Lecture 3 - Fokker .Planckequation @
Last time,
we studied the Langevin equationdvMdz = - dv + Rlt ) + Fextlt )
We solved this equation by assuming a given realization
of the random force Rltj and then take ensemble
averages .Another approach is to find the probability
distribution PC v. tl of finding the velocity in the range
&,
vtdv } at time t.
The equation describing the
time evolution of Plv ) is the Fokker - Planck equation .
Let as derive the Fokker - Planck equation starting
from the general Langevin equation of variable ✓
dv- =
flv) + ylt )dt
where ylt ) is a Gaussian white noise with
< nlti > = 0
{ nltlylt' ) > = Tdlt - t
'
)
as before.
µ=te : The variable v is Velocity in the Brownian
motion case,
but Langevin equation > describe
many processes and ✓ then stands for whatever
Variable is the relevant one to that process .
→ But first,
we need to digress a bit into the general
theory of stochastic processes .
3-2Stochastic processes
- some definitions
See for example van Kampen,
or D. Aroras
Xlt ) a stochastic variable,
discrete or continues,
that depends
On the parameter t which we will take as time.
The time series { XK ) } is called a stochasticpocesss
The probability PC x. t ) that XLE ) =× is defuel
asp( x. + , = <
SIX- XCH ) >
where < . > is the ensemble average of the random
process . The joint probability distribution
PK , ,t , ;×z ,tz ; ... ;xn,tn|={c( ×,
- XKD ) ... SCXN - Xltn ) ) )
The conditional probability of observing ×,
at t,
,
Xz at tz,
...
,and Xµ at
µ given that we had
J,
at I, Sz at Ez ,
...
,Ym at Tm is denoted
P ( ×, ,ti ; . . .
jxw.tn/y, ,t , ; .
. .
, Ym ,em )
Here we assum t,
7 tz 7.
. .7 tµ > T
,> . - ' > Em
For example,
we must have
P ( ×, ,t , ;×o ; to )= PCX
, ,t ,1×0 to ) PCX . to )
3-3Markov processes
In a Markov process the conditional probability for
transitioning into a state only depends on the
last state,
i.e.,
P( ×, ,t , ;
. .. )×n,InlY , ,tx ; ...
, Sm ,Tm )
= PK, ,t , ; ...
, kn.tn/Yi,ei )
A Markov process therefore liecksnmy : onlythe most recent state determines what happens
after that,
the further past historydoes not
matter .
By iteration we obtain :
PC ×, ,t , ; ... , Xn .tw ) = PC ×
, ,t,1×,tz ; ... ;Xµ .tw/Pkz,tz;...iXn.tn )
= PCX, ,t , 1×2 ,tz ) PC
xz.tn/3.ts...Xw.tw)P(xs,t3;...,Xn,tw)=...=P(x,,t,1xz,ti)Plxz,tz1x3,t3)-..PlXn..,tN-ilXn.tn)PKN,tw
)
⇒ All probabilities are determined by the transition
probabilityP(×n+ , ,tne ,I Xntn ) and PK ,t )
.
3- 4
Chapman - Kolmogorov equation
By definition,
we have
PC ×, ,t , ) = fdx ,
PC ×, ,t . ;k ,
-4 )
= fdxz PC ×, ,t , 1×2 ,tz ) Plxz ,tz )
and
PC ×, ,t , 1×3 ,tz ) = fdxz PCX
, ,t , 1×2,#z ;X3 ,ts)P(
k.tl/s.ts)
For a Markov process this implies
PCX, ,t , 1×3 ,ts ) = Jdxzpk, ,t , 1×2 ,tDP(
xretzlx, ,t ,
)
[Chapman - kolmogorov relation
Some examples Of Markov processes ;
- Random walk where each step is independent
and identically distributed.
Is Brownian motion a Markov process?
Need to be more specific .On time scales larger than
Emicro Such that ( RIE ) Rlt' ) 7 = TS I t - t
'
),
vlt ) as
described by the Langevin equation is Markovian.
what about ×( E) ? Not in general,
butyes if we Observe
on time scales large compared with Eu,
the velocityCorrelation time
.
3-5Fokker - Planck equation -
continued
Since the Langevin equation is a first order time
differential equation and ylt ) is Gaussian the
process is Markovian : The probability p( v. tlvotol
of having velocity v at t,
given that it had
Velocity Vo at to only depends on v• ( and not
on the velocity at other times ! ).
Q : Why would this not be true for a second
order differential equation ?
A
Markovprocess satisfies
p(xtlxto) =fdx'PlxtIX.ElPlxit'lxo.to)
where t 't [ to ,t ].
We now want to usethis equation to derive
a differential equation for the probabilityPC v. t Ivo .toD
.
The idea is best dcscriped in a picturePlx '
.
t ) PC × ,tt8t )n \
,X 1
1
'xty*-. • i* i• ×.
1
..
,
AH.i• i
I I1
|
, i,
I I 1 )
to t ttst
3-6
To be general ,
then,
we assume we have a Stochastic
Variable xlt ) and define
Sxlt) = × ( ttot ) - X ( t )
and assume
<orxlt) > = F. ( x ) ort
< (8×(+1727= Fzcx ) St
< (8×1+17") = OC ( oti ) n ) 2
We how write the Chapman - kolmogorovrelation as :
PC x. ttstlxoto ) = fdx' PC x. ttstlx'
.tl PC ×' .tl xo.to ) # )
Need to calculate the transition probability
PC x. ttstlx '
,t ) = ( 8C × - Sxk ) - x
') )
= { 1 + < sxlti > ¥ , + 'z< ( s×kn27ad÷zt ... }8k . x' )
[Taylor expansion
= { 1 + F. (d) st 8"
( × - x' ) +
tzfzcx' )
Star"tx . a) + ... }
where osmcx - x' ) = ftfn s ( x - x
' )
Plugging this into # and using
;fcx') 8
" )( x - xydx
'= t is
"
f' " '
( × )-
is
3-7
which you can prove by integrating by parts ,we get
( suppressingthe Conditional xo.to )
0 a?
PC x. ttot ) = Plxitl -
f×[F. a) PK ,t ) ] St +
tzonffzlx)P( at )]5t
+ Occotpj
Taking the limit ort -70 this gives
The Fokker - Planck equation :
01k¥=
- f×[ F. ( × )P( x. t ) ]+tz¥n[ EKIPKIH ]2t
This can be put into the form of a ( outinwdzequation for P :
¥.
+ f×j=o
where
j = Fi G) P - tz f×[ EK )P ]
( see also alternative derivation in Jack 's notes )
3-8Fokker -
Planck equation & Brownian motion
For the Langevin equation we have from our earlier results
x → ✓
- rt< vct ) > = voe ( P
.
2- 7)
C Lt ,t' 1 = < ult ) ✓ ( t'
) ) - < ✓ (E) v ( t'
) )
= Imzetktt'
1
zty[ etrminlt't "
-1 } ( p.2.io )
Using this we get ( Exercise : convince yourself this is corret )
F,
=( Vcttotl -
v ( t ) )- = - Xxrlt )
art
Fz =
< ( duct ' 12 > p- =
-
st m2
or
8¥ = rofrcvp) +Fmioofp
Note : A simpler wayto get this without knowing the full
solution is simply integrate the Langevin eqnlion . Namby
mda÷ = - art 7 't )* ort
ttot
⇒ f. =t<orca ) =tS<date >=tfds(Enns ) +4 's ' 7'
in )ot
st St -
t + o
=- JRCE )
and
< @up , = ( ( .ructiotttmtft"n 's ) ds )'
)
= ftp.ds#kds'cycs)ylsy > + okoti )
= Fist toasts'
) =) Fz =
Im2
3-9
This Fokker - Planck equation describes the time evolution of
the probability distribution Plu ).
At long times we should
reach equilibrium where this becomes the Maxwell
distribution
Peglv ) = ne×p( - Y÷ )B
Let's check that this is consistent .
In the Steady
slateop
- =O
at
The right hand side gives :
rfucvpeatt'÷mi¥Peg={r+Enzi÷}frCvPq)=oB
pif T = -
or T=
ZMKBTJ
as usualZKTMB - ( p .
2-8 )
Why is < ( svi"
) = oceti ) ?
In order for our general derivation of the Fokker -
Planck equation to hold,
we needed < ( Su ,"
) for n > 2
to be o(@t5 ) . why is this + me ? Because we assumed
Gaussian white noise
3- to
Gaussian white noise
It is generallynot enough to specify only the mean
< yltl ) and the variance < nltlylt'
) > to fully specify
a probabilitydistribution function .
For Gaussian noise,
however,
it is,
Since all higher order correlation function
are determined by these two . For example
< y , ).
. . ,7zn)=E IT < gin ;) ( Wick 's theorem )all
contractions
For example "
< nmzysnu ,> = 4,727<43%7+4,1 .
> < nzna > + < nine ) < nil , >
etc .
This is true for all Gaussian noise.
Gaussian white noise is further specified by having
( nltlylt'
) ) x 8h - t'
1
and therefore a constant power spectrum
For the Langevin equation ,
this means in particle that :
< ( ohm > ~ ( of )"
and < ( of " "
> ~etF' '
3-11
Spectral properties
A stationary process satisfies time translation invariance :
PIX, ,t , ;×z ,tz ;
.. .gl/n,tn)=P(x,,titt;xz,tztt;...;Xn.twtt )
for all T and N.
In particular
PC x. t ) = PC x )
PK, ,ti ; × , ,tz ) = PCX
, ,t ,- tz ;×z ,°)
Suppose we are interested in knowing the power Spectre -
Of a Stationary process .
T.beDefine
£,
( w ) = fdtxlt ) eiwt- Hz
The spectral function
s+( w ) = ttslxtcw )P >
Does the limit SCW ) = STCW ) exist ?
Th Tlz- iwlt - t
' )< IITCW )P ) = Sat fdt '
< xlt )x It ' ) > e
- tlz th -
C ( E - t' ) ( stationary )
X
T=t - t'
t=±c£+ty}=§dtei"(E) ( T - lei )
T - 12
and therefore
SC w ) = ftp.%deeiutcce ) ( I - ¥ )oC t.ie )
is
= fdeeiiut c (E)- a
[proving convergence nontrivial part of derivation
,
see Annas P .
50 05 van Kampen
This is the Wiener - khinchin theorem
co
S ( w ) = S dt e-in "
< xlttt )x( t ) >
- a
Exampks*white noise : C (e) = TSIE ) ⇒ Slw ) = T
* Brown noise : velocity correlates of Brownian motion
C (e) = Ae- rkl
•
- iwt - TIMde⇒ SCW ) = AS e
- is
= a[ ii. + Is ]=a÷ .
slw )^
a white See Jack 's notes fr
alternative derivation and↳ Brown
more detailed physical>
W
examples