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Similarity solutions of Fokker-Planck equation with time-dependent coefficients

In this work, we consider the solvability of the Fokker-Planck equation with both time-dependent drift and diffusion coefficients by means of the similarity method. By the introduction of the similarity variable, the Fokker-Planck equation is reduced to an ordinary differential equation. Adopting the natural requirement that the probability current density vanishes at the boundary, the resulted ordinary differential equation turns out to be integrable, and the probability density function can be given in closed form. New examples of exactly solvable Fokker-Planck equations are presented, and their properties analyzed.


I. INTRODUCTION
The Fokker-Planck equation (FPE) is one of the basic tools which is widely used for studying the effect of fluctuations in macroscopic systems [1]. It has been employed in many areas: physics, chemistry, hydrology, biology, even finance, and others. Because of its broad applicability, it is therefore of great interest to obtain solutions of the FPE for various physical situations. Many methods, including analytical, approximate and numerical ones, have been developed to solve the FPE.
Generally, it is not easy to find analytic solutions of the FPE. Exact analytical solutions of the FPE are known for only a few cases, such as linear drift and constant diffusion coefficients [1]. In most cases, one can only solve the equation approximately, or numerically. The well-known analytic methods for solving the FPE include a change of variables, perturbation expansion, eigenfunction expansion, variational approach, Green's function, path integral, moment method, and the continued-fraction method (for a review of these methods, see eg. [1]). Symmetry methods have also been quite useful in solving the FPE [2]. Two interesting analytic approximation approaches are the WKB analysis [1] and the normal mode analysis [3]. The finite-difference [4] and finite-element methods [5] are among some useful numerical methods. Most of these methods, however, are concerned only with FPEs with time-independent diffusion and drift coefficients.
Solving the FPEs with time-dependent drift and/or diffusion coefficient is in general an even more difficult task. It is therefore not surprising that the number of papers on such kind of FPE is far less than that on the FPE with time-independent coefficients. Some recent works on the FPE with time-dependent diffusion coefficients appear in [6][7][8], and works involving time-dependent drift coefficients can be found in [9][10][11]. Refs. [12][13][14] consider FPEs with both time-dependent diffusion and drift coefficients. The symmetry properties of the one-dimensional FPE with arbitrary coefficients of drift and diffusion are investigated in [14]. Such properties may in some cases allow one to transform the FPE into one with constant coefficients. It is proved that symmetry group of these equations can be of one, two, four or six parameters and the criteria are also obtained.
In our previous work [11], we have considered, within the framework of a perturbative approach, the similarity solutions of a class of FPEs which have constant diffusion coefficients and small time-dependent drift coefficients. Motivated by our previous work, in this paper we would like to study the solvability of the FPE with both timedependent drift and diffusion coefficients by means of the similarity method.
One advantage of the similarity method is that it allows one to reduce the order of a partial differential equation [15]. Thus the FPE can be transformed into an ordinary differential equation which may be easier to solve. However, for the FPE to admit similarity solutions, it must possess proper scaling property under certain scaling transformation of the basic variables. This all boils down to the scaling behaviors of the drift and diffusion coefficients. To study these scaling behaviors is the main aim of the present work.
The plan of the paper is as follows. Sect. II discusses the scaling form of the FPE. Then in Sect. III the similarity method is applied to reduce the FPE into an ordinary differential equation, and the exact solution of the probability density expressed in closed form. Some new examples of exactly solvable FPEs with similarity solutions are presented in Sect. IV and V. Sect. VI concludes the paper.

II. SCALING OF FOKKER-PLANCK EQUATION
The general form of the FPE in (1 + 1)-dimension is where W (x, t) is the probability distribution function, D (1) (x, t) is the drift coefficient and D (2) (x, t) the diffusion coefficient. The drift coefficient represents the external force acting on the particle, while the diffusion coefficient accounts for the effect of fluctuation. W (x, t) as a probability distribution function should be normalized, i.e., domain W (x, t) dx = 1 for t ≥ 0. The domains we shall consider in this paper are the real line x ∈ (−∞, ∞), and the half lines x ∈ [0, ∞) and (−∞, 0].
As mentioned in the Introduction, it is difficult, if not impossible, to find exact solutions of the general FPE with time-dependent drift and diffusion coefficients. Here we shall be content with a more modest aim by seeking a special class of solutions, namely, the similarity solutions of the FPE. Such solutions are possible, provided that the FPE possesses certain scaling symmetry. Below we shall show that, if both the drift and diffusion coefficients assume proper scaling forms, then the FPE can be solved with the similarity method.
Consider the scale transformationx where ε, a and b are real parameters. Suppose under this transformation, the probability density function and the two coefficients scale as Here c, d and e are also some real parameters. Written in the transformed variables, eq.(1) becomes One sees that if the scaling indices satisfy b = a − d = 2a − e, then eq.(4) has the same functional form as eq.(1). In this case, the FPE admits similarity solutions. We shall present such solutions below.

III. SIMILARITY METHOD
The similarity method is a very useful method for solving a partial differential equation which possesses proper scaling behavior. One advantage of the similarity method is to reduce the order of a partial differential equation through some new independent variables (called similarity variables), which are certain combinations of the old independent variables such that they are scaling invariant, i.e., no appearance of parameter ε, as a scaling transformation is performed.
In our case, the second order FPE can be transformed into an ordinary differential equation which may be easier to solve. Here there is only one similarity variable z, which can be defined as For a , b = 0, one has α = 0 , ∞. In what follows, we derive the scaling forms and closed form solutions of the probability and current density functions.

A. Probability distribution function
The general scaling form of the probability density function W (x, t) is W (x, t) = (t δ1 /x λ1 )y(z), where λ 1 and δ 1 are two real parameters and y(z) is a scale-invariant function under the same transformation (2). From the assumed scaling behavior of W (x, t) in (3), we have −c/a = λ 1 − (δ 1 /α). Without loss of generality and for clarity of presentation, we hereby assume the parameters (λ 1 , δ 1 ) = (0, αc/a). This gives The normalization of the distribution function is For the above relation to hold at all t ≥ 0, the power of t should vanish, and so one must have c = −a, and thus Next we determine the scaling forms of the drift and diffusion coefficients D (1) (x, t) and D (2) (x, t). Following the same way of determining the transformation between W (x, t) and y(z), we can have D (1) For simplicity, we set λ 2 = 0. This gives δ 2 = α − 1, where the relations between the scaling exponents, namely, b = a − d and α = a/b, have been used. A similar procedure is applied to determine D (2) (x, t) in terms of ρ 2 (z). To summarize, the scaling forms of the coefficients are With eqs. (5), (6) and (9), the FPE is reduced to where the prime denotes the derivative with respect to z. Thus it is seen that the solvability of the FPE, eq. (1), under the similarity method depends solely on that of eq. (10). An easy way to find exact solutions of eq. (10) is to relate it to either the hypergeometric equation or the confluent hypergeometric equation. This requires that ρ 1 (z) be a linear function of z, and ρ 2 (z) a (linear) quadratic function of z for the (confluent) hypergeometric case. However, it turns out that ρ 1 (z) and ρ 2 (z) need not be so restricted in order to make eq. (10) exactly solvable. This is because eq. (10) is exactly integrable. Integrating it once, we get where C is an integration constant.
To fix the constant C, we consider the boundary conditions of the probability density W (x, t) and the associated probability current density J(x, t).

B. Relation between probability density and probability current density
From the continuity equation we have Using eqs. (5), (8) and (9), we get From eq. (11), we can reduce the above equation to where W (x, t) = t −α y(z) and z = x/t α have been used in obtaining the second expression.
For the domains which we are interested in, i.e., the whole line and the half lines, the natural boundary conditions are These conditions imply that C = 0, and that J(x, t) is proportional to W (x, t) and x, Eq. (17) can also be obtained by scaling consideration as follows. Under the scale transformation, the scaling behavior of J(x, t) takes the formJ(x,t) = ε h J(x, t), where h = −b is determined from expression (13) with the help of (2), (3), the relation b = a − d = 2a − e and c = −a. The general scaling form of the probability current is J(x, t) = (t δ3 /x λ3 ) Σ(z). Without loss of generality, we choose the set of parameters (λ 3 , δ 3 ) = (0, −1). This gives Inserting the similarity solution W (x, t) = t −α y(z) and (18) into eq. (12), one finds αy(z) + αzy ′ (z) = Σ ′ (z). This can be integrated to give Σ(z) = [αy(z) + αzy ′ (z)] dz + constant = αzy(z) + constant. Adopting the conditions (16) then gives eq. (17).
C. Analytic expression of similarity solution W (x, t) With C = 0, eq. (11) is reduced to This equation is easily solved to give Thus the probability density function W (x, t) is given by where A is the normalization constant. It is interesting to see that the similarity solution of the FPE can be given in such an analytic closed form. Exact similarity solutions of the FPE can be obtained as long as ρ 1 (z) and ρ 2 (z) are such that the function f (z) in eq. (21) is an integrable function and the resulted W (x, t) is normalizable. Equivalently, for any integrable function f (z) such that W (x, t) is normalizable, if one can find a function ρ 2 (z) (ρ 1 (z) is then determined by f (z) and ρ 2 (z)), then one obtains an exactly solvable FPE with similarity solution given by (21). Our discussions so far indicate a general way to construct exactly solvable FPE with similarity solutions. We shall not attempt to exhaust all possibilities here. Instead, we will only present a few interesting cases. We will mainly be concerned with cases related to f (z) having the forms (i) f (z) = Az + B and (ii) f (z) = A + B/z, where A and B are real constants. The corresponding W (x, t) are respectively. Possible choices of ρ 1 (z) and ρ 2 (z) that give f (z) in form (i) and (ii) are, respectively, (i) ρ 1 (z) = µ 1 z+µ 2 , ρ 2 (z) = µ 4 , and (ii) ρ 1 (z) = µ 1 z + µ 2 , ρ 2 (z) = µ 3 z. We will discuss these two cases in Sect. IV and V.
The solution (26) is a Gaussian (or normal) distribution in x. The full width at half maximum (FWHM) of the solution is related to the parameters α, µ 1 , µ 4 and time t as C 1 (µ 4 t 2α )/(α − µ 1 ), where C 1 = 2 √ 2 ln 2. Hence, from eq. (24), one can see that the FWHM is affected by the coefficient µ 1 of the term x/t of the drift coefficient and the coefficient µ 4 of the term t 2α−1 of the diffusion coefficient respectively. Also, it can be seen that the location of the peak of the solution, x = (µ 2 t α )/(α − µ 1 ), depends on the parameters α, µ 1 , µ 2 and time t. From eq. (24), it is seen that the location of the peak of the probability density can only be influenced by the drift coefficient. The parameter µ 2 in the drift coefficient plays an important role in the determination of the location of the peak. For instance, if µ 2 = 0, then the peak stays at the origin and will not change with time. The value of the peak is (α − µ 1 )/(2πµ 4 t 2α ), and is dependent on the parameters α, µ 1 , µ 4 and time t. Thus it is affected by both the drift and diffusion coefficients.
One notes that at fixed time t, W (x, t) with µ 2 and −µ 2 are the mirror images to each other with respect to the y-axis as the rest of the parameters are kept fixed. Furthermore, when the parameters (2α − µ 1 , −µ 4 ) take the place of (µ 1 , µ 4 ) with the rest unchanged, the corresponding two W (x, t)'s are the mirror images to each other. One then finds that W (x, t) is invariant as the parameters (µ 1 , µ 2 , µ 4 ) are replaced by (2α − µ 1 , −µ 2 , −µ 4 ).
As shown previously, the FWHM and the peak value are both related to (µ 4 t 2α )/(α − µ 1 ). The parameter α can be either positive or negative. When α > 0, the FWHM of the solution (26) is getting broader, the peak is turning smaller and is moving away from the origin with time. On the other hand, if α < 0, then the FWHM of the solution will shrink, the peak value will become higher and move toward the origin as time elapses. The situation where the FWHM is neither expanding nor contracting is impossible because α = 0.
respectively. From the figures above and the analysis in Sect. IV. A, one sees that the peak of the probability distribution W (x, t) moves to the right as time increases. This is due to the presence of the drift force. Furthermore, owing to the presence of µ 1 and µ 4 in the drift and diffusion coefficient, the FWHM of W (x, t) is getting wider, and the peak value is getting smaller.
FIG . 1 shows the evolution of the solution of the FPE with a set of time-dependent drift and diffusion coefficient. In subsections C and D, we shall study the difference in behavior of the solutions as one of the coefficients changes from being time-independent to being time-dependent, while the other coefficient is being kept fixed.
In the following, one example is given to illustrate the solution (27)  From (17) and (27), one can find that the maximum probability current J(x, t) occurs at x = −µ 4 t α /µ 2 , which moves with time.

C. Examples with the same drift but different diffusion coefficients
Let us take µ 2 = 0, and consider two different values of α, namely, α = 1/2 and α = 1. The case with α = 1/2 corresponds to D (1) (x, t) = µ 1 x/t and D (2) (x, t) = µ 4 , with the probability distribution The other case, α = 1, leads to These two examples have the same drift force, and differ only in the diffusion coefficients: constant in the first case, and linear in time in the second. One can thus study how the time-dependent diffusion modifies the behavior of the system with constant diffusion. As µ 2 = 0, the peak of the probability distribution will not move with time. Comparing eqs. (28) and (29), one sees that the FWHM is changed from C 1 (2µ 4 t)/(1 − 2µ 1 ) to C 1 (µ 4 t 2 )/(1 − µ 1 ), and the value of the peak is changed from (1 − 2µ 1 )/(4πµ 4 t) to (1 − µ 1 )/(2πµ 4 t 2 ). The two distributions coincide only at time t c = 2(1−µ 1 )/(1−2µ 1 ).
The evolution of the solutions (28)

D. Examples with the same diffusion but different drift coefficients
When the set of parameters α = 1/2, and µ 1 = µ 2 = 0 is taken, one has D (1) (x, t) = 0, D (2) (x, t) = µ 4 and probability distribution The other set of parameters α = 1/2 and µ 1 = 0 leads to D (1) (x, t) = µ 2 / √ t, D (2) (x, t) = µ 4 and the probability distribution In this case, (30) is the solution of the diffusion equation with a constant diffusion coefficient µ 4 . The solution (30) demonstrates only the effect of the diffusion. Eq. (31) gives the solution with additional drift µ 2 / √ t. The probability distribution (31) moves with time but keeps its FWHM at the value of 4 (ln 2) µ 4 t , the same as the solution (30). It is easy to see in (31) that when µ 2 takes the positive (negative) value, W (x, t) will move in the +x (−x) direction as time elapses, and the peak value 1/(4πµ 4 t) gets smaller with time.
The evolution of these solutions (30) and (31) are plotted in FIG. 6 and FIG. 7 respectively with two sets of the parameters, which are the same except µ 2 .
The next interesting exactly solvable example we shall discuss is an FPE with ρ 1 (z) = µ 1 z + µ 2 and ρ 2 (z) = µ 3 z. The corresponding drift and diffusion coefficients are Eq. (21) is integrable and gives The form of W (x, t) implies that the domain of x is defined only on half-line. For definiteness we shall take x ∈ [0, ∞). Normalizability of W (x, t) then requires It is seen that the distribution W (x, t) is invariant under the changes (µ 2 , µ 3 , x) and (−µ 2 , −µ 3 , −x) (other parameters being the same). This means that the distributions W (x, t) with the parameters (µ 2 , µ 3 ) and (−µ 2 , −µ 3 ) are mirror images of each other with respect to the y-axis. It is also worth noticing that for a given α, one will have the same solutions for different drift and diffusion coefficients as long as the quantity (µ 1 − α)/µ 3 remains unchanged. Furthermore, the system is invariant under the transformations α → −α, µ 1 → µ 1 − 2α, t → 1/t, and µ 2 , µ 3 and µ 4 unchanged. Hence the distribution with (α, µ 1 ) at time t is the same as the distribution with (−α, µ 1 − 2α) at time 1/t. Solution (33) with µ 2 = µ 3 presents an interesting stochastic process. In this case, W (x, t) becomes the exponential function, whose peak is always located at the origin. Its peak value is |(µ 1 − α)/(µ 3 t α )|, which is dependent on the parameters α, µ 1 µ 3 and time t, and hence is affected by both the drift and diffusion coefficients. The peak at x = 0 is increasing (decreasing) as t increases for α < 0 (α > 0). That means, by an appropriate choice of the drift and diffusion parameters, one can have a situation where the probability function is accumulating at the origin. For such situation, the effect of the drift force is stronger than that of the diffusion, causing the distribution to be pushed toward the origin as time elapses. An example of such situation is depicted in FIG. 8, which demonstrates the evolution of solution (33) with α = −2 and µ 2 = µ 3 .

VI. SUMMARY AND DISCUSSIONS
The FPE is an important equation in many different areas. Analytic solutions of the FPEs with both time-dependent drift and diffusion coefficients are generally difficult to obtain. In this paper, we have presented a general way to construct exact similarity solutions of the FPE. Such similarity solutions exist when the FPE possesses proper scaling behavior.
The similarity method makes use of the scaling-invariant property of the FPE. By the introduction of the similarity variable, the FPE can be reduced to an ordinary differential equation, which may be easier to solve. The general expression of the ordinary differential equation corresponding to the FPE with time-dependent drift and diffusion coefficients is given in this paper. It is interesting to find, by the natural requirement that the probability current density vanishes at the boundary, that the resulting ordinary differential equation is integrable, and the probability density function can be given in closed form. We have presented several new examples of exactly solvable Fokker-Planck equations with time-dependent coefficients. Symmetry properties of the solutions are also discussed.
Of course there are many other possibilities not presented here. These can be worked out easily by the reader following the method presented in this paper. Now we would like to briefly discuss whether there exist FPEs with similarity solutions which are related to the so-called quasi-exactly solvable (QES) equations. QES systems give rise to a new class of spectral problems for which it is possible to determine analytically a part of the spectrum but not the whole spectrum [16][17][18][19][20]. Thus QES systems are intermediate to the exactly solvable systems and the non-solvable ones. One-dimensional QES systems related to the sl(2) Lie algebra have been classified into ten classes in [17]. For example, the Class I QES model is given by the equation [17,20] µz 2 y ′′ (z) − 2az 2 − (2b + µ)z − 2c y ′ (z) + 2aN z + E µ y(z) = 0, N = 0, 1, 2, . . .
Note that the parameter N appears in the z-dependent term in the coefficient of y(z). For each N = 0, 1, 2, . . . , there are only N + 1 exactly known solutions, and hence it is only QES. Now one is interested to know whether there exists an FPE whose similarity solution is related to the sl(2)-based QES models. To answer this question, let us note that if a given second order differential equation of the form P (z)y ′′ + Q(z)y ′ + R(z)y = 0 can be cast in the form in eq. (10), then one must have Q ′ (z) = P ′′ (z) + R(z). It is easy to see that eq. (35) does not satisfy this requirement, and thus there is no FPE which can be related to the Class I QES model. In fact, it can be checked that this is also true for the other classes in [17].