation. As expected, it can be concluded that the
results obtained using the functions derived from the original Schrödinger type equation provide better
solutions than those obtained by the proposed approximate method. However, this approach proves to
be very restrictive, since the number of potentials with exact solutions to the Schrödinger equation is
very small.
©M.T. Araujo, E. Drigo Filho, 2015 43003-1
http://dx.doi.org/10.5488/CMP.18.43003
http://www.icmp.lviv.ua/journal
M.T. Araujo, E. Drigo Filho
In this paper, an improvement to the previously proposed solution [8] is suggested, producing a more
accurate result for cases of symmetrical partially-confining potentials. To check the validity of the results,
the error on the results is estimated through direct substitution of the approximate solution into the
Fokker-Planck equation.
In the study of the truncated harmonic potential, the amount of particles that escape from the poten-
tial well are calculated and a phase transition for the systems being studied is identified.
In section 2, there is a brief discussion of the Fokker-Planck equation and the approximate solution
method is presented, with the changes suggested in relation to the previously proposed solution [8]. In
sections 3 and 4, the proposed method is applied to two partially-confining potentials. In the first case,
the region of the well potential is described by a harmonic potential and, for the second case, the well is
given by a non-harmonic polynomial potential. Finally, the conclusions are presented in section 5.
2. The Fokker-Planck equation
Given the importance and difficulty of solving the Fokker-Planck equation (FPE), different methods
have been proposed to study this equation. Such methods include numerical treatments [10, 11] and
mapping the FPE onto a Schrödinger type equation [2, 7, 12]. In the latter case, the expression of P (x, t )
is given by a series of functions as follows:
P (x, t ) =
∞∑
n=0
anφ0(x)φn(x)e−t |λn |, (2.1)
where the eigenfunctions φn(x) and the eigenvalues λn are the solution of the Schrödinger type equationobtained from the FPE (1.1):
λnφn =−φn
2
{
d f (x)
dx
+ f 2(x)
2Q
}
+Q
d2φn
dx2 . (2.2)
Comparing equation (2.2) with the Schrödinger equation, it can be seen that the term in parentheses
is equivalent to a potential function. Thus, this term is commonly called the effective potential, Vef(x):
Vef(x) = 1
2
{
d f (x)
dx
+ f 2(x)
2Q
}
. (2.3)
The probability distribution indicated by equation (2.1) assumes that the initial condition for the time
t = 0 is the probability distribution expressed by a delta function P (x,0) = δ(x).
In order to obtain the numerical solution shown in equation (2.1), it is necessary to define a criterion
to the cut off in the sum to truncate the series. Another problem arises when one cannot get the solution of
the corresponding Schrödinger equation (2.2). To get around this last problem, an approximate analytical
solution composed of two parts was suggested in a previous paper [8]. This solution involves the discrete
levels of the problem and the other one uses a Gaussian distribution for all continuous states. Although
the solution proposed in [8] agrees with the numerical results, the adopted approach can be improved
upon.
The attention here is focused on specific issues involving symmetrical and partially confined poten-
tials, where it is not possible to get the complete solution of the Schrödinger type equation (2.2). Thus, for
these types of potential, a similar treatment to that of reference [9], for example, is unfeasible. For the
potentials studied, there are regions where the spectrum is continuous and a region where there may be
bound states (i.e., a discrete spectrum). For x > d and x <−d , it is assumed that the potential is constant
and the spectrum is continuous. In −d < x < d , there is a potential well and the spectrum becomes dis-
crete. It is also assumed that the potential is continuous, especially at the points x =±d that correspond
to the intersections between the region where the potential is constant and the well potential region (fig-
ures 1, 6 and 8 exemplify the type of the studied potential). The probability distribution in such cases is
calculated separately for each region of the potential.
P (x, t ) =
NIρI(x, t ), −d > x,
NIIρII(x, t ), −d É x É d ,
NIIIρIII(x, t ), x > d .
(2.4)
43003-2
Approximate solution for Fokker-Planck equation
The function ρ(x, t ) in (2.4) is the probability distribution for each region; NI, NII and NIII are relatedto the normalization in each piece of the probability distribution. For regions (I and III) of the continuous
spectrum, the ρ functions are given by a Gaussian function, as suggested previously [8]
ρI(x, t ) = ρIII(x, t ) = 1√
4Qπt
e−x2/
4Qt (2.5)
and for region II, the function ρ is expressed by the series indicated in (2.1) with a limited number of
eigenvalues ( j )
ρII(x, t ) =
j∑
i=0
aiφ0(x)φi (x)e−t |λi |. (2.6)
The number j corresponds to the number of discrete eigenvalues present in the potential well under
analysis.
As the functions ρI(x, t ) and ρIII(x, t ) are equal, because of the symmetry of the problem, one can
assume that the normalization parameters NI and NIII are the same. It can also be assumed that at theinterface points between the potentials (±d ), the distribution ρ(x, t ), equation (2.4) should be continuous,
i.e., ρII(d , t ) = ρIII(d , t ) and ρI(−d , t ) = ρII(−d , t ). Thus, the condition of the continuity of the distribution
and the symmetry of the problem imply that NI = NIII and
NII(t ) = NI e−d 2/
4Qt√
4πQt
{
j∑
i=0
aiφ0(d)φi (d)e−t |λi |
}−1
. (2.7)
This relationship (2.7) shows that the normalization NII depends on NI and also, in general, dependson time. To simplify the notation, we will rewrite equation (2.7) as NII(t ) = NIg (t ), such that:
g (t ) = e−d 2/
4Qt√
4πQt
{
j∑
i=0
aiφ0(d)φi (d)e−t |λi |
}−1
. (2.8)
Therefore, the overall probability distribution for a problem with the discussed features (a partially
confining and symmetric potential) is obtained by equation (2.4), subject to the condition (2.7), i.e.,
P (x, t ) =
NI 1p
4Qπt
e−x2/
4Qt ,
NIg (t )
j∑
i=0
aiφ0(x)φi (x)e−t |λi |,
NI 1p
4Qπt
e−x2/
4Qt ,
−d > x,
−d É x É d ,
x > d .
(2.9)
Applying the normalization condition to the probability distribution (2.9), one gets
NI(t ) =
2√
4πQt
∞∫
d
e−x2/
4Qt dx + g (t )
d∫
−d
j∑
i=0
aiφ0(x)φi (x)e−t |λi |dx
−1
. (2.10)
According to this result, the normalization parameter NI is dependent on time and the probabilitydistribution (2.9) should be normalized for each time value. Looking at the approximate solution given
by equation (2.9), one can see that when d is close to zero the system approximated to a free particle
system and the probability distribution approximates to a Gaussian. On the other hand, when d is very
large (d →∞), i.e., the size of the system tends to an infinite well, the solution of the problem is given by
the usual discrete series of functions, as shown in equation (2.1).
The difference between the solution given by (2.9) and that shown in reference [8] is that here, the
Gaussian is used only in areas where the potential is constant, while in reference [8], it was suggested
that it should be included in all regions of the space. Numerical results show that the new approach is
more suitable, leading to more accurate results.
In general, when there is a partial confinement, a temporal dependency arises already in the coeffi-
cient NI . Under these conditions, the probability distribution for large times in the region of the potential
43003-3
M.T. Araujo, E. Drigo Filho
well (region II) may go to zero, which indicates the escape of particles from the minimum region of the
potential.
From the proposed solution it is suggested that the escape of particles from the potential well can be
quantified by the value Y (t ,Q) defined by:
Y (t ,Q) = NI(t )g (t )
d∫
−d
j∑
i=0
aiφ0(x)φi (x)e−t |λi |dx. (2.11)
The function Y (t ,Q) gives the number of particles within the confinement region for each time t and
different values of the diffusion coefficient (Q). The functions NI(t ) and g (t ) are given by expressions
(2.10) and (2.8), respectively.
Equation (2.11) also allows the evaluation of the influence of temperature in the escape process of
the particles in the well. Assuming that the temperature is proportional to the diffusion coefficient [2],
the calculation of the population for different values of Q allows for the analysis of the evolution of the
system in terms of temperature. This information allows, in principle, the study of the thermodynamic
properties of the system, such as phase transitions.
To check the accuracy of the proposed method, we introduce the function ε(x, t ) based on the Fokker-
Planck equation (1.1)
ε(x, t ) = ∂P (x, t )
∂t
−
{
− ∂
∂x
[
f (x)P (x, t )
]+Q
∂2P (x, t )
∂x2
}
. (2.12)
This function provides a quantitative parameter to check if the solution approximates to the actual
solution of the problem. If the solution is accurate, then ε(x, t ) = 0. In this way, the further this function
ε(x, t ) is close to zero, the better is the proposed function P (x, t ) to describe the real solution for the
system under study.
Substituting the solution presented in (2.9) into expression (2.12), it can be seen that for x < −d and
x > d , where f (x) is zero, we obtain
εI(x, t ) = 1√
4Qπt
e−x2/
4Qt dNI(t )
dt
, (2.13)
and for −d < x < d , the expression ε(x, t ) is obtained by direct substitution of the solution in this region
[equation (2.9)] in equation (2.12).
Since the construction of ε(x, t ) involves derivatives of the probability distribution P (x, t ), it is worth
noting that, close to the points where the derivative is discontinuous, the use of this criterion is impaired
and should be used with caution. Thus, in this study, ε(x, t ) was not defined for values of x close to
±d . However, the use of the function ε(x, t ) to evaluate the solution avoids comparisons with solutions
obtained by other methods which could, in itself, introduce an additional error.
3. Truncated harmonic oscillator
In this section, we apply the approximate solution of the FPE to a model with a truncated harmonic
oscillator, whose strength is given by
f (x) =
{ −kx,
0,
−d É x É d ,
−d > x and x > d
(3.1)
with (±d ) being the interface points between the harmonic potential and the constant potential. Sub-
stituting this expression of force into equation (2.3), it can be seen that the effective potential can be
identified by
V (x) =
{
v0,
k2x2
4Q − k
2 ,
−d > x and x > d ,
−d É x É d ,
(3.2)
where v0 = k2d 2/4Q −k/2 is a constant value chosen such that the potential function is continuous.
43003-4
Approximate solution for Fokker-Planck equation
Figure 1. Comparative graph of the truncated harmonic potential (solid line) and the usual harmonic
potential (dashed line) forQ = 1, k = 1.4, d = 1.5.
Figure 1 shows the harmonic potential graph (dotted line) and the truncated harmonic potential,
equation (3.2) (continuous line). For the potential studied, the solid line in figure 1 shows that there are
two distinct regions: one is a potential well region described by a harmonic potential and the other is
described by a constant potential.
The harmonic oscillator problem is a case in which equation (2.2) has a full analytical solution [2] and
the probability distribution is given by:
P (x, t ) =
∞∑
n=0
(
1
2nn!
√
α
π
)
e−αx2
Hn
(p
αx
)
Hn(0)e−t |λn |, (3.3)
where Hn are Hermite polynomials, α = k/2Q and k is a constant. The eigenvalues λn are equal to
nk (n = 0,1,2, . . . ,∞). Therefore, replacing the function f (x), equation (3.1), in the FPE (2.2), using the
approach presented in the previous section, the proposed solution for this example is given by:
P (x, t )=
NI 1p
4Qπt
e−x2/
4Qt , −d > x and x > d ,
NIg (t )
j∑
n=0
(
1
2n n!
√
α
π
)
e−αx2
Hn
(p
αx
)
Hn(0)e−t |λn |, −d É x É d
(3.4)
with j being the maximum number of discrete states in the potential well region. The function g (t ) in
this case is found by substituting the discrete functions of equation (3.4) in equation (2.8) and thus
g (t ) = e−d 2/
4Qt√
4πQt
{
j∑
n=0
(
1
2nn!
√
α
π
)
e−αx2
Hn
(p
αx
)
Hn(0)e−t |λn |
}−1
(3.5)
and NI(t ) is obtained by normalization, equation (2.10),
NI(t ) =
2√
4πQt
∞∫
d
e−x2/
4Qt dx + g (t )
d∫
−d
j∑
i=0
(
1
2i i !
√
α
π
)
e−αx2
Hi
(p
αx
)
Hi (0)e−t |λi |dx
−1
. (3.6)
It is assumed that, within the well, the truncation of the potential only slightly alters the original so-
lutions of the harmonic oscillator. Thus, for the region between −d É x É d , the eigenfunctions [equation
(3.3)] and eigenvalues (λn = nk) are the same as for the harmonic potential. The only difference is that
the number of terms of the series was limited taking into account the height of the potential well.
43003-5
M.T. Araujo, E. Drigo Filho
(a) (b)
Figure 2. (a) The probability distribution (3.4) versus x for different values of time. The parameters used
are: v0 = 0.5, d = 1.55, k = 1.4,Q = 1. (b) Estimated error ε(x, t ) for each solution.
Figure 2 shows the approximate probability distribution (3.4) for the truncated harmonic potential
[figure 2 (a)] and the error associatedwith this solution given by the function ε(x, t ) [figure 2 (b)], equation
(2.12), for different values of time. In the construction of figure 2, the values for the constants k = 1,Q = 1,
d = 1.55 v0 = 0.5were used. In this example there is only one discrete level in the well with an eigenvalue
of zero, λ0 = 0.
Observing the figure 2 (a), one can see that the probability distribution is greater in the region of
minimum potential, even for extended periods of time. Since the given solution was constructed using
an approximate method, one can see from figure 2 (b) that the probability distribution (3.4) does not
completely satisfy the Fokker-Planck equation, in other words, ε(x, t ) , 0. However, it is noted that the
calculated errors are small and decrease as time increases. The larger relative errors appear when the
probability distribution is calculated within the region of the well potential.
In the vicinity of the interface points (±d ), the error of the solution shows a discontinuity. This discon-
tinuity is expected since the potential behavior studied is composed by joining different functions, and
their profile (figure 1) is not smooth for the whole curve.
Through the probability distribution P (x, t ), equation (3.4), the variation of the number of particles
in the region of minimum potential can be calculated, equation (2.11). The curves in figure 3 show the
variation in the number of particles in the region of the potential well for different values of the diffusion
coefficient.
In the definition of the potential used, equation (3.2), the depth of the well (related to v0) dependson the diffusion coefficient (Q) and the interface point (d ).Thus, to maintain the fixed value of v0 (equalto 0.5) for different values of Q , it is necessary to change the value of d . Maintaining a fixed value v0ensures that, within the potential well, the number of eigenvalues j that are solutions of the Schrödinger
type equation (2.2) is fixed. In the example discussed here, v0 = 0.5, there is only one eigenvalue of this
kind.
Since it is assumed that the diffusion coefficient is proportional to temperature [2], lower values ofQ
represent lower system temperatures. Figure 3 shows that the decrease in the population of the region of
the potential well depends on the value ofQ , i.e., it depends on the temperature.
Initially the number of particles in the region of the well has the maximum value and with the in-
crease in time this number of particles decreases. This drop in the number of particles is more pro-
nounced for larger values of Q. This behavior is expected and consistent with the behavior of a system
subject to a non-confining potential.
Figure 4 represents the behavior of the function Y (t ,Q), equation (2.11), for a very large time value
43003-6
Approximate solution for Fokker-Planck equation
Figure 3. Curves describing the population near the
minimum of potential, equation (2.11) versus time
t for k = 1.4 and the well depth v0 = 0.5.
Figure 4. Curve of Y (t ,Q), equation (2.11) versusQ
for the truncated harmonic oscillator with only one
state within the well [equation (2.9) with j = 0] and
for a large value of time (t = 104).
(t = 104). In this figure, it can be seen that for small values ofQ (typicallyQ lower than 0.1), the population
is confined to the minimum potential region. On the other hand, for larger values of Q , the number of
particles within the well of potential decreases to zero. This behavior shows a phase transition in which
the particles remain in the well of potential at low temperatures, and at high temperatures the potential
well becomes emptied.
Another example can be got by increasing the depth of the potential well for v0 = k2d 2/4Q −k/2 = 1.5
with k = 1.4 and d = 2.1. The solution of (3.4) under these conditions gives two terms of the series (λ0 = 0
and λ1 = k) and the probability distribution is represented in figure 5, along with the associated error
ε(x, t ) [equation (2.12)].
(a) (b)
Figure 5. (a) P (x, t ), equation (3.4) versus x for different values of time for k = 1.4 and d = 2.1,Q = 1 and
v0 = 1.5. (b) Estimated error ε(x, t ) for each time value.
43003-7
M.T. Araujo, E. Drigo Filho
Figure 6. Curve of Y (t ,Q), equation (2.11) versusQ for the truncated harmonic oscillator with two states
within the well [equation (2.9) with j = 0] and long time (t = 104).
Comparison of figures 2 (a) and 5 (a) shows that the curve of the probability distribution is smooth
and the peak is more pronounced when the depth of the well is increased [figure 5 (a). As previously
discussed, the calculation of the error ε(x, t ) shows a discontinuity at the points (x =±d ). It can be seen
that, as time increases, there is a decrease in the calculated error indicating that the proposed solution is
best for longer times.
In figure 6, the evolution of the number of particles as a function of the diffusion coefficient Q for a
long time, t = 104 is shown. Again, one can see a phase transition in the system. However, this transition
is less sudden and its effects are noticeable at values ofQ higher than in the previous case (figure 4). This
effect is a result of the increased depth of the well, which makes the particle escape more difficult.
4. A non-harmonic polynomial potential
As a second example, the force associated with the system is assumed to be given by
f (x) =
{
0, −d > x and x > d ,
−ax3 −bx, −d É x É d ,
(4.1)
where a and b are two constants. On substituting this expression f (x), equation (4.1), in the Schrödinger
type equation, the effective potential, equation (2.3), can be written as,
Vef(x) =
{
v0, −d > x and x > d ,
a2x6
4Q + abx4
2Q +
(
b2
4Q − 3a
2
)
x2 − b
2 , −d É x É d ,
(4.2)
where the constants are chosen to ensure the continuity of the potential. For the region corresponding to
x > d and x <−d , the potential Vef(x) has a constant value v0 equal to
v0 = a2d 6
4Q
+ abd 4
2Q
+
(
b2
4Q
− 3a
2
)
d 2 − b
2
. (4.3)
Figure 7 shows the curve of the partially confining potential given by equation (4.2). The value of v0was fixed equal to 1 and the values of constants a and b were adjusted to allow just one minimum inside
the potential well, the values used are a = 0.45 and b = 1.75. For these values of a and b, if Q = 1, the
intersection points are d =±1.37.
In general, for non-harmonic polynomial potentials, the Schrödinger equation (2.2) has no exact/ana-
lytical solution. However, it is possible to determine part of the solution (partially soluble potential [13,
43003-8
Approximate solution for Fokker-Planck equation
Figure 7. Curve of potential (4.2) versus x for a = 0.45, b = 1.75, the depth v0 = 1,Q = 1 and d = 1.37.
14]). In such cases, the approach introduced in this work can be used, approximating the solution for the
original potential to that of the truncated potential and building P (x, t ) from function (2.9), that is,
P (x, t ) =
NI 1p
4Qπt
e−x2/
tQ , −d > x and x > d ,
NIg (t )
j∑
i=0
aiφ0(x)φi (x)e−t |λi |, −d É x É d .
(4.4)
In equation (4.4) the functions φi (x) are chosen in order to satisfy the Schrödinger equation (2.2) with
the potential (4.2). The region of the well potential, described by equation (4.2), does not give a general
analytic/exact solution to all eigenfunctions φi (x). In this case, just the ground state is determined. How-
ever, depending on the well depth more eigenfunctions are necessary. Then, one possibility to get around
this problem is to use other approximate methods, for example, the variational method [15].
Considering the potential characteristics studied and the parameters used (a = 0.45, b = 1.75, Q = 1,
d =±1.37 and v0 = 1), there is only one state in the potential well region. Thus, only the first term of the
series, should be considered in the region −d É x É d :
P (x, t ) = NIg (t )φ0(x)2e−tλ0 . (4.5)
Then, in this example, when the potential is not truncated, just the ground state solution is analytically
determined. In this case the eigenvalue (λ0) is equal to 0 and the function φ2
0(x) is the same adopted
solution of stationary ground state of the Schrödinger type equation when the potential well is infinite.
Thus, the function to be used in equation (4.5) is:
φ2
0(x) ∝ exp
{
− a
4Q
x4 − b
2Q
x2
}
. (4.6)
Therefore, the probability distribution for the force given by equation (4.1) is represented as,
P (x, t ) =
NI(t ) 1p
4Qπt
e−x2/
tQ , −d > x and x > d ,
NI(t )g (t )e−
a
4Q x4− b
2Q x2
, −d É x É d ,
(4.7)
where the function g (t ) is given by
g (t ) = e−d 2/
Qt√
4πQt
ev0/Q (4.8)
43003-9
M.T. Araujo, E. Drigo Filho
(a) (b)
Figure 8. a) The probability distribution (4.7) versus x, for truncated non-harmonic potential (4.2). b)
Error ε(x, t ) of the approximate solution. The numeric constants used are a = 0.45, b = 1.75, Q = 1 and
v0 = 1.
and the normalization NI(t ) is obtained from equation (2.10),
NI(t ) =
2√
4πQt
∞∫
d
e−x2/
tQ dx + g (t )
d∫
−d
e−
a
4Q x4− b
2Q x2
dx
−1
. (4.9)
Figure 8 shows the probability distribution (4.7) and the associated error ε(x, t ), equations (2.12), for
different values of time. The potential (4.2) has the constants a and b equal to 0.45 and 1.75, respectively.
For numerical calculations, the depth of the well was fixed as v0 = 1, which implies a unique eigen-
value λ0 = 0. The interface points between the regions are d =±1.37 and the value used for the diffusion
coefficient to construct the curves shown in figure 8 isQ = 1.
It can be seen from figure 8 (a) that the probability distribution P (x, t ) has a peak in the region of
the potential well even for very long times. Initially, there is a very distinct peak probability (t = 1) and,
as time passes, this peak decreases and there is an increase in the width of the curve P (x, t ) at its base.
The increased width of the probability distribution indicates the escape of the particles from the central
region to the region of constant potential.
In the same way as for the harmonic case, the suggested solution is substituted in the Fokker-Planck
equation to evaluate the error of the proposed method by determining the function ε(x, t ) [figure 8 (b)].
One can see that the approximate solution is better for large values of time than for shorter times. It is
also noted that the largest error is in the region within the potential well and is smaller in the side regions
where the potential is constant.
5. Conclusion
This paper presents an approximate analytical solution to the Fokker-Planck equation for partially
confining potentials. The suggested solution corresponds to an adaptation of a previous proposal [8]
from the same authors. Here, there is suggested the removal of the Gaussian function from the region
of potential well, which permits a greater numerical accuracy of solution. This can be noted by using the
expression ε(x, t ), equation (2.12).
In all the cases studied, following the initial condition P (x,0) = δ(x) and with the values of the con-
stants as given in the examples, the probability distribution has a peak in the central region of the po-
43003-10
Approximate solution for Fokker-Planck equation
tential well. As the time increases, the curves P (x, t ) show a widening and a reduction in height. The
calculation of ε(x, t ), equation (2.12), as a way of assessing the accuracy of the approximate solutions in
each example, indicates that the solutions have smaller errors for longer values of time than for shorter
times.
The approach outlined above allows for the study of a large number of problems whose solution
proves difficult or impossible to obtain by othermethods. For example, the truncated potentials discussed
here could not be handled by the procedure given in reference [9], since it is not possible to get the
exact/analytical solutions of the associated Schrödinger type equation.
The suggested solution method has the advantage of being extended to classes of partially confining
systems that do not have an exact analytical solution. Thus, an analytical expression, albeit approximate,
of the probability distribution provides important information, allowing the study of a much larger num-
ber of systems. In addition to this, the use of the test function ε(x, t ) [equation (2.12)] permits a quantita-
tive measure of the accuracy of the result.
Analyzing the first example studied, i.e., the truncated harmonic potential with different depths, a
transition phase can be identified involving the escape of particles from the well region of the potential.
At low temperatures, the particles are trapped, while for higher temperatures the particles can escape.
This escape leads to the emptying of the well. These results are very reliable, since they are obtained for
long periods of time, a condition at which the proposed method turns out to be more accurate.
As afinal remark, one observes that the approach introduced here can be addressed to thewell known
problem of the diffusion controlled escaping from a potential well [16, 17]. In this kind of problem, the
calculation of the rate coefficients has a central importance [18] and the calculation developed in the
present work can be used to compute these quantities. Particularly, the escape rate problem is hard to
analyze when the system is trapped in a potential well which correspond to the only point of minimum
in the potential [19]. In this context, the proposed function Y (t ,Q), equation (2.11), can be useful.
Acknowledgements
The authors acknowledge the financial support by the Brazilian agency CNPq (Proj. ESN
No. 233776/2014-1 and Proj. PDE No. 232865/2014-0) and the financial support of the Spanish MINECO
(Project MTM2014-57129-C2-1-P) and Junta de Castilla y León (UIC 011).
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Наближений розв’язок рiвняння Фоккера-Планка
М.Т. Аройо1,2, Е. Дрiго Фiльо1,3
1 Iнститут бiологiї, UNESP— державний унiверситет штату Сан-Паулу, Сан-Жозе-ду-Рiу-Прету, Бразилiя
2 Iнститут фiзики, Унiверситет м. Аугсбург, Нiмеччина
3 Факультет теоретичної фiзики, атомної фiзики та оптики та Iнститут математики,
Вальядолiдський унiверситет, 47011 Вальядолiд, Iспанiя
У цiй статтi запропоновано наближений розв’язок спецiального класу рiвняння Фоккера-Планка. Розв’я-
зок базується на зв’язку з рiвнянням типу Шредингера з частково обмеженим i симетричним потенцi-
алом. Щоб оцiнити точнiсть розв’язку, запропоновано функцiю похибок, яка отримана з оригiнального
рiвняння Фоккера-Планка. Використовуючи запропонований метод, проаналiзовано два приклади, а са-
ме, утятий гармонiчний потенцiал i негармонiчний полiном. Окрiм цього, для утятого гармонiчного по-
тенцiалу обговорено поведiнку системи в залежностi вiд температури.
Ключовi слова: рiвняння Фоккера-Планка, рiвняння Шредингера, наближений розв’язок
43003-12
http://dx.doi.org/10.1103/PhysRevLett.108.038302
http://dx.doi.org/10.1103/PhysRevE.62.4688
Introduction
The Fokker-Planck equation
Truncated harmonic oscillator
A non-harmonic polynomial potential
Conclusion
|