Random walks with waiting times depending on the preceding jump length

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!