Oct 5, 2011 - âStrangeâ KineticS (Shlesinger, Zaslavsky, Klafter, ... attraction of one-sided. -stable Lévy law. HSC SM .... Strange diffusion on DNA. Lévy flights ...
Hugo Steinhaus Center WUT, , 5 Oct 2011
“Normal” DE:
t
f (t , r )
D f (t , r )
Fractional DE:
f (t , r ) J.B. Perrin, 1909
t
D
/2
f (t , r )
Brockmann, Hufnagel, Geisel, 2006
Natural and Modified Forms of Distributed Order Fractional Diffusion Equations Aleksei Chechkin Akhiezer Institute for Theoretical Physics, Kharkov, Ukraine
HSC SM, Wroclaw
In collaboration with Rudolf Gorenflo, FU Berlin Igor Sokolov, HU Berlin Joseph Klafter, TAU Tel Aviv Vsevolod Gonchar, AITP, Kharkov
Nickolai Korabel, BIU, Ramat Gan Review: A. Chechkin, I. Sokolov, J. Klafter, in: Fractional Dynamics, S.C. Lim, R. Metzler, J. Klafter (Eds), World Scientific ( 2011 ? )
HSC SM, Wroclaw
Outline Introduction: fractional kinetics, relation to random walk scheme, time and space fractional derivatives Time and space fractional diffusion equations in normal and modified forms, equivalence of the two forms Distributed order fractional derivative Distributed order fractional diffusion equations (DODE) in normal and modified forms non-equivalence of the two forms different regimes of anomalous diffusion
Summary: the Table
HSC SM, Wroclaw
Fractional Kinetics: Diffusion and Kinetic Equations with Fractional Derivatives “Strange” KineticS (Shlesinger, Zaslavsky, Klafter, 1993): connected with deviations
“Normal” Kinetics Diffusion law
Fractional Kinetics x2 (t )
x2 (t )
t
x2 (t )
t
,
ln t ,
1
0
Relaxation
Exponential
Non-exponential “Lévy flights in time”
Stationary state
MaxwellBoltzmann equilibrium
Confined Lévy flights as non-Boltzmann stationarity
Simplest Types of “Fractionalisation” f t
time – fractional diff eq
t
, 0
1
2
D
f
HSC SM, Wroclaw
DIFFUSION EQ
x2
Caputo/Riemann-Liouville derivative
t
space – fractional diff eq 2
x
2
, 0
2
|x|
multi-dimensional / anisotropic
Riesz derivative : symmetric combination of left/right side RL derivatives time-space / velocity fractional
Question : Derivation ? Answer: from Generalized Master Equation and/or and/or Generalized Langevin Description
HSC SM, Wroclaw
Underlying physical picture: Random walk with long jumps and long waiting times Random walk x(t) , PDF f(x,t), consisting of Random jumps :
i
x(ti ) x(ti 1 )
Random waiting times :
Jump PDF :
ti ti 1
i
( )
Waiting time PDF :
i=1,2,…
w( )
Question : Diffusion equation for f(x,t) in the long time – space limit ? Answer : Depends on the asymptotic behavior of 2
mean square displacement
mean waiting time
d 0
d
w( )
2
( )
( ) and w( ) .
either finite or infinite
either finite or infinite
HSC SM, Wroclaw
Underlying physical picture: Random walk with long jumps and/or long waiting times 2
( )
finite
Ordinary DE
finite
w( ) 1/ 1 0
f t
Belongs to domain of attraction of one-sided -stable Lévy law
2
f
x
2
f t
Time fractional DE
1,
t
2,
2
Space fractional DE
D
f
1
1/
0
Belongs to domain of attraction of symmetric stable Lévy law
2
f
x
2
D
D
f |x|
Space-time fractional DE
f t
D
f |x|
HSC SM, Wroclaw
Time fractional derivatives in Caputo and Riemann-Liouville forms t
Fractional integral
1 (t ( )
J f (t ) :
)
1
f ( )d
R+
,
0
definition
Caputo
d
f (t ) : J 1
dt
d f (t ) dt
t
d
f :
dt
1 d t (1 )
Riemann-Liouville
0
