Approximation Methods for SDEs
ÉIABATICREMOVALFASTVARIABLES
The Browniansmoke particleobeys NLII
I
my pi t noise duetocollisionwith air molecules
massxaciderationtlinear
drag
r given
by Stokesdrag onsphere viscousflow
noise is alsounderstoodfromstatistical physics
NONDIMENSIONALISATION
Results is LANGEVINEQUATIONS
axe tvedt XE x position
Ve faveat
fedBy Ve x velocity
Intersted in the system when actors totimescale at
long timescale compared
ofo uprocess
forVe
Aim find an equationforposition Xt thatdoesn'tinclude Vt the SLOW Dynamics
Start bywriting downtheFPE ofthe Langevin equations
for poxv t petDcap for Ep
Here a ex v t
Yee E E gg
so E E Et E oof
I D Y 00 Ep
ftp.crij entriesof E
Pt ut tf Pro
pet KRAMER'S EQUATION
VII CupgPI
IDEA find approximate asymptotic solutionby makingan expansion
p xv t e Cxv E E'pacx.v.tt
p x v E Eps
Insert in Kramer's equation and equatepowers
of E
potepsterpat etherpotepstepat potepstepat potepstepat
Order up pov v O LEADINGORDER EQUATION
12
Po probability p por o as lui o Integrate
decaycondition
Upetpov cease
, P V po pox e e
Here PCxt is an UNDETERMINED Function
of ix t
Order upsetPalu VPox
I
Rex e ve v2
this equation has a Solvabilitycondition because flitsav fcvpstpsdvdv fupstp.it
L
and I've O decayconditioners
cftp.t
Try to solve using meenhmogeneosen PsYx.t
eu4a sameas p butwithdifferentPs
For P I try Ps Fix E ve
Psu Fox t s Va e
PstPsv Fox E e M
Cups Padre Fox e ve F Px
Ps Rex t vé PsexE e
Order 1 CupatPau u PoetVpsx
M
Ptlx.be tC Px xtJV2e xtCPzCx t ve
for our expansion tocondition
bevalid this equationmustbe solvable
Ritson o
applysolvability
Pt Fet Pal éI Pak e Ivedi o
Pe Pm SMOLUCHOWSKI DIFFUSION EQUATION
This is the FPE axe Dawe SLOW Dynamics
of
different Brownianprocess
EXAMPLE HAKEN'S SYSTEM
axe yetEXEYE att Ed Bst
yea
dye att
YE BEI E dBat
AIM eliminate Ye
findthe slow dynamics for Xt
FPE B FK o
E E Bt Gk
e rye g
t
1
ft
ÉIABATICREMOVALFASTVARIABLES
The Browniansmoke particleobeys NLII
I
my pi t noise duetocollisionwith air molecules
massxaciderationtlinear
drag
r given
by Stokesdrag onsphere viscousflow
noise is alsounderstoodfromstatistical physics
NONDIMENSIONALISATION
Results is LANGEVINEQUATIONS
axe tvedt XE x position
Ve faveat
fedBy Ve x velocity
Intersted in the system when actors totimescale at
long timescale compared
ofo uprocess
forVe
Aim find an equationforposition Xt thatdoesn'tinclude Vt the SLOW Dynamics
Start bywriting downtheFPE ofthe Langevin equations
for poxv t petDcap for Ep
Here a ex v t
Yee E E gg
so E E Et E oof
I D Y 00 Ep
ftp.crij entriesof E
Pt ut tf Pro
pet KRAMER'S EQUATION
VII CupgPI
IDEA find approximate asymptotic solutionby makingan expansion
p xv t e Cxv E E'pacx.v.tt
p x v E Eps
Insert in Kramer's equation and equatepowers
of E
potepsterpat etherpotepstepat potepstepat potepstepat
Order up pov v O LEADINGORDER EQUATION
12
Po probability p por o as lui o Integrate
decaycondition
Upetpov cease
, P V po pox e e
Here PCxt is an UNDETERMINED Function
of ix t
Order upsetPalu VPox
I
Rex e ve v2
this equation has a Solvabilitycondition because flitsav fcvpstpsdvdv fupstp.it
L
and I've O decayconditioners
cftp.t
Try to solve using meenhmogeneosen PsYx.t
eu4a sameas p butwithdifferentPs
For P I try Ps Fix E ve
Psu Fox t s Va e
PstPsv Fox E e M
Cups Padre Fox e ve F Px
Ps Rex t vé PsexE e
Order 1 CupatPau u PoetVpsx
M
Ptlx.be tC Px xtJV2e xtCPzCx t ve
for our expansion tocondition
bevalid this equationmustbe solvable
Ritson o
applysolvability
Pt Fet Pal éI Pak e Ivedi o
Pe Pm SMOLUCHOWSKI DIFFUSION EQUATION
This is the FPE axe Dawe SLOW Dynamics
of
different Brownianprocess
EXAMPLE HAKEN'S SYSTEM
axe yetEXEYE att Ed Bst
yea
dye att
YE BEI E dBat
AIM eliminate Ye
findthe slow dynamics for Xt
FPE B FK o
E E Bt Gk
e rye g
t
1
ft
