random walks with waiting times depending on the preceding jump length
DESCRIPTION
Random walks with waiting times depending on the preceding jump length. V.Yu.Zaburdaev MPI for Dynamics and Self-Organization, Göttingen, Germany. Standard Random Walk Model. x. x. x+x’. The probability to jump into. is. The probability to jump away at. is. - PowerPoint PPT PresentationTRANSCRIPT
Random walks with waiting times depending on the preceding jump length
V.Yu.Zaburdaev
MPI for Dynamics and Self-Organization, Göttingen, Germany
Standard Random Walk Model
xx x+x’
)(f
The probability to jump into ]','[ dxxxxx dxxg |)'(|is
The probability to jump away at ],[ dttt dttf )(is
The probability to “survive” until t : t
dftF0
)(1)(
)'(xg
Generalized Random Walk Model
xx x+x’
|)|,( yf
The probability to jump into ]','[ dxxxxx dxxg |)'(|is
The probability to jump away at , ],[ dttt dtytf |)|,(
The probability to “survive” until t : t
dyfytF0
|)|,(1|)|,(
)(yg
x-y
provided it arrived from distance y
Coupled transition kernel
|),(| yg )(f |),(| yg |)|,( yf
“Physiological” example:
For a longer jump, more time is necessary to recover
Mean resting time is a function of a jump distance |)(| yr
|)(|e|)(|
1|)|,( y
r
r
yyf
.0const,,|||)(| 00 yyr
In the simplest case |,||)(| yyr analogy with /1const
M.F.Shlesinger, J.Klafter et al. (1982,1987), E.Barkai (2002), M.Meerschaert et al. (2002), S.A.Trigger et al. (2005)
Microscopic details
In the given point (x,t) there are particles which arrived there at different times and from different points, therefore they will fly away also at different times.
The outgoing flow of particles from a given point is not a simple function of the concentration alone.
dydytxNtxnt
0
),,,(),(
dydyF
yfytxNtxQ
t
0 |)|,(
|)|,(),,,(),(
(1)
(2)
)()0,(),(|)|,()(),( 0
0
xntFdtyxQyFydygtxnt
(3)
Coupled transition kernel (equations)
t
dydytxNtxn0
),,,(),(
dydyF
yfytxNtxQ
t
0 |)|,(
|)|,(),,,(),(
)()0,(),(|)|,()(),( 0
0
xntFdtyxQyFydygtxnt
)]()()(),()(|)[|,( 0 tyyxntyxQygyFN
Coupled transition kernel (equations II)
kp
pkpk yfyg
fnQ
|)}(|)({1
)0(0,
kpkp
pkpkpk nF
yfyg
fyFygnn 0
0, )0(
|)}(|)({1
)0(|)}(|)({
.0|,||)(||)),(|(|)|,( yyyyf rr
kk
yp
kpkpk n
yg
yFygnn 0||
0, }e)({1
|)}(|)({
“Finite velocity” Green’s function with /1G
Finite velocity
xx x+x’
const
G.Zumofen, J.Klafter et al. (1993), V.Yu.Zaburdaev, K.V.Chukbar (2002), E.Barkai (2002), M.Meerschaert et al. (2002), I.M.Sokolov, R.Metzler (2003)
,)(1 /||
0,
kxp
p
kppk gef
nFn
,|)|1(
1)(
12 x
xg 0/
0
e1
)(
f
,1,2 x
||/2 x
xD'''' 2kp
,2 x ,12/1,|| x '''' 2kp 2/1tx
,2/10,|| x '')()('' 22
ik
pik
p tx
Back to the coupled model
/1,|)}(|)({,,, kppkpk yFygGn
2/32/1
|)|(
,4/1
t
xtG
1
)()(0
xxn
Coupling & finite velocity together
;}e)({1
|)}(|)({/||||
/||0
,k
ypyp
kpyp
kpk yg
yFeygnn
)1/( eff
dydzzgy
tyxQny
fly
||
)(||
,1
2/1
1
)()(0
eff
xxn
flytotal nnn
Conclusions
• Introduction of microscopic details is necessary for the understanding and correct description of CTRW
• They allow e.g. to solve a problem of coupled transition kernel and take into account the finite velocity of walking particles simultaneously
• These features open a possibility for applications in biological problems where recovery processes and finite velocity of motion are presented: foraging movements of animals, motion of zooplankton, …
V.Yu.Zaburdaev, J.Stat.Phys. (on-line first) (2006)
Thank you for attention!