Particular solution for anomalous diffusion equation with source term

This article has been downloaded from IOPscience. Please scroll down to see the full text article.

2011 J. Phys.: Conf. Ser. 285 012031

(http://iopscience.iop.org/1742-6596/285/1/012031)

Download details:

IP Address: 200.145.3.40

The article was downloaded on 18/07/2013 at 21:30

Please note that terms and conditions apply.

View the table of contents for this issue, or go to the journal homepage for more

Home Search Collections Journals About Contact us My IOPscience

http://iopscience.iop.org/page/terms
http://iopscience.iop.org/1742-6596/285/1
http://iopscience.iop.org/1742-6596
http://iopscience.iop.org/
http://iopscience.iop.org/search
http://iopscience.iop.org/collections
http://iopscience.iop.org/journals
http://iopscience.iop.org/page/aboutioppublishing
http://iopscience.iop.org/contact
http://iopscience.iop.org/myiopscience


Particular solution for anomalous diffusion equation with source 

term 

M T Araujo
1
, E D Filho

2 

1Physics Department, Unesp, São José do Rio Preto, SP, Brazil 
2Physics Department, Unesp, São José do Rio Preto, SP, Brazil 

 

E-mail: 1marcelot-araujo@hotmail.com and 2edrigof@gmail.com  

 

 
Abstract: In literature the phenomenon of diffusion has been widely studied, however for 

nonextensive systems which are governed by a nonlinear stochastic dynamic, there are a few 

soluble models. The purpose of this study is to present the solution of the nonlinear Fokker-

Planck equation for a model of potential with barrier considering a term of absorption. Systems 

of this nature can be observed in various chemical or biological processes and their solution 

enriches the studies of existing nonextensive systems. 

1. Introduction 

A large number of phenomena can be related to kinetic processes, this is the case, for example, of 

electrochemical reactions in metals or in membranes. In these cases, the atoms or particles that interact 

are in the interface between the environments, the medium and the interior of the material. In many 

cases, this difference between the environments leads to a diffusion process in which particles of the 

medium are absorbed into the structure. This absorption can occur at a constant rate or, in some cases 

it can present temporal dependence [1].  

The Fokker-Planck equation is widely used in literature to study stochastic processes of diffusion, 

where it is possible to consider the presence of forces acting on the interaction between the bodies [2]. 

Its representation is: 

 [ ]
2

2

( , ) ( , )
( ) ( , ) ,

P x t P x t
f x P x t Q

t x x

∂ ∂∂
= ⋅ +

∂ ∂ ∂
 

where P(x,t) is the distribution of probabilities or a density of particles depending on the problem 

concerned. Q is the coefficient of diffusion and f(x) is the external force that acts in the environment. 

The distribution of probability obtained from equation (1) maximizes the entropy of Gibbs-

Boltzmann ln( )i ii
S P P=∑  and offers a very good description of the behavior of systems 

characterized by fluctuations which are proportional to the time variation, where there are short-range 

interactions among particles [3]. 

When considering anomalous systems, where the fluctuations are disproportionate to the time and 

where there is the presence of long-range interactions among particles, the common Fokker-Planck 

equation (1) is not able to describe well the model [4,5]. In order to solve this problem, the Fokker-

Planck equation must be rewritten as: 
2

2

( , ) ( , )
( ) ( , )

P x t P x t
f x P x t Q

t x x

µ ν
µ∂ ∂∂

 = − ⋅ + ∂ ∂ ∂
 

As it can be observed, the difference between (1) and (2) is the presence of the indexes µ and n in 

P(x,t) [6]. These indexes are real constants that must be adjusted according to the system analyzed and 

are constantly related to the process of anomalous diffusion in biological systems [7 – 10]. When µ = 

1, the probability provided by (2) is intimately associated with nonextensive systems, because it 

maximizes the entropy proposed by Tsallis [11, 12], 

[ ]
2

2

( , ) ( , )
( ) ( , )

P x t P x t
f x P x t Q

t x x

ν∂ ∂∂
= − ⋅ +

∂ ∂ ∂
. 

(2) 

(3) 

(1) 

Dynamic Days South America 2010 IOP Publishing
Journal of Physics: Conference Series 285 (2011) 012031 doi:10.1088/1742-6596/285/1/012031

Published under licence by IOP Publishing Ltd 1



In the problem addressed, we will make a brief review of nonlinear diffusion in a simple system 

consisting of two regions separated by a barrier of potential. In section 2 we present the solution of the 

nonlinear Fokker-Planck equation (NLFP) with an additional term, which is interpreted as a source 

term or absorption term that favors the passage of particles through the potential barrier. In section 3 

we discuss about the results obtained and, finally, we present the conclusions. 

 

2. PURPOSE 
In this paper, the function P(x,t) is interpreted as a density of particles of a medium. An additional 

term, proportional to the function µ(t) is included in equation (3) and is related to the absorption rate of 

the model. This way, we obtain: 

( )
( ) ( )

( )
( )

2

2

, ,
, ( , )

P x t P x t
P x t f x Q t P x t

t x x

υ

µ
∂ ∂∂

= −   + − ∂ ∂ ∂
 

The nonlinearity of the Fokker-Planck equation is due to the exponent ν present in the second 

derivative. Equation (1) is employed in the study of nonextensive systems, where there are long-range 

interactions and the thermodynamic relations meet the proposal introduced by Tsallis [5, 13]. 

 

 
Figure 1: Representation of the potential considered with k1 = 1 and k2 = 0.25. 

 

In this study, the function f(x) is given by the derivative of the potential: 

( ) 2 3

1 2 ,V x k x k x= −  

whose characteristic graph is shown in Figure 1. The solution for this system is the form: 

( ) ( ) ( ){ }1/( 1)
2, ( ) 1 ( , ,1 )P x t D t t x M x t

υ
υ β

−
= − − −    

where D(t), β(t) and M(x,t) are functions dependent on the time to be defined later. For a long time, the 

ansatz above indicates the distribution of the stationary probability of the Fokker-Planck equation. As 

we are considering a nonextensive system, the solution needs to be given this way, because this 

function maximizes the Tsallis’ entropy and allows us to obtain a good description of the environment 

through relevant thermodynamic information as, for example, the free energy. 

Applying (6) in (4) we obtain a system of equations, 

( )

1

2

2 2

2 1

1

2 3 1

1 2 2

( )
)    2 ( ) 2 ( ) ( ) ( ) ( ) 0

)  6 ( ) ( ) ( ) 0

( )1
)  4 4 ( ) ( ) 0

( )

( )
) 2 3 6 ( ) ( ) 4

( )

dD t
i D t k QD t t t D t

dt

M
ii k D t x QD t t

x

d t
iii x k QD t t

t dt

d tM M M M M
iv k x k x k x Q t D t x

t t dt x x x

ν

ν

ν

ν

ν β µ

ν β

β
ν β

β

β
νβ

β

−

−

− + + =

∂
− − =

∂

 
− + = 

 

∂ ∂ ∂ ∂ − − + − − − +  ∂ ∂ ∂ ∂ 

2

0










    =    

 

(6) 

(5) 

(4) 

I II 

Dynamic Days South America 2010 IOP Publishing
Journal of Physics: Conference Series 285 (2011) 012031 doi:10.1088/1742-6596/285/1/012031

2



Disregarding the relations ii) and iv), we go back to the system presented in [14], when the force is 

proportionate to x, only less than the source term µ(t). By isolating the term 2 ( ) ( )Q t D t νν β from i) and 

substituting in iii), we find the following expression: 

( ) ( )1 2
2 ( ) 0.

( ) ( )

d t dD t
t

t dt D t dt

β
µ

β
− − =

 
By integrating (7) with respect to t, we observe that β(t) is defined as: 

2 2 ( )0

2
0

( ) ( ) ,T tt D t e
D

β
β =  

where the function T(t) is equal to the integration of µ(t) in a determined time interval.  

The equation ii) provides the function M(x,t), which satisfies the problem. To that effect, 

integrating twice over x is enough. To facilitate, we will replace the function β(t) by definition (8). 

Therefore, the expression is: 
3 2 ( )

2( , )
( )

T tk x e
M x t

D tγ

−−
= , 

where γ is a constant equal to 
0 0Q Dνβ , adopted to facilitate the notation. 

Thus, after some manipulations we obtain the following expression: 
3 2 ( ) 3 2 ( ) 3 2 ( )

3 32 2 1 2

2 22 1 1

( 1) 6( )2
6 12 0

( )( ) ( ) ( )

T t T t T t
k x e k x e k k x edD tdD

k x k x
dt D t dtD t D t D t

ν ν ν

ν
γ γ γ

− − −

+ + +

+
− + − − + = , 

or: 

2 2 ( )

1

( 1)
2 ( ) 2 ( ) 0

3

T tdD
k D t D t e

dt

νν
γ +−

− − + = . 

Until now, from a system of four equations (i, ii, iii and iv), we have defined β(t) and M(x,t) in 

terms of D(t) and µ(t), which were not found yet. In order to obtain these expressions, it is necessary to 

compare equations (11) and i): 

2 2 ( )
1

2 2 ( )
1

( )
2 ( ) 2 ( ) ( ) ( ) 0

( 1)
2 ( ) 2 ( ) 0

3

T t

T t

dD t
D t k D t e t D t

dt

dD
k D t D t e

dt

ν

ν

γ µ

ν
γ

+

+


− + + =


−− − + =


 

As we can observe, both expressions are similar except for the derivative in relation to D(t) and the 

product ( ) ( )t D tµ ⋅ . By subtracting one from the other, we have: 

( ) ( 1)
( ) ( ) 0

3

dD t dD
t D t

dt dt

ν
µ

−
+ + =  

and by rearranging the terms of (13), we obtain a relation for µ(t) in function of D(t): 

( 2)
( ) .

3 ( )

dD
t

D t dt

ν
µ

+
= −

 
Through the relation found above, we can rewrite T(t) as 

( ) ( 2)/3
( ) ln ( ) .T t D t

ν− +
=  

By using (15), the exponential term is replaced by, 

( ) 2( 2 ) /3
ln ( )2 ( ) 2( 2)/3( )

D tT te e D t
ν

ν
− +

− += = . 

and by substituting the relation above  in (11) we obtain an equation which is only dependent on the 

function D(t), which, being solved provides the other functions: β(t), M(x,t) and µ(t). Due to the 

simplicity, we will work with the second equation of system (12) and, for this reason, we have 

rewritten it as: 

(7) 

(8) 

(9) 

(10) 

(11) 

(12) 

(13) 

(14) 

(15) 

(16) 

Dynamic Days South America 2010 IOP Publishing
Journal of Physics: Conference Series 285 (2011) 012031 doi:10.1088/1742-6596/285/1/012031

3



( 1)/3
( 1)/3

12 2 0
dD

k D
dt

ν
ν γ

− −
− −− + = . 

The expression of D(t) that satisfies the above is 

1

3/( 1)

20 0

2 2
1 0 1 0

( ) 1 k tQ Q
D t e

k D k D

ν
νβ νβ

− −

⋅
   

= + −      
. 

Accordingly with the relations (8), (9) and (14), the functions β(t), M(x,t) e µ(t) are represented as: 

1

2

20 0 0

2 2 2
0 1 0 1 0

( ) 1 k tQ Q
t e

D k D k D

β νβ νβ
β ⋅

   
= + −      

, 

( )1

1

20 0
1 2 2

0 1 0

( 2)
( ) 2 1 1

( 1)

k tQ Q
t k e

D k D

νβ νβν
µ

ν

−

− ⋅   +  
= − − − −    −     

, 

and 

1

2

2 32 1 0 0 0

2 2

0 1 0 1 0

( , ) 1
k tk k D Q Q

M x t e x
Q k D k D

νβ νβ
νβ

⋅
  −  

= + −  
   

. 

Having defined all the expressions of ansatz (6) the probability is expressed as: 

1 1

3/ 1 2

2 20 0 0 0 0

2 2 2 2 2
1 0 1 0 0 1 0 1 0

( , ) 1 1 ( 1) 1k t k tQ Q Q Q
P x t e e

k D k D D k D k D

ν
νβ νβ β νβ νβ

ν

− −

⋅
        

= + − ⋅ − − + − ⋅ ⋅                    

1

1/ 1
2

22 32 1 0 0 0

2 2
0 1 0 1 0

1 .k tk k D Q Q
x e x

Q k D k D

ν

νβ νβ
νβ

−

⋅
      ⋅ − + − ⋅          

 

 

3. RESULTS AND DISCUSSION 
Figures 2 and 3present the graphs of the distribution P(x,t) obtained in different time intervals, with x 

ranging from -1.5 to 2, which corresponds to the region of stable potential minimum (region I). The 

values assigned to constants were Q = 0,5 and k1 = 1, k2 = 0.25, ν = 1.5 for Figure 2 and k1 = 0.2, k2 = 

0.05, ν = 0.5 for Figure 3. 

The time taken by the particles density to reach zero, considering these values of the constants, was 

of approximately 1 unit, when ν is 1.5 and 2 units for ν equal 0,5.  

 

 

 

 

 

 

 

 

 

(17) 

(18) 

(19) 

(20) 

(21) 

(22) 

Dynamic Days South America 2010 IOP Publishing
Journal of Physics: Conference Series 285 (2011) 012031 doi:10.1088/1742-6596/285/1/012031

4



 
Figure 2: Density of particles vs. space and time for k1 = 1, k2 = 0.25 and ν = 1.5.  

 

 

 
Figure 3: Density of particles vs. space and time for k1 = 0.2, k2 = 0.05 and ν = 0.5.  

 

The inclusion of a term referring to an absorption rate has been already discussed in some papers 

(see references [15] and [16]). However, in these cases, the function f(x) corresponds to a harmonic 

potential. In this model, a disturbance term was inserted in the harmonic potential, creating a barrier 

and two state levels. This change has generated a specific temporal dependence for this case in the 

absorption rate (20).  

Temporal dependences of the kind presented here (20) can be observed in many biochemical 

systems [17, 18] when analyzing the absorption rates, creating the possibility of approximating this 

theoretical model with data of reactions or interactions obtained in the laboratory. In these cases, the 

limitation consists of the necessity of adjusting the nonlinear parameter (ν) to the system being treated, 

as it can be seen when comparing Figures 2 and 3.  

P(x,t) 

P(x,t) 

Dynamic Days South America 2010 IOP Publishing
Journal of Physics: Conference Series 285 (2011) 012031 doi:10.1088/1742-6596/285/1/012031

5



The values of k1 and k2 in Figure 3 are smaller than in Figure 2, reducing the size of the barrier and 

facilitating the diffusion between regions I and II of Figure 1. The value of nonlinear parameter is 

altered due to the change in the size of the barrier, guaranteeing a physically acceptable behavior of 

the solution.  

 

4. CONCLUSION 

The graphs made from ansatz have suggested an expected behavior of the system, i.e., as time passes 

there is a tendency for particles to go from a region of higher potential to a region of lower potential.  

This diffusion process is favored not only by the nonlinear factor of the Fokker-Planck equation, 

but also by the addition of the source term µ(t), which helps the particles to transpose the barrier. The 

incorporation of this temporal function in the Fokker-Planck nonlinear equation led to the solution of a 

simple system of potential barrier.  

In literature there are a few soluble models for NLFP, which hinders the application of the 

nonextensive formalism in the study of chemical or biological processes. 

 

ACKNOWLEDGMENTS 

The authors thank Capes (Coordination for the Improvement of Higher Education Personnel) for the 

financial support. 

 

References 
 

[1] Vemulapalli G K 1993 Physical Chemistry ed Prentice-Hall (New Jersey: Inc. A Simon & Schuster Company 

Englewood Cliffs).  

[2] Risken H 1989 The Fokker-Planck Equation (Berlim: Springer). 

[3] Tomé T 2006 Braz. Jour. Phys. 36 - 4A 1285 - 89. 

[4] Cohen E G D 2002 Physica A 305 19 – 26. 

[5] Tsallis C 1999 Braz. Jour. Phys 29 1-35. 

[6] Frank T D 2005 Nonlinear Fokker-Planck Equations – Fundamental and Applications  (Berlim: Springer). 

[7] Pedron I T and Mendes R S 2005 Rev. Bras. Ens. Fís. 27 251 – 258. 

[8] Adler R S 1978 Jour. Chem. Phys. 69 2849 – 63. 

[9] Ott A, Bouchaud J P, Langevin D and Urbac W 1990 Phys. Rev. Lett. 65 2201 – 04. 

[10] Borland L 1998 Phys. Rev. E 57 6634 – 42. 

[11] Tatsuaki W, Takeshi S 2001 Phys. A 301 284 – 90. 

[12] Tsallis C 2009 Introduction to Nonextensive Statistical Mechanics Rio de Janeiro (Rio de Janeiro: Springer). 

[13] Shiino M 2001 J. Math. Phys. 42 2540-53. 

[14] Plastino A R and Plastino A 1995 Physica A 222 347 – 54. 

[15] Drazer G, Wio H S and Tsallis C 2000 Phys. Rev. E 61 1417-22. 

[16] Pedron I T, Mendes R S, Malacarne L C and Lenzi E K 2002 Phys. Rev. E 65 0411081-6. 

[17] Crank J 1956 The Mathematics of Diffusion ed Oxford University Press (London). 

[18] Hansen C M 2010 Eur. Polym. J. 46 651-662. 

 

Dynamic Days South America 2010 IOP Publishing
Journal of Physics: Conference Series 285 (2011) 012031 doi:10.1088/1742-6596/285/1/012031

6