How to solve Fokker-Planck equation treating mixed eigenvalue spectrum?

An analogy of the Fokker-Planck equation (FPE) with the Schrödinger equation allows us to use quantum mechanics technique to find the analytical solution of the FPE in a number of cases. However, previous studies have been limited to the Schrodinger potential with discrete eigenvalue spectrum. Here...

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Дата:	2013
Автори:	Brics, M., Kaupuzs, J., Mahnke, R.
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Опубліковано:	Інститут фізики конденсованих систем НАН України 2013
Назва видання:	Condensed Matter Physics
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Цитувати:	How to solve Fokker-Planck equation treating mixed eigenvalue spectrum? / M. Brics, J. Kaupuzs, R. Mahnke // Condensed Matter Physics. — 2013. — Т. 16, № 1. — С.13002:1–13. — Бібліогр.: 13 назв. — англ.

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spelling	irk-123456789-1210732017-06-14T03:05:11Z How to solve Fokker-Planck equation treating mixed eigenvalue spectrum? Brics, M. Kaupuzs, J. Mahnke, R. An analogy of the Fokker-Planck equation (FPE) with the Schrödinger equation allows us to use quantum mechanics technique to find the analytical solution of the FPE in a number of cases. However, previous studies have been limited to the Schrodinger potential with discrete eigenvalue spectrum. Here we will show how this approach can be applied also for mixed eigenvalue spectrum with bounded and free states. We solve the FPE with boundaries located at x=±L/2 and take the limit L→∞, considering examples with constant Schrödinger potential and with Pöschl-Teller potential. An oversimplified approach has been earlier proposed by M.T. Araujo and E. Drigo Filho. A detailed investigation of the two examples shows that the correct solution, obtained in this paper, is consistent with the expected Fokker-Planck dynamics. Аналогiя рiвняння Фоккера-Планка (FPE) з рiвнянням Шредингера дозволяє використати метод квантової механiки для знаходження аналiтичного розв’язку FPE для низки випадкiв. Проте, попереднi дослiдження обмежувалися потенцiалом Шредингера з дискретним спектром власних значень. Тут ми покажемо, як цей пiдхiд можна також застосувати до спектру змiшаних власних значень зi зв’язаними i вiльними станами. Ми розв’язуємо FPE з границями, що знаходяться при x = ±L/2 i беремо границю L → ∞, розглядаючи приклади з постiйним потенцiалом Шредингера i потанцiалом Пешля-Теллера. Спрощений пiдхiд ранiше запропонували M.T. Араухо та E. Дрiго Фiльйо. Детальне дослiдження двох прикладiв показує, що коректний розв’язок, отриманий в цiй статтi, узгоджується з очiкуваною динамiкою Фоккера-Планка. 2013 Article How to solve Fokker-Planck equation treating mixed eigenvalue spectrum? / M. Brics, J. Kaupuzs, R. Mahnke // Condensed Matter Physics. — 2013. — Т. 16, № 1. — С.13002:1–13. — Бібліогр.: 13 назв. — англ. 1607-324X PACS: 05.10.Gg DOI:10.5488/CMP.16.13002 arXiv:1303.5211 http://dspace.nbuv.gov.ua/handle/123456789/121073 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	An analogy of the Fokker-Planck equation (FPE) with the Schrödinger equation allows us to use quantum mechanics technique to find the analytical solution of the FPE in a number of cases. However, previous studies have been limited to the Schrodinger potential with discrete eigenvalue spectrum. Here we will show how this approach can be applied also for mixed eigenvalue spectrum with bounded and free states. We solve the FPE with boundaries located at x=±L/2 and take the limit L→∞, considering examples with constant Schrödinger potential and with Pöschl-Teller potential. An oversimplified approach has been earlier proposed by M.T. Araujo and E. Drigo Filho. A detailed investigation of the two examples shows that the correct solution, obtained in this paper, is consistent with the expected Fokker-Planck dynamics.
format	Article
author	Brics, M. Kaupuzs, J. Mahnke, R.
spellingShingle	Brics, M. Kaupuzs, J. Mahnke, R. How to solve Fokker-Planck equation treating mixed eigenvalue spectrum? Condensed Matter Physics
author_facet	Brics, M. Kaupuzs, J. Mahnke, R.
author_sort	Brics, M.
title	How to solve Fokker-Planck equation treating mixed eigenvalue spectrum?
title_short	How to solve Fokker-Planck equation treating mixed eigenvalue spectrum?
title_full	How to solve Fokker-Planck equation treating mixed eigenvalue spectrum?
title_fullStr	How to solve Fokker-Planck equation treating mixed eigenvalue spectrum?
title_full_unstemmed	How to solve Fokker-Planck equation treating mixed eigenvalue spectrum?
title_sort	how to solve fokker-planck equation treating mixed eigenvalue spectrum?
publisher	Інститут фізики конденсованих систем НАН України
publishDate	2013
url	http://dspace.nbuv.gov.ua/handle/123456789/121073
citation_txt	How to solve Fokker-Planck equation treating mixed eigenvalue spectrum? / M. Brics, J. Kaupuzs, R. Mahnke // Condensed Matter Physics. — 2013. — Т. 16, № 1. — С.13002:1–13. — Бібліогр.: 13 назв. — англ.
series	Condensed Matter Physics
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first_indexed	2025-07-08T19:08:55Z
last_indexed	2025-07-08T19:08:55Z
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fulltext	Condensed Matter Physics, 2013, Vol. 16, No 1, 13002: 1–13 DOI: 10.5488/CMP.16.13002 http://www.icmp.lviv.ua/journal How to solve Fokker-Planck equation treating mixed eigenvalue spectrum? M. Brics1, J. Kaupužs2, R. Mahnke1 1 Institute of Physics, Rostock University, D–18051 Rostock, Germany 2 Institute of Mathematics and Computer Science, University of Latvia, LV–1459 Riga, Latvia Received July 3, 2012, in final form August 24, 2012 An analogy of the Fokker-Planck equation (FPE) with the Schrödinger equation allows us to use quantum me- chanics technique to find the analytical solution of the FPE in a number of cases. However, previous studies have been limited to the Schrödinger potential with a discrete eigenvalue spectrum. Here, we will show how this ap- proach can be also applied to a mixed eigenvalue spectrum with bounded and free states. We solve the FPE with boundaries located at x =±L/2 and take the limit L →∞, considering the examples with constant Schrödinger potential and with Pöschl-Teller potential. An oversimplified approach was proposed earlier by M.T. Araujo and E. Drigo Filho. A detailed investigation of the two examples shows that the correct solution, obtained in this paper, is consistent with the expected Fokker-Planck dynamics. Key words: Fokker-Planck equation, Schrödinger equation, Pöschl-Teller potential PACS: 05.10.Gg 1. Introduction The one–dimensional Fokker-Planck equation (FPE) for the probability density p(x, t), depending on variable x and time t , assumes the generic form [1–7] ∂p(x, t) ∂t =− ∂ ∂x [ f (x, t)p(x, t) ] + ∂2 ∂x2 [ D(x, t) 2 p(x, t) ] . (1.1) Here, the drift coefficient or force f (x, t) and the diffusion coefficient D(x, t) depend on x and t in general. The Fokker-Planck equation is related to the Smoluchowski equation. Starting with pioneering works by Marian Smoluchowski [1, 2], these equations have been historically used to describe the Brownian-like motion of particles. The Smoluchowski equation describes the high-friction limit, whereas the Fokker- Planck equation refers to the general case. The FPE provides a very useful tool for modelling a wide variety of stochastic phenomena arising in physics, chemistry, biology, finance, traffic flow, etc. [3–6]. Given the importance of the Fokker-Planck equation, different analytical and numerical methods have been proposed for its solution. As it is well known, the stationary solution of FPE can be given in a closed form if the condition of a detailed balance holds. The study of the time-dependent solution is a much more complicated problem. The FPE (1.1) with a general time-dependence and a special x-dependence of the drift and diffusion coefficients has been studied analytically in [7] using Lie algebra. This method is applicable when the Fokker-Planck equation has a definite algebraic structure, which makes it possible to employ the Lie algebra and the Wei-Norman theorem. Generally, there are only a few exactly solvable cases. A simple example is a system with con- stant diffusion coefficient and harmonic interaction of the form f (x) =−dV (x)/dx with harmonic poten- tial V (x) ∼ x2. The case with double-well potential is already quite non-trivial and requires a numerical approach [8]. The known relation between the Fokker-Planck equation and the Schrödinger equation can also be used. This approach allows us to apply the well known methods of quantum mechanics. In particular, © M. Brics, J. Kaupužs, R. Mahnke, 2013 13002-1 http://dx.doi.org/10.5488/CMP.16.13002 http://www.icmp.lviv.ua/journal M. Brics, J. Kaupužs, R. Mahnke analytical solutions can be found in the cases, where the eigenvalues and eigenfunctions for the consid- ered Schrödinger potential are known. For a general Schrödinger potential, numerical treatments used in quantum mechanics, such as the Crank-Nicolson time propagation with implicit Numerov’s method for second order derivatives [9], are very useful. To apply it to Schrödinger-type equation, we just need to replace the real time step ∆t by an imaginary time step ∆t →−i∆t . In quantum mechanics, this is called imaginary time propagation and is used for calculation of both ground states and excited states. The an- alytical studies of mapping the FPE to Schrödinger equation have been so far restricted to a treatment of discrete eigenstates. An attempt has been made in [10] to extend this approach to the potentials with a mixed (discrete and continuous) eigenvalue spectrum. However, we have found a basic error in this treatment, indicated explicitly in the end of section 4.3. The aim of our work is to show how the problem with mixed eigenvalue spectrum can be treated correctly. We will show this in two examples: one with constant Schrödinger potential and another with Pöschl-Teller potential. The same example has been incorrectly treated in [10]. To avoid any confusion one has to note that the Pöschl-Teller potential is referred to as Rosen-Morse potential in [10]. 2. Solution of FPE with constant diffusion coefficient We start our consideration with the one-dimensional Fokker-Planck equation (1.1) in the following formulation ∂p(x, t) ∂t =− ∂ ∂x [ f (x)p(x, t) ] + D 2 ∂2p(x, t) ∂x2 (2.1) for the probability density distribution p(x, t), depending on the variable x and time t . Here, f (x) is the nonlinear force and D is the diffusion coefficient, which is now assumed to be constant. We consider natural boundary conditions lim x→±∞ p(x, t) = lim x→±∞ ∂p(x, t) ∂x = 0 (2.2) and take the most frequently used initial condition p(x, t = 0) = δ(x − x0) (2.3) in the form of the δ-function. This FPE (2.1) can be transformed into an equation of Schrödinger type (see section 2.2). Unfortunately, the well known relation [see equation (2.25)], derived for the discrete eigenvalue spectrum, cannot be applied if this equation has a continuous or mixed eigenvalue spectrum. To overcome this problem, we follow a properly corrected treatment of [10]. Namely, we solve the FPE with boundaries located at x =±L/2 and then take the limit L →∞ (see section 2.3). This approach is used in quantum mechanics to describe unbounded states. To keep a closer touch with quantum mechanics, here we will use the boundary conditions p(x =±L/2, t) = 0, further referred to as absorbing boundaries. 2.1. The stationary solution The stationary solution pst(x) is the long-time limit of p(x, t) at t →∞, which follows from the equa- tion 0 = d dx [ f (x)pst(x) ] − D 2 d2pst(x) dx2 . (2.4) The force f (x) can be expressed in terms of the potential V (x) via f (x) =−dV (x)/dx. It yields 0 =− d dx [ dV (x) dx pst(x)+ D 2 dpst(x) dx ] . (2.5) Due to the natural boundary conditions, we have zero flux jst(x) ≡− dV (x) dx pst(x)− D 2 dpst(x) dx =C with C = 0 . (2.6) 13002-2 How to solve Fokker-Planck equation? Thus, we have dpst(x) dx =− 2 D dV (x) dx pst(x) , (2.7) dpst(x) pst(x) =− 2 D dV (x) , (2.8) which yields the stationary solution pst(x) =N −1Y (x) , (2.9) where Y (x) ≡ exp [ − 2 D V (x) ] (2.10) has the meaning of an unnormalized stationary solution only in case of natural boundaries and N is the normalization constant N = +∞∫ −∞ dx exp [ − 2 D V (x) ] . (2.11) This function Y (x) is further used to construct a time-dependent solution. 2.2. The time-dependent solution with discrete eigenvalues Here, we derive a time-dependent solution, starting with the transformation p(x, t) → q(x, t) defined by p(x, t) = Y 1/2(x) q(x, t) ≡ exp [ − 2 D V (x) 2 ] q(x, t) . (2.12) This transformation removes the first derivative in the original Fokker-Planck equation and generates the equation of Schrödinger type for the function q(x, t), i. e., ∂q(x, t) ∂t =−VS(x)q(x, t)+ D 2 ∂2q(x, t) ∂x2 , (2.13) where VS(x) =− { 1 2 d2V (x) dx2 − 2 D [ 1 2 dV (x) dx ]2} (2.14) is the so-called Schrödinger potential. In the case of discrete eigenvalues, we apply the superposition ansatz q(x, t) = ∞∑ n=0 an(t)ψn(x) . (2.15) After inserting (2.15) into (2.13), we get the eigenvalue problem D 2 d2ψn (x) dx2 −VS(x)ψn (x) =−λnψn (x) (2.16) for eigenfunctions ψn(x) and eigenvalues λn Ê 0 with time-dependent coefficients an(t) given by an(t)= an (0)exp(−λn t) . (2.17) According to this, equation (2.15) can be written as q(x, t) = ∞∑ n=0 an(0)e−λn tψn (x) . (2.18) The eigenfunctions ψn(x) are orthonormal, i. e., +∞∫ −∞ ψn (x)ψm(x)dx = δnm (2.19) 13002-3 M. Brics, J. Kaupužs, R. Mahnke and satisfy the closure condition (completeness relation) ∞∑ n=0 ψn (x′)ψn(x) = δ(x − x′) . (2.20) Equation (2.16) can be written as a Schrödinger-type eigenvalue equation with Hermitian Hamilton op- erator H : H ψn(x) =λnψn(x) with H =− D 2 d2 dx2 +VS(x) . (2.21) The coefficients an(0) in (2.18) are calculated using the initial condition p(x, t = 0) = Y 1/2(x)q(x, t = 0) = δ(x − x0) . (2.22) According to (2.18), this relation can be written as Y −1/2(x)δ(x − x0) = ∞∑ m=0 am(0)ψm (x) . (2.23) In the following, we multiply both sides of this equation by ψn (x) and integrate over x from −∞ to +∞. Taking into account (2.19), it yields the so far unknown coefficients an(0) = Y −1/2(x0)ψn(x0) . (2.24) The final result of this calculation reads p(x, t) = √ Y (x) Y (x0) ∞∑ n=0 e−λn tψn(x0)ψn (x) . (2.25) Note that this method can also be used for other boundary conditions. The solution in the general form of (2.25) is well known from older studies, e. g., [11] and can be found in many textbooks, e. g., [3, 4]. 2.3. The time-dependent solution with mixed eigenvalue spectrum Consider now the problem with two absorbing boundaries located at x =±L/2 instead of the natural boundary conditions. In this case, we have a discrete eigenvalue spectrum, and equation (2.25) can be used (with summation over exclusively those eigenfunctions which satisfy the boundary conditions in a box of length L) to calculate the probability distribution pL(x, t), i. e., pL(x, t) = √ Y (x) Y (x0) ∞∑ n=0 e−λn,L tψn,L(x0)ψn,L(x) , (2.26) where λn,L are eigenvalues andψn,L(x) are the corresponding eigenfunctions, which fulfill the boundary conditions. Let us split this infinite sum into two parts: for λn,L <λcon and λn,L Êλcon, where λcon is the smallest continuum eigenvalue in the case of natural boundaries. This eigenvalue spectrum is shown schematically in figure 1, where the value of λcon is shown by a horizontal dotted line, the eigenvalues λn,L <λcon — by solid lines and the eigenvalues λn,L Êλcon — by dashed lines. Let M(L) be the maximal value of n for which λn,L < λcon and kn−M(L),L = [ 2(λn,L −λcon)/D ]1/2 for n > M(L) and ψcon kn−M(L),L (x) = ψn,L(x) for n > M(L). Hence, we have pL(x, t) = √ Y (x) Y (x0) M(L)∑ n=0 e−λn,L tψn,L (x0)ψn,L(x) + √ Y (x) Y (x0) e−λcon t ∞∑ m=1 e − 1 2 Dk2 m,L tψcon km,L (x0)ψcon km,L (x) . (2.27) The solution with natural boundaries is the limit case L →∞ p(x, t) = lim L→∞ pL(x, t) (2.28) 13002-4 How to solve Fokker-Planck equation? Figure 1. A schematic view of the eigenvalue spectrum for the problem with two absorbing boundaries at x =±L/2. The Schrödinger potential VS(x) together with the boundaries at x =±L/2 is indicated by a solid curve and vertical lines. or p(x, t) = √ Y (x) Y (x0) N−1∑ n=0 e−λn tψn(x0)ψn(x) + √ Y (x) Y (x0) e−λcon t lim L→∞ ∞∑ m=1 e − 1 2 Dk2 m,L tψcon km,L (x0)ψcon km,L (x) , (2.29) where N = limL→∞ M(L) is the number of bounded states in the case with natural boundaries. Since the eigenfunctions cannot be normalized at L →∞, it is appropriate to write equation (2.29) for unnormal- ized eigenfunctions ψ̄con km,L (x), p(x, t) = √ Y (x) Y (x0) N−1∑ n=0 e−λn tψn (x0)ψn(x) + √ Y (x) Y (x0) e−λcon t lim L→∞ ∞∑ m=1 e − 1 2 Dk2 m,L t N −1 ∆kL ︸︷︷︸ g−1(k ,L) ψ̄con km,L (x0)ψ̄con km,L (x)∆kL , (2.30) where the normalization constant N is given by N = L/2∫ −L/2 dx \|ψ̄con k (x)\|2 (2.31) and the expression under infinite sum is divided and multiplied by ∆kL = km+1,L −km,L . The infinite sum can be split into two parts: one with odd m and the other with even m. If the Schrödinger potential is symmetric, then one of these two parts contains only odd eigenfunctions ψ̄o k (x), whereas the other part has only even eigenfunctions ψ̄e k (x). In the limit L →∞, these two sums can be represented by corresponding integrals, yielding p(x, t) = √ Y (x) Y (x0) N−1∑ n=0 e−λn tψn (x0)ψn(x) + √ Y (x) Y (x0) e−λcon t ∞∫ 0 dk e− 1 2 Dk2 t g−1 e (k)ψ̄e k (x0)ψ̄e k (x) + √ Y (x) Y (x0) e−λcon t ∞∫ 0 dk e− 1 2 Dk2 t g−1 o (k)ψ̄o k (x0)ψ̄o k (x) , (2.32) 13002-5 M. Brics, J. Kaupužs, R. Mahnke where ge(k) = lim L→∞  2∆kL L/2∫ −L/2 dx \|ψ̄e k (x)\|2   , (2.33) go(k) = lim L→∞  2∆kL L/2∫ −L/2 dx \|ψ̄o k (x)\|2   . (2.34) This representation is useful if the eigenvalues and eigenfunctions are known. 3. The analytical solution of FPE with constant force Let us consider a constant force term. In this case, the Fokker-Planck equation (2.1) reads ∂p(x, t) ∂t =−vdrift ∂p(x, t) ∂x + D 2 ∂2p(x, t) ∂x2 . (3.1) This is a drift-diffusion problem for the potential V (x) =−vdriftx normalized to V (x = 0) = 0. No station- ary solution exists for this problem, because the normalization constant N in equation (2.11) diverges in this case. Nevertheless, the transformation (2.12) p(x, t) = Y (x)1/2q(x, t) with Y (x) = exp [ − 2 D V (x) 2 ] = exp [vdrift D x ] (3.2) can be used here to obtain an equation of Schrödinger type (2.13) with constant Schrödinger potential VS = 1 2D v2 drift . (3.3) The stationary Schrödinger-type equation corresponding to (2.21) reads d2ψn(x) dx2 − [ v2 drift D2 − 2 D λn ] ψn(x) = 0. (3.4) Let us now add two absorbing boundaries located at x =±L/2, where ψ(x =±L/2) = 0. Only in the case of real kn = [ 2λn /D − v2 drift /D2 ]1/2 > 0 equation (3.4) has non-trivial solutions ψn (x) = A cos(kn x)+B sin(kn x) , (3.5) which satisfy the boundary conditions. These solutions are ψn,L (x) =    √ 2 L cos ( kn,L x ) if n is even, √ 2 L sin ( kn,L x ) if n is odd, (3.6) where n = 0,1,2, . . . and kn,L = π L (n+1) . (3.7) According to (3.6)–(3.7), we have from (2.33) and (2.34) ge(k) = go(k) =π . (3.8) Taking into account that λcon = lim L→∞ min{λn,L} = lim L→∞ min { D 2 k2 n,L + v2 drift 2D } = v2 drift 2D (3.9) 13002-6 How to solve Fokker-Planck equation? holds, we obtain from equation (2.32) the expression p(x, t) = exp [ 1 D vdrift(x − x0) ] exp [ − v2 drift 2D t ] × 1 π ∞∫ 0 dke− 1 2 Dk2t [ cos(kx)cos(kx0)+ sin(kx)sin(kx0) ] . (3.10) Using the well known identities cos(kx)cos(kx0)+ sin(kx)sin(kx0) = cos[k(x − x0)] (3.11) and ∞∫ 0 dk e−αk2 cos(βk) = √ π 4α e−β 2/4α , (3.12) after simplification we obtain the well known result p(x, t) = 1 p 2Dt exp [ − (x − x0 − vdriftt)2 2Dt ] , (3.13) which describes a moving and broadening Gaussian profile. 4. Fokker-Planck dynamics with Pöschl-Teller potential Here, as a particular example we consider the force f (x) =−b tanh (αx) (4.1) with some positive constants b and α. This corresponds to the diffusion problem in the potential V (x) = b α ln (coshαx) , (4.2) normalized to V (x = 0) = 0. Figure 2 shows that this potential is actually a smoothed version of the V-shaped potential. The corre- sponding Schrödinger potential in this case is VS(x) = b2 2D − ( b2 2D + bα 2 ) 1 cosh2(αx) . (4.3) Figure 2. Graphical representation of equation (4.2) for b = 1 and several values of parameter α. 13002-7 M. Brics, J. Kaupužs, R. Mahnke Figure 3. Pöschl-Teller potential (4.4) for V0 = 1 and several values of the parameter α. If we compare it [see equation (4.4) and figure 3] with the well known Pöschl-Teller potential VPT(x) =VS(x)− b2 2D =− V0 cosh2 (αx) , (4.4) we see that equation (4.3) represents the shifted by b2/2D Pöschl-Teller potential with V0 = b2/2D+bα/2. Aswe can see from figure 3, the Pöschl-Teller potential gives amixed (discrete and continuous) eigenvalue spectrum. Therefore, equation (2.25) cannot be directly used to solve the FPE. We have to use (2.32). The eigenvalue equation (2.16) for the potential (4.3) reads D 2 d2ψn(x) dx2 − [ b2 2D − ( b2 2D + bα 2 ) 1 cosh2(αx) ] ψn (x) =−λnψn(x) . (4.5) By introducing dimensionless variables x̃ = αx, l̃ = b/Dα and λ̃n = 2λn /Dα2 − l̃ 2, we write (4.5) in a dimensionless form − d2ψn (x̃) dx̃2 − l̃ ( l̃ +1 ) 1 cosh2 x̃ ψn(x̃) = λ̃nψn (x̃) . (4.6) Analytical solutions for both bounded and unbounded eigenfunctions of equation (4.6) are known and can be found in [12, 13]. 4.1. Bounded solutions for Pöschl-Teller potential The equation (4.6) has N = max{m ∈N \|m < l̃+1} bounded states n = 0,1,2, . . . , N−1, whereN is a set of all natural numbers N= {0,1,2, . . .}. Here, we consider the eigenfunctions with λ̃n = 0 as unbounded, because they cannot be normalized. The eigenvalues can be calculated from the following equation [12] λ̃n =−(l̃ −n)2 , for n < N ; n ∈N . (4.7) Note that at least one bounded state with λ̃0 = −l̃ 2 always exists for l̃ > 0, which corresponds to λ0 = 0. The bounded eigenfunctions are known [12] ψn (x̃) = cosh−l̃ (x̃)× { Ne(n)F ( − 1 2 n, 1 2 n− l̃ ; 1 2 ;−sinh2 x̃ ) if n is even, No(n)sinh(x̃)F ( 1 2 − n 2 , n 2 + 1 2 − l̃ ; 3 2 ;−sinh2 x̃ ) if n is odd, (4.8) where F denotes a hypergeometric function, which can be represented by Gaussian hypergeometric se- ries F(α,β;γ;ζ) = Γ(γ) Γ(α)Γ(β) ∞∑ k=0 Γ(α+k)Γ(β+k) Γ(γ+k) ζn n! . (4.9) 13002-8 How to solve Fokker-Planck equation? The normalization constants are Ne(n) = [ 2 ( l̃ −n ) ( l̃ − 1 2 n ) (n+1) 1 B ( 1 2 , l̃ − 1 2 n ) B ( 1 2 ,1+ 1 2 n ) ]1/2 , (4.10) No(n) = [ 2 ( l̃ −n ) l̃ − 1 2 (n+1) 1 B ( 3 2 , l̃ − 1 2 (n+1) ) B ( 1 2 , 1 2 (n+1) ) ]1/2 , (4.11) where B(a,b) is the beta function B(a,b) = Γ(a)Γ(b)/Γ(a +b). 4.2. Unbounded solutions for Pöschl-Teller potential The unbounded solutions have a continuous eigenvalue spectrum with 0 É λ̃ <∞. Thus, we can in- troduce k̃ = λ̃1/2 (with k̃ = k/α). The Pöschl-Teller potential is symmetric. Therefore, the eigenfunctions are the even and odd functions known from [13] ψ̄k̃,l̃ (x̃) = A ·ψe k̃ ,l̃ (x̃)+B ·ψo k̃ ,l̃ (x̃) , (4.12) ψ̄e k̃,l̃ (x̃) = (cosh x̃)l̃+1 F ( r, s; 1 2 ;−sinh2 x̃ ) , (4.13) ψo k̃,l̃ (x̃) = (cosh x̃)l̃+1 sinh(x̃)F ( r + 1 2 , s + 1 2 ; 3 2 ;−sinh2 x̃ ) , (4.14) where A and B are constants, and r = 1 2 ( l̃ +1+ ik̃ ) , s = 1 2 ( l̃ +1− ik̃ ) . (4.15) Since these are unbounded solutions, eigenfunctions cannot be normalized within x ∈ (−∞;+∞). As we see, the eigenfunctions are rather complicated in general case. The expressions become es- sentially simpler for integer values of l̃ . Therefore, without loosing the general idea, we will show the solutions of the Fokker-Planck equation for l̃ = 1 and l̃ = 2. 4.3. The solution of FPE for Pöschl-Teller potential with parameter l̃ = 1 For l̃ = 1 (which implies b = αD) we have only one bounded state with the eigenvalue λ̃0 = −1 and the eigenfunction [equation (4.8) for n = 0] ψ0(x̃)= 1 p 2cosh(x̃) . (4.16) The unbounded eigenfunctions (4.13) and (4.14) are ψ̄e k̃ (x̃) = cos(k̃ x̃)− 1 k̃ tanh(x̃)sin(k̃ x̃) , (4.17) ψ̄o k̃ (x̃) = sin(k̃ x̃)+ 1 k̃ tanh(x̃)cos(k̃ x̃) . (4.18) As proposed in section 2, we add two absorbing boundaries located at x̃ =±L̃/2. Due to these bound- ary conditions, we have only discrete values of k̃ . Let us denote them by k̃L̃,m for even functions and by κ̃L̃,m for odd functions. The values of k̃L̃,m and κ̃L̃,m , obtained from the boundary conditions, are positive solutions of the transcendent equations k̃L̃,m = tanh(L̃/2)tan(k̃L̃,m L̃/2), (4.19) κ̃L̃,m tan(κ̃L̃,m) =− tanh(L̃/2), (4.20) 13002-9 M. Brics, J. Kaupužs, R. Mahnke where m = 1,2,3, . . . denotes the m-th smallest positive solution. The equations for normalized eigenfunc- tions now read as ψe k̃L̃,m (x̃)=N −1/2 e (k̃L̃,m , L̃) · [ cos(k̃L̃,m x̃)− 1 k̃L̃,m tanh(x̃)sin(k̃L̃,m x̃) ] , (4.21) ψ0 κ̃L̃,m (x̃)=N −1/2 o (κ̃L̃,m , L̃) · [ sin(κ̃L̃,m x̃)+ 1 κ̃L̃,m tanh(x̃)cos(κ̃L̃,m x̃) ] , (4.22) where normalization constants for odd and even eigenfunctions are Ne(k̃ , L̃) = ( k̃2 +1 )[ k̃L̃ − sin(k̃L̃) ] 2k̃3 , (4.23) No(k̃, L̃)= ( k̃2 +1 )[ k̃L̃ + sin(k̃L̃) ] 2k̃3 . (4.24) In the limit case L̃ →∞, equations (4.19)–(4.20) for the allowed k̃ values, as well as equations (4.23)– (4.24) for the normalization constants simplify to k̃L̃→∞,m = 2mπ L̃ , ∆k̃L̃→∞ = 2π L , (4.25) κ̃L̃→∞,m = (2m −1)π L̃ , ∆κ̃L̃→∞ = 2π L , (4.26) Ne(k̃, L̃ →∞) =No(k̃, L̃ →∞) = L 2 k̃2 +1 k̃2 , (4.27) and we also have ge(k̃) =∆k̃L̃→∞ ·Ne(k̃, L̃ →∞) =π k̃2 +1 k̃2 , (4.28) go(κ̃) =∆κ̃L̃→∞ ·No(κ̃, L̃ →∞) =π κ̃2 +1 κ̃2 . (4.29) Inserting these relations as well as λcon = l̃ 2α2D/2 (following from λ̃con = 2λcon/Dα2− l̃ 2 = 0) into (2.32), we finally obtain the time-dependent solution of the Fokker-Planck equation p(x, t) = 1 2cosh2(αx) + cosh(αx0) πcosh(αx) e− 1 2 Dα2t ∞∫ 0 dk̃ e− 1 2 Dα2k̃2t k̃2 k̃2 +1 ψ̄e k̃ (αx)ψ̄e k̃ (αx0) + cosh(αx0) πcosh(αx) e− 1 2 Dα2t ∞∫ 0 dk̃ e− 1 2 Dα2k̃2t k̃2 k̃2 +1 ψ̄o k̃ (αx)ψ̄o k̃ (αx0) . (4.30) If the initial condition is given by x0 = 0, then ψo k̃ (0) = 0 and ψe k̃ (0) = 1 hold, which allows us to obtain a simpler expression p(x, t) = 1 2cosh2(αx) + 1 πcosh(αx) e− 1 2 Dα2t × ∞∫ 0 dk̃ e− 1 2 Dα2k̃2t k̃2 k̃2 +1 [ cos(k̃αx)− 1 k̃ tanh(αx)sin(k̃αx) ] . (4.31) The solution for parameters b = 2, D = 2 and α = 1, corresponding to l̃ = 1, with the initial loca- tion of the delta-peak at x0 = 5 is shown in figure 4 for different time moments t . As we can see, the 13002-10 How to solve Fokker-Planck equation? Figure 4. The probability distribution at different time moments t , calculated for the parameters b = 2, D = 2 and α= 1 (l̃ = 1) starting at x0 = 5. probability distribution moves to the left. It broadens at the beginning. For larger times, it becomes nar- rower again and converges to the stationary solution pst(x) = limt→∞ p(x, t) = [cosh2(αx)]−1 = ψ0(x)2 [see equations (4.30) and (4.16)], which is a symmetric distribution around x = 0. The stationary solution is practically reached at t = 10. This behavior is expected from the drift-diffusion dynamics. For small times t → 0, we have a delta-peak located at x = x0 in accordance with the given initial condition (2.3). For comparison, the “general solution” of [10] does not satisfy this initial condition due to a wrong construction, where the contribution of bounded states is simply summed up with a Gaus- sian probability density profile (calculated with an error). The latter corresponds to unbounded states for zero Schrödinger potential at L →∞, as it is evident from (3.13) and (3.3) at vdrift = 0. Therefore, the result appears to be correct only at t → ∞ when the Gaussian part vanishes. It is clear that the whole set of eigenfunctions should be calculated self-consistently for the given potential to obtain a correct and meaningful result, since only in this case the completeness relation (2.20) holds and all different eigen- functions are orthogonal. Thus, the basic error of [10] is that some of the eigenfunctions are calculated for zero Schrödinger potential in [10], whereas all of them should be calculated for the true Schrödinger potential. 4.4. The solution of FPE for Pöschl-Teller potential with parameter l̃ = 2 For l̃ = 2 (which implies b = 2αD) we have two bounded states with eigenvalues λ̃0 =−4 and λ̃1 =−1. The corresponding eigenfunctions are ψ0(x̃) = p 3 2cosh2(x̃) , (4.32) ψ1(x̃) = √ 3 2 sinh(x̃) cosh2(x̃) . (4.33) The unbounded eigenfunctions are ψ̄e k̃ (x̃) = [ 1+ k̃2 −3tanh2(x̃) ] cos(k̃ x̃)−3k̃ tanh(x̃)sin(k̃ x̃) , (4.34) ψ̄o k̃ (x̃) = [ 1+ k̃2 −3tanh2(x̃) ] sin(k̃ x̃)+3k̃ tanh(x̃)cos(k̃ x̃) . (4.35) By adding again two absorbing boundaries at x̃ = ±L̃/2, we have discrete values of k̃ , i. e., k̃L̃,m for even functions and κ̃L̃,m for odd functions. In the limit L̃ → ∞, we again obtain the classical infinite- square-well relations for eigenstates: k̃L̃→∞,m = (2m −1)π L̃ , (4.36) κ̃L̃→∞,m = 2mπ L̃ . (4.37) 13002-11 M. Brics, J. Kaupužs, R. Mahnke Figure 5. The probability distribution at different time moments t , calculated for the parameters b = 2, D = 4 and α= 1 (l̃ = 2) starting at x0 = 5. The normalization constants in this case are Ne ( k̃ , L̃ →∞ ) =No ( k̃, L̃ →∞ ) = L 2 ( k̃2 +4 )( k̃2 +1 ) . (4.38) By applying the same steps as in the case of l̃ = 1, we obtain the solution p(x, t) = 3 4cosh4(αx) + 3 2 sinh(αx)sinh(αx0) cosh4(αx) e− 3 2 Dα2t + cosh2(αx0) πcosh2(αx) e−2Dα2t ∞∫ 0 dk̃ e− 1 2 Dα2k̃2t 1 k̃2 +5k̃2 +4 ψe k̃ (αx)ψe k̃ (αx0) + cosh2(αx0) πcosh2(αx) e−2Dα2t ∞∫ 0 dk̃ e− 1 2 Dα2k̃2t 1 k̃2 +5k̃2 +4 ψo k̃ (αx)ψo k̃ (αx0) . (4.39) The solution for parameters b = 4, D = 2 and α = 1, corresponding to l̃ = 2, with the initial condition given by x0 = 5 is shown in figure 5 for different time moments t . The evolution of the probability distri- bution is very similar to that one shown in figure 4 for l̃ = 1, with the only essential difference that the dynamics is faster and the distribution is somewhat narrower due to a deeper potential well. 5. Conclusions Using the analogy of the Fokker-Planck equation with the Schrödinger equation, it has been shown how the time-dependent solution can be constructed in the case of mixed eigenvalue spectrum with free and bounded states. The method is based on the idea of introducing two absorbing boundaries at x =±L/2, considering the limit L →∞ afterwards. Although this idea is similar to the one proposed ear- lier in [10], it is obvious that the problem is quite non-trivial, so that the oversimplified (i.e., erroneous) approach of [10] cannot be used— see discussion in the end of section 4.3. Analytical solutions have been found and analyzed in two examples of the Schrödinger potential being constant (constant force) and a shifted Pöschl-Teller potential. For the latter potential, the analytical solutions have been compared with the results of the Crank-Nicolson numerical integration method, and the agreement within an error of 10−7 has been found. The time evolution of the calculated probability distribution in these examples is consistent with the usual drift-diffusion dynamics. Acknowledgement The authors M. B. and J. K. thank for financial support from Academic Exchange Office at Rostock Uni- versity having made it possible to continue our long-standing collaboration between Rostock (Germany) and Riga (Latvia). 13002-12 How to solve Fokker-Planck equation? References 1. Smoluchowski M., Theory of the Brownian movements. Bulletin de l’Academie des Sciences de Cracovie, 1906, 577–602. 2. Smoluchowski M., Irregularity in the distribution of gaseous molecules and its influence. Boltzmann Festschrift, 1904, 626–641. 3. Risken H., The Fokker-Planck Equation, Springer, Berlin, 1984. 4. Gardiner C.W., Handbook of Stochastic Methods, Springer, Berlin, 2004. 5. Mahnke R., Kaupužs J., Lubashevsky I., Physics of Stochastic Processes. How Randomness Acts in Time, Wiley– VCH, Weinheim, 2009. 6. Schadschneider A., Chowdhury D., Nishinari K., Stochastic Transport in Complex Systems. From Molecules to Vehicles, Elsevier, Amsterdam, 2011. 7. Lo C.F., Eur. Phys. J. B, 2011, 84, 131–136; doi:10.1140/epjb/e2011-20723-7. 8. Liebe Ch., About Physics of Traffic Flow: Empirical Data and Dynamical Models, PhD thesis, Rostock University, 2010. 9. Bauer D., Koval P., Comput. Phys. Commun., 2006, 174, 396–421; doi:10.1016/j.cpc.2005.11.001. 10. Araujo M.T., Filho E.D., J. Stat. Phys., 2012, 146, 610–619; doi:10.1007/s10955-011-0411-8. 11. Barrett J.F., Lampard D.G., IEEE Trans. Inf. Theory, 1955, 1, 10–15; doi:10.1109/TIT.1955.1055122. 12. Nieto M.M., Phys. Rev. A, 1978, 17, 1273–1283; doi:10.1103/PhysRevA.17.1273. 13. Lekner J., Am. J. Phys., 2007, 75, 1151–1157; doi:10.1119/1.2787015. Як розв’язати рiвняння Фоккера-Планка, використовуючи спектр змiшаних власних значень? М. Брицс1, Я. Каупузс2, Р. Манке1 1 Iнститут фiзики, Унiверситет м. Росток, D–18051 Росток, Нiмеччина 2 Iнститут математики i комп’ютерних наук, Латвiйський унiверситет, LV–1459 Рига, Латвiя Аналогiя рiвняння Фоккера-Планка (FPE) з рiвнянням Шредингера дозволяє використати метод квантової механiки для знаходження аналiтичного розв’язку FPE для низки випадкiв. Проте, попереднi дослiдження обмежувалися потенцiалом Шредингера з дискретним спектром власних значень. Тут ми покажемо, як цей пiдхiд можна також застосувати до спектру змiшаних власних значень зi зв’язаними i вiльними ста- нами. Ми розв’язуємо FPE з границями, що знаходяться при x = ±L/2 i беремо границю L →∞, розгля- даючи приклади з постiйним потенцiалом Шредингера i потанцiалом Пешля-Теллера. Спрощений пiдхiд ранiше запропонували M.T. Араухо та E. Дрiго Фiльйо. Детальне дослiдження двох прикладiв показує, що коректний розв’язок, отриманий в цiй статтi, узгоджується з очiкуваною динамiкою Фоккера-Планка. Ключовi слова: рiвняння Фоккера-Планка, рiвняння Шредингера, потенцiал Пешля-Теллера 13002-13 http://dx.doi.org/10.1140/epjb/e2011-20723-7 http://dx.doi.org/10.1016/j.cpc.2005.11.001 http://dx.doi.org/10.1007/s10955-011-0411-8 http://dx.doi.org/10.1109/TIT.1955.1055122 http://dx.doi.org/10.1103/PhysRevA.17.1273 http://dx.doi.org/10.1119/1.2787015 Introduction Solution of FPE with constant diffusion coefficient The stationary solution The time-dependent solution with discrete eigenvalues The time-dependent solution with mixed eigenvalue spectrum The analytical solution of FPE with constant force Fokker-Planck dynamics with Pöschl-Teller potential Bounded solutions for Pöschl-Teller potential Unbounded solutions for Pöschl-Teller potential The solution of FPE for Pöschl-Teller potential with parameter =1 The solution of FPE for Pöschl-Teller potential with parameter =2 Conclusions

How to solve Fokker-Planck equation treating mixed eigenvalue spectrum?

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