2- ILecture 2 - Brownian motion

Originally : random movement of Pollen Particles in a fluid

1827 - Robert Brown observes random irregular nation

of pollen particles in water

1905 - Einstein provides a theory of random collisions

from molecule ) =) heat I molecular m . tier

( Einstein did not know about Brown 's exp . )

How do wemodel this ?

Let's start with the crudest description :

Random walk

Xltl

f.

See Python simulation

-t

Position xlt ) only measured at discrete times tn=n at

xlttotl = xltlt DXN-

random variable drawn from

somedistribution Plax )

Assume Bxn are independent , identically distributed

4 Dxn ) = 0 and <@×n)27 =02 < A

in -1

thenX #= DX

.+ DX

, + ... + Dan ,= E DXK

4=0

Xlt ) sum of large number of independent identically2-2

distributed random variables

⇒Xlt ) normally distributed for large t

[ central limit theorem ( see digression next two

PK ) = f- e

- ztoI( × - mx )2Pages )

Zito ,2

Let's calculate M×and ox

Mx = { ×Lt77 = E 4D×k > = 0

R

of= @Ltl > = E < D×kb×e7

Kel

= § < CDXKB + E < B×nD×e7

-htl -

oh =< Dxa > < d×e>=o since

Dxu and Dxe independent

= no2 = oltto

EZDT

with diffusion coefficient D= ¥t

=D Pcxit ) = # e- ¥tYITDT

Solution of diffusion equation ( last lecture )

Assumptions : no Interaction between Brownian Particles

⇒ low density. BXK independent ⇒ Dt > 7 tmic Collision time

2-3Central limit therein ( small digression )

Assume Xi IEI, ...

,N a set of identically

distributed

random variables with mean µ = < xi > and standard

deviation o2 = < ( Sxi )?

) = < ⇐ - < × . > )2 ).

Define a new random variable

N

y - E Xiit 1

then the probability distribution of yIs

Plg ) = < Sly - §Xi ) ) = fdx ,... dxnpcx , ) . ...

Pkn) 8 ( y

- § 4 )

Now use that 8 ( y -

go ) =fake'k(TT° )

Plus) = ITSdxipkijfdkeik'T ' ×il

ikxi= fdk exp ( N en fdxplx ) e - iky )

= folk exp ( Nlnglk ) - iky )

where glk ) = <

ei°k*> is the generating function of PK )

such that n

(K× )

lugck ) =[

nT<<x

"

) )

n

is the cumulent generating function

< < × > ) = < x ) = or

4×2 ) ) = < ×2 ) - < × >2

= 02 etc .

2-4

This then gives

Plgl = fdkexp ( ikly - Mn ) - nzlioz ) + fdkenn§zt¥" a ×

""

= E#ie×p(' III ) + out

which is Gaussian distributed with

My = Nn

of = Not

2- 5

Langevin Equation ( Paul Langevin 1908 )

Let's try to make a bit more microscopic description

of the dynamics of the Brownian particle .

we assume

that the mass of the Brownian particle M > > Macon ,

the mass of the atoms in the Surrounding liquid .

The atoms then are bombarding the particle from

all directions with each collision taking a time

tm , .← .

much smaller than the characteristic

time scale of motion for the Particle . we mode

then the impact of the fhid by a random force

RK ) and the motion by a stochastic Newton of :

( For simplicity of notation I 'll do this in ID,

no

essential physics is lost by doing this here )dv

M -= RIE )

dt

where< RLH > = 0 "

white noise"

< Rlt7R( t'

) 7=PSlt - t'

) since Fouriertransit

of S is a constant.

How do we solve such stochastic differential equations ?

There is a large body at mathematical work and

various levels of sophistication in obtaining solutions.

solve vlt ) for a given Rltl,

then average

Two main approaches :{ calculate Plait )

See reference material for details of this,

we

will just go ahead and try to understand things

as we move forward .

we begin by simply taking expectation values,

i.e.,

2-6

d< v >M - = h Rlt ) >

at

or 1

hvltl > = vo +- fdt

'

< Rlt'

) > = v.

m

.-

0

Can this be correct ?

No,

at large times we Should reach equilibration

in which Lvlt ) ) =o.

So,

what did wemiss ?

Inertial fiction / dissipation ! we expect the liquid to

resist the motion of the particle .

There must

therefore be a systematic part to the force from

the atoms in addition to the rarlo - part .

This has to depend on velocity and if the velocity

is in some sense small enough ,then

Flt ) =- xv+ Rlt ) lx > ° )

If in addition there is an external force Fe×+ ,we

obtain the Langevin equation

made =- xvt Rltlt text

NI For a macroscopic spherical particle the friction is

given by Stoke , law : ×= Gina ,

with y the viscosityand a the radius ,

2- 7

Solving the Langevin equation

mi = - xv + Rltltfext

Let 'S first tale text =o and look again at the

average :

Matsu > = - au >

- rt with r=±< ✓ LED = voe m

⇒ -

This makes sad Tv= Yr velocity relaxation line

we can therefore go on and look at the

general solution.

For a given realization RIH

of the random force,

the solution is

t

- y ( t - s ) 1

✓ it )=Irtv

.+ Se a @G) + Feats) ]ds * ,

0

As we can verify by inspection.

mdaI= . ym @ tv. + § e-

" ← "

fnfu, )tFe×+]ds]tRl#tfexttt )

w -=D

VLE )

We Can use this solution now to calculate averages

and velocity correlates,

as well as the positiont

Xlt ) = to + Svcs )dS

0

For now we take Feet = 0.

2- 8

What more do we expect ?

In equilibrium we expect thematization,

are thus

I zm< v2 ) = I KBT

Using # ) we have

( ( dual )'

) = < Kvltl - ettu. ] }

= tmz §ds§ds ' e-8/2+-5 's "

< Rls ) t= In § e-

28 't → )ds

Pscs - sy

=zEgn[ i. in ] EY I.

2inIn order for everything to come together ,

this requires

±- = KDI or T = ZRMKBT

2h28 M

This is a great and deep result,

so let's make sure

we understand it and that it makes sense .

* TXT the higher the temperature the larger

the fuctuations,

or randomness in

the tree R KRZ > AT )

* pxj Fluctuation ← > dissipation

Equilibrium requires a fire balance

between fluctuations and dissipation .

2- 9

This is a general result,

sometimes referred to as

The fuctuation - dissipation theorem .

we will see It inmany guises during this course

.

A note about averages

we have been using the average

< A 7 = JPCR )A[ RHI ] DR

functional average over the noise Rlt )

Other averages +

At = ±, falt ) dt timeaverage

0

< A >eg Ordinary equilibrium average

( e.g. with g~ e- Pit)

A process is ergot if

lin It = < A >T - ) is

2- 10

Velocity corrdator

An object of great interest is the velocity correlates

c Ct ,t'

) I < ✓ (E) v ( t'

) ) - < ✓ Lt ) > < v ( t'

) )

= mt, §ds § dsi e-Nt 's ) ' " t

'

' " )< Rls )r1s' ) )

t'

sa

r -

rlttty+285+

-. - -

i

- - -t > t

':

tefalse1 m21 of

=Iertttt't;g[ eat

'

. ifI MZI >

t'

Si t ,t' → co

→ T - rlt - t'

)- e

It - t 't finite Zrm '

or T

( Ct,

t'

)=- Imi E81 ¥1

= C Lt - t'

)t ,t' → as

It # I finite

Note that this only depends on t.tl !

We expect this since at long times we reach an

equilibrium steady state ( time independent )

Note : If we integrate this result

A a

SCCE) dt = f Efate

Mldt = Iyer =2YBmI

- p - a

aFriction coefficient

or

£ = ytm = z÷gf f CC t ) dt related to correlation

function .

- a

Genel !

fluctuation - dissipation -

2- 11

PositionLet's now calculate the position

t

X It ) = Xo + f v Is)ds

0

= x. + ogtdsfersv .

. , § ds ' it's - s'

' tnrls ' )]

= X. t VI ( i - e-tt

) ttm§ds§ds' Rls' ) et 's 's

' )

and

< xlt ) > = xo + Vg ( , - est )

what about correlations ?

< xlt ) xlt 's > - < xlti > ( xlt 'D

= Ii §ds,§ds ,

'

§dszjjdsi e-* '

→'

'' → " is " )< Risi)Risi ) >

Take tzt'

( xhti ) - < xcti >2

= If §ds§ds ' e-" 5+5

'

)§dg§ds,

' is" 't '

'

'as

,.si ,

Infinity,

⇐ Yi's.im#eEso < ( sxi ) = Emi §ds§ds. em 's " [ eormin

" 's "-1 ]

2- 12

Koxi> = Emiftds ftds ' [ et " ' 5'

l- e-

" st ''

1

]

=Fni{ totalIds'

e- " "' '

+ Does' insist ]

. Iasi "

foes' ers

'

}=

T

In.[ t + Flirt . i ) - 's ( i . est ,

'

]

÷ZKBI t = ZDE

yrm e-

T = WMKBT with D= KBI Diffusion coefficientXM