A subordination approach to modelling of subdiffusion inspace-time-dependent force fields

Aleksander WeronMarcin Magdziarz

Hugo Steinhaus CenterWrocław University of Technology

Jerusalem 28.03.2008

Contents

Fractional Fokker-Planck equation (FFPE)• Definition and basic properties• Subordinated Langevin approach• Method of computer simulation

FFPE with jumps Fractional Klein-Kramers equation FFPE with time-dependent force fields Subdiffusion with space-time-dependent force

Fractional Fokker-Planck (Smoluchowski) equation

The equation

0<<1, describes anomalous diffusion (subdiffusion) in the presence of an external potential V(x), [1].

0 Dt

1-α – fractional derivative of Riemann-Liouville type – friction constant K – anomalous diffusion coefficient

[1] R. Metzler, E. Barkai, and J. Klafter, Phys. Rev. Lett. 82, 3563 (1999). R. Metzler and J. Klafter, Phys. Rep. 339, 1 (2000).

,

FFPE - limit case α1

For α1, FFPE reduces to the standard

Fokker-Planck (Smoluchowski) equation

whose solution is the PDF corresponding to the following Itô stochastic differential equation

Here, B(t) is the standard Brownian motion.

,

.

Subordinated Langevin approach

Claim 1. The solution w(x,t) of the FFPE is equal

to the PDF of the process

Y(t)=X(St),

where the parent process X() is given by the Itô

stochastic differential equation (Langevin equation)

and St is the so-called inverse -stable subordinator independent of X().

[2] M. Magdziarz, A. Weron and K. Weron, Phys. Rev. E, 75 016708 (2007)

The inverse -stable subordinator St is defined as

where U() is the strictly increasing -stable Lévy motion with the Laplace transform

The role of St is analogous to the role of the

fractional derivative 0 Dt1-α in the FFPE.

Subordinated Langevin approach

Computer simulation – I Step

Using the standard method of summing up the increments of the process U(), we get:

where =t, j are i.i.d. positive -stable random variables

V - uniformly distributed on (-/2, /2) and W - exponentiallydistribution with mean one.

(*)

The iteration (*) ends when U() crosses the time

horizon T. We approximate the values St0, ..., StN

,

using the relation with

Using the Euler scheme, we approximate the diffusion

Computer simulation – II Step

for k=1, ..., L. Here L is the first integer that exceedsand are i.i.d. standard normal random variables.

Finally, using the linear interpolation, we get

for

Fig. Sample realizations

of: (a) the subordinated

process X(St),

(b) the diffusion X(),

(c) the subordinator St .

Note the similarities

between the constant

intervals of X(St) and

St and the similarities

between X(St) and

X() in the remaining

domain. Here =0.6.

Fig. Evolution in time of (a) the subordinated process X(St), (b) the Brownian motion X(t). The cusp shape of the PDF in the first case is characteristic for the subordinator St . Here =0.6.

Fig. Estimated quantile lines and two sample paths of the process X(St) with constant potential V(x)=const. Every quantile line is of the formwhich confirms that the process is /2 self-similar. Here =0.6.

FFPE with jumps

The equation

0<<1, 0<≤2, describes competition between subdiffusionand Lévy flight in the presence of an external potential V(x).

0 Dt1-α – fractional derivative of Riemann-Liouville type

– friction constant K – anomalous diffusion coefficient – Riesz fractional derivative

[1] R. Metzler and J. Klafter, Phys. Rep. 339, 1 (2000).

,

FFPE with jumps – limit cases

For =2 we recover the FFPE discussed previously

For 1, solution of the FFPE with jumps is equal to the PDF of the diffusion

driven by the symmetric -stable Lévy motion .

For =2 and 1, we obtain the standard Fokker-Planck (Smoluchowski) equation.

FFPE with jumps –Subordinated Langevin approach

Claim 2. The solution w(x,t) of the FFPE with jumps is

equal to the PDF of the process

Y(t)=X(St),

where the parent process X() is given by the Itô

stochastic differential equation (Langevin equation)

and St is the -stable subordinator independent of X().

[3] M. Magdziarz and A. Weron, Phys. Rev. E, 75 056702 (2007).

Fig. Sample paths of:

(a) the subordinated process X(St),

(b) the diffusion X(),

(c) the subordinator St .

The interplay between

long rests and long

jumps is distinct.

Here =0.7 and =1.3.

(a)

(b)

(c)

Fig. Comparison of three sample realizations of th process X(St) for three different parameters .

The constant intervals are repeated, while the jumps of the process dependent on the parameter are different.

The smaller the longer jumps.

Here =0.7.

Fig. Comparison of three sample realizations of the process X(St) for three different parameters .

The height of the jumps is repeated, while the waiting times (constant intervals) depend on .

Here =1.3.

Fig. Comparison of the estimated PDFs of the process X(St) for two different parameters and fixed parameter .

The log-log scale window confirms that in both cases the tails decay as a power law. Here =1.4.

The FKK equation

Fractional Klein-Kramers equation

0<<1, describes position x and velocity v of a particle of mass m exhibiting subdiffusion in an external force F(x). kBT – Boltzmann temperature – friction constant

[4] R. Metzler and J. Klafter, Phys. Rev. E 61, 6308 (2000); E. Barkai and R.J. Silbey, J. Phys. Chem. B 104, 3866 (2000);

R. Metzler, I.M. Sokolov, Europhys. Lett. 58, 482 (2002).

Fractional Klein-Kramers equation –Subordinated Langevin approach

Claim 3. The solution W(x,v,t) of the FKKE is equal to thePDF of the process

Y(t)=(X(St),V((St)),

where the parent process (X(), V()) is given by the 2-dim.Itô stochastic differential equation (Langevin equation)

[5] M. Magdziarz and A. Weron, Phys. Rev. E, 76, 066708 (2007).

Fig. Exemplary sample paths (red lines) and estimated quantile lines (blue lines) corresponding to the processes X(St) and V(St) in the presence of double-well potential. Here =0.9, m=kBT==1.

Fig. Comparison of the estimated and theoretical stationary solution of the FKKE. Here =0.9, m=kBT==1.

FFPE with time-dependent force

0<<1, describes subdiffusion in the presence of an external time-dependent force F(t).

The fractional operatort Dt

1-α in the above equation appears to the right of F(t), therefore, it does not modify the time-dependent force.

The equation

[6] I.M. Sokolov and J. Klafter, Phys. Rev. Lett. 97, 140602 (2006).

FFPE with time-dependent force –Subordinated Langevin approach

Claim 4. The solution w(x,t) of the FFPE with the force F(t) is

equal to the PDF of the process

Y(t)=X(St),

where the parent process X() is given by the subordinated

stochastic differential equation (Langevin equation)

U() is the strictly increasing -stable Levy motion and St

is its inverse.

[7] M.Magdziarz, A.Weron, preprint (2008).

The process Y(t) admits an equivalent representation

thus, it consist essentially of two contributions:

the stochastic integral depending on the external time-dependent force F(t), and

the force-free pure subdiffusive part B(St).

FFPE with time-dependent force –Subordinated Langevin approach

Fig. Estimated solutions of the FFPE with F(t)=sin(t). The results were obtained via Monte Carlo methods based on the corresponding Langevin process Y(t). Here =0.8.

Fig. Two simulated trajectories (red lines) and nine quantile lines (blue lines) of the process Y(t) with F(t)=sin(t) and =0.8.

Subdiffusion in space-time dependent force

Claim 5. The Langevin picture of subdiffusion in arbitrary space-time-dependent force F(x,t) takes the form:

Y(t)=X(St),

where the parent process X() is given by the subordinated stochastic differential equation

[8] A. Weron, M. Magdziarz and K. Weron, Phys. Rev. E 77, (2008).

[9] C.Heinsalu,et al. , Phys.Rev.Lett. 99, 120602 (2007)

The FFPE for this case is not rigorously derived yet.

Fig. Simulated trajectory of the process Y(t) with space-time-dependent force F(x,t)= -cx(-1)[t]. After each time unit, the sign of the force changes,switching the motion of the particle with characteristic moves towards origin, when the force F(x,t) takes the harmonic form.

„There is no applied mathematics in form of a ready doctrine. It originates in the contact of mathematical thought with the surrounding world, but only when both mathematical spirit and the matter are in a flexible state”

Hugo Steinhaus (1887-1972)

Conclusion