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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
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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 D t 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 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.,.
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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(S t ), where the parent process X( ) is given by the Itô stochastic differential equation (Langevin equation) and S t 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 S t is defined as where U( ) is the strictly increasing -stable Lévy motion with the Laplace transform The role of S t is analogous to the role of the fractional derivative 0 D t 1-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 - exponentially distribution with mean one. (*) The iteration (*) ends when U( ) crosses the time horizon T. We approximate the values S t 0,..., S t N, 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 exceeds andare 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(S t ), (b) the diffusion X( ), (c) the subordinator S t. Note the similarities between the constant intervals of X(S t ) and S t and the similarities between X(S t ) and X( ) in the remaining domain. Here =0.6.
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Fig. Evolution in time of (a) the subordinated process X(S t ), (b) the Brownian motion X(t). The cusp shape of the PDF in the first case is characteristic for the subordinator S t. Here =0.6.
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Fig. Estimated quantile lines and two sample paths of the process X(S t ) 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.
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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 D t 1-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(S t ), where the parent process X( ) is given by the Itô stochastic differential equation (Langevin equation) and S t 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(S t ), (b) the diffusion X( ), (c) the subordinator S t. 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(S t ) 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(S t ) 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(S t ) 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). k B T – 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 the PDF of the process Y(t)=(X(S t ),V((S t )), 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(S t ) and V(S t ) in the presence of double-well potential. Here =0.9, m=k B T= =1.
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Fig. Comparison of the estimated and theoretical stationary solution of the FKKE. Here =0.9, m=k B T= =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 D t 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(S t ), where the parent process X( ) is given by the subordinated stochastic differential equation (Langevin equation) U( ) is the strictly increasing -stable Levy motion and S t 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(S t ). 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(S t ), 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
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