Transcript AW_MM_Jerozolima_2008

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
• 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 Dt1-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 α1
 For α1, 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)
Subordinated Langevin approach
 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.
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 - exponentially
distribution with mean one.
 The iteration (*) ends when U() crosses the time
horizon T. We approximate the values St0, ..., StN ,
using the relation
with
Computer simulation – II Step
 Using the Euler scheme, we approximate the diffusion
for k=1, ..., L. Here L is the first integer that exceeds
are i.i.d. standard normal random variables.
and
 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 form
which confirms that the process is /2 self-similar. Here =0.6.
FFPE with jumps
The equation
,
0<<1, 0<≤2, describes competition between subdiffusion
and 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).
(a)
Fig. Sample paths of:
(a) the subordinated
process X(St),
(b) the diffusion X(),
(c) the subordinator St .
(b)
The interplay between
long rests and long
jumps is distinct.
Here =0.7 and  =1.3.
(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.
Fractional Klein-Kramers equation
The FKK 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 the
PDF 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
The equation
0<<1, describes subdiffusion in the presence of an
external time-dependent force F(t).
The fractional operatort Dt1-α in the above equation
appears to the right of F(t), therefore, it does not modify
the time-dependent force.
[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).
FFPE with time-dependent force –
Subordinated Langevin approach
The process Y(t) admits an equivalent representation
thus, it consist essentially of two contributions:
 the stochastic integral depending on the external timedependent force F(t), and
 the force-free pure subdiffusive part B(St).
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
The FFPE for this case is not rigorously derived yet.
[8] A. Weron, M. Magdziarz and K. Weron, Phys. Rev. E 77, (2008).
[9] C.Heinsalu,et al. , Phys.Rev.Lett. 99, 120602 (2007)
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.
Conclusion
„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)