Originally Pollen Particles in fluid - KTH Teoretisk...
Transcript of Originally Pollen Particles in fluid - KTH Teoretisk...
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