a subordination approach to modelling of subdiffusion in space-time-dependent force fields
DESCRIPTION
A subordination approach to modelling of subdiffusion in space-time-dependent force fields. Aleksander Weron Marcin Magdziarz Hugo Steinhaus Center Wrocław University of Technology Jerusalem 28.03.2008. Contents. Fractional Fokker-Planck equation (FFPE) Definition and basic properties - PowerPoint PPT PresentationTRANSCRIPT
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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
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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
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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).
,
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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.
,
.
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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)
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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
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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
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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
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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.
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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.
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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.
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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).
,
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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.
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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).
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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)
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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.
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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.
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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.
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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).
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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).
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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.
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Fig. Comparison of the estimated and theoretical stationary solution of the FKKE. Here =0.9, m=kBT==1.
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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).
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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).
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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
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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.
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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.
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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.
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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.
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„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