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