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Procesy stochastyczne, proste modele i zastosowania
•błądzenie przypadkowe, ruch Browna
•równanie dyfuzji
•stochastyczne równania różniczkowe
Błądzenie przypadkowe w jednym wymiarze: cząstka wykonuje ruch w prawo z prawdopodobieństwem p i w lewo z prawdopodobieństwem q
2ln21ln
2
ln222
)1(2
ln21
2
)1(2
ln21
2ln)2/1(),(ln
qmN
pmNmNmNNmNmN
NmNmNNNNNmp
)(2ln21ln)
21(!ln 1 NONNNN
By using Stirling’s formula
22
,22
2
mNqlmN
mNprmNmNNpmmm
NqmmNq
NpmmNpNpq
NqmNqmNq
NpmNpmNp
qmNqpmNp
NNNmp
21ln
21
2
21ln
21
2)2ln(
21
)2
1(ln21
2
)2
1(ln21
2
ln2
ln2
2ln21ln
21),(ln
In total N steps the position m on a 1-dim lattice is achieved with the probabilityp(m,N): l steps have been performed to the left (with probability q) and r steps to the right (with probability p)! r=m+l, m+2l=N
Expanding the logarithm,
Npqpqm
NpqmNpqNmp
yields
xOxxx
4)(
4)(
21)2ln(
21),(ln
)(21)1ln(
2
32
))((4
)()()(
4
2/1
22
2
NpONpq
pqmOmNpq
We want to approximate thedistribution in its center and up to fluctuations around the mean value,
so we can neglect the last term in theabove equation for
Npqm
NpqNmp
4)(
21exp
422),(
2
Continuum limit...
Probability of findinga random walker in aninterval of width 2
around a position x attime t.
We require now:
x
constDtxpq
tx
2)(2
0,0
dxDtxx
Dtdxtxp
2)(
21exp
221),(
2
txpqD
tNtxmxxmx
2)(2
,
Dtxx
DtxtNxmp
2)(
21exp
222),(
2
constvtxptx
ttxpxNpmxtx
)21(2,0,0
)21(2)
21(2)(
Dtvtx
Dtxtxp
2)(
21exp
222),(
2
With starting condition
and boundary condition),(),(),(
0),()()0,(
2
2
txpx
Dtxpx
vtxpt
txpxxp
x
txxvpDp
txxvqDq
txqxvq
txpxpv
txp
txpp
txpqD
NmqpNmppNmp
2
2
22
222
)()(
)()(
)()()1()1(
)()1()()1)(12()(2
),1(),1()1,(
A master equation for thediscrete random walker
Fokker-Planck-Smoluchowskiequation
),(1)(
2
)(),1(),(2),1(
),1(),1(),()1,(
2
2
Nmptx
D
xNmpNmpNmpD
NmpxvqNmp
xvp
tNmpNmp
),(1)(
2)(
),1(),(2),1(
),(),1(
),1(),(),()1,(
),(1)(
2)(
),1(),(2),1(
),1(),1(),()1,(
2
2
2
2
Nmpxvq
xvp
txD
xNmpNmpNmpD
xNmpNmpvq
xNmpNmpvp
tNmpNmp
Nmptx
Dx
NmpNmpNmpD
NmpxvqNmp
xvp
tNmpNmp
Reinsert v and D
into the lastterm of eq.
Taking the continuum limit and keeping v and D constant, we arrive again at the Fick-diffusionequation!
Błądzenie przypadkowe z symetrycznym prawdopodobieństwemp=q=1/2 (…nieco inne rozwiązanie)
Simple polymer models... Random walk revisited.
Thermodynamics....
F
