Elements of Kinetic Theory - Berkeley Cosmology...

Phys 112 (S2006) 9 Kinetic theory 1 B. Sadoulet

Elements of Kinetic TheoryStatistical mechanics

General description + computation of macroscopic quantitiesEquilibrium: Detailed Balance/ EquipartitionFluctuations

DiffusionMean free pathBrownian motionDiffusion against a density gradient

Drift in a fieldEinstein equation

Balance between diffusion and driftEinstein relationConstancy of chemical potential

Other transport phenomenaHeat transportMomentum transport=viscosity

Johnson noiseMostly in Kittel and Kroemer Chap. 14


Thermodynamic quantitiesPressure cf. Kittel and Kroemer Chapter 14 p. 391

If the particles have specular reflection by the wall, the momentum transfer for a particle arriving at angle θ is

Integration on angles gives

that we would like to compare with the energy density�

P =Force

dA=

d!p!tdA

= d"0

2#

$ d cos% 2p cos% v cos% n p( )p2dp

0

&

$0

1

$!

�

non relativistic! pv = 2" ! pressure P =2

3u (energy density)

u =3

2

N

V# ! P =

N

V# = same pressure as thermodynamic definition =

#$%

$V U,N

ultra relativistic ! pv = " ! P =1

3u

2 pcos!

�

2

3! 2" pv n p( )p

2dp

0

#

$!

�

u = ! n p( )d3p

0

"

# = 4$ ! n p( )p2dp

0

"

#


Detailed BalanceConsider 2 boxes filled with gas in communication through a

small apertureEach supposed in thermal equilibriumTemperature T1, T2Concentration n1, n2

What is gained by one box= what is lost by the other one

Detailed balance argumentAt equilibrium: No net flux=> gains of box 1= losses of box 1

In particular:Outgoing flux of particles = Incoming flux of particles

Outgoing energy flux = Incoming energy flux

Only way for ideal gas

=> equality of temperature and chemical potential (and pressure)

Equipartition of energycan be thought as equilibrium between 3 degrees of freedom each having

! n1,"1( ) =! n2 ,"2( )

J n1,!1( ) = J n2,! 2( )n1 = n2 !1 = !2

For ideal gas

n1,2 p( )d 3p =

n1,2

VnQ !1,2( )exp "

# p( )

!1,2

$

%&'

()d

3p

h3

!1 d.f. =1

2"


FluctuationsMicroscopic exchanges

In a system in equilibrium, exchanges still go on at the microscopiclevel! They are just balanced!

Macroscopic quantities= sum of microscopic quantities ≈ N x mean

But with a finite system, relative fluctuations on such sum is of theorder 1/√ N

e.g., fluctuation on total energy

Computation with partition function; By substitution we can see that

Similarly when there is exchange of particles

�

U = ! = !sps

s

" = !s

e#! s

$

Zs

"

!E

2= E " E( )2 = E

2 " E2

= #s2 e

"#s$

Zs

% " #se

"#s$

Zs

%&

'

( ( (

)

*

+ + +

2

U = E =! 2" log Z

"!

!E2= E2 " E 2

=#2$U

$#=

#2$#2$log Z

$#

$#

Variancenot entropy!

�

N =!" logZ

"µ! ,V

�

!N

2= N

2" N

2=

#$#$ logZ

$µ

$µ=#$ N

$µ


How do systems come in equilibrium?Energy transfer

If 2 gases at different temperatures are put in contact, moleculesof the hotter gas have in average higher energy and transfer netenergy to the lower temperature gas => temperature equilibrium.

Energy transport by diffusion. Not instantaneous!=> thermal conductivityHeat transfer equation

Similarly, transfer between wall of gas enclosure and gas.=>e.g., black body radiation: equilibrium between walls and photons

inside cavity

Momentum transferif shear between 2 fluid volumes

related to viscosity (see later)

Particle transferIf a gas system 1 is put in contact with another gas system 2 where

the concentration of gas molecules is lower , the higher density insystem 1 will favor diffusion of molecules to system 2 =>concentration equilibrium

=> Diffusion equation


Scattering Mean Free PathInteraction cross section

Consider a beam of particles incident on a targetProbability of interaction in a slab of thickness dz =

Cross section : dimension = areaExample: hard spheres

Mean free pathProbability of interaction in interval dz

Survival probabilityParticle enters medium at z=0. l is the attenuation length

! n dz

!

! = "d2

dl =

1

!n

dz

dz

l

N z + dz( ) = N z( ) 1!dz

l

"#$

%&'

( The survival probability varies as:

P z + dz( ) = P z( ) 1!dz

l

"#$

%&' )

dP

dz= !

P z( )

l) P z( ) = exp !

z

l

"#$

%&'

Prob z( )dz = exp !z

l

"#$

%&'dz

l

Probability of interaction between z, z+dz

d


Diffusion: No Concentration GradientBrownian motion

Succession of scatters:Consider a particle of speed vAssume isotropic scattering, no concentration gradient

=> Average displacement between two scatters along z axis

constant concentration => l does not depend on θ

Average displacement squared between two scatters along z axis

= Variance

Number of scatters for particles of speed v per unit time

=> Evolution of variance with timeAverage on distribution of velocities

= Diffusion coefficient

!zbetween scatters

= scos" e#

s

lds

l$

d cos"2

d%2&

= 0!

z

!z2

between scatters= scos"( )2 e

#s

lds

l$

d cos"2

d%2&

=2

3l2

dNscatters

dt=v

l

d !z2

fixed v

dt=

2lv

3

d !z2

dt=2 lv

3=2

3

lvf v( )dv"f v( )dv"

= 2D =d !x2

dt=d !y2

dt

D =lv

3


Diffusion: Concentration GradientSuppose that we have concentration gradient along the z axis

to first order in

=> l depends on θ=> Probability of survival along direction θ is such that

=> Probability of interaction between s and s+ds

The mean displacement along the z axis between collisions (keeping only first order inrelative gradient)

z!

Psurvival s + ds( ) = Psurvival s( ) 1!ds

l

"

# $

%

& '

1

l= !n"

1

l=1

lo

1+1

no

dno

dzz

#

$ %

&

' ( =

1

lo

1+1

no

dno

dzscos)

#

$ %

&

' (

�

dPsurvival

s( ) = !Psurvival

s( )

lo

1+1

no

dno

dzscos"

#

$ %

&

' ( ds

! Psurvival

s, cos"( ) = exp #1

lo

s +s2

2

1

no

dno

dzcos"

$

%&'

()$

%&'

()* e

#s

lo 1#s2

2lo

1

no

dno

dzcos"

$

%&'

()

Pinteract = Psurvival s( )ds

l=ds

lo

e

! slo 1!

s2

2lo

1

no

dno

dzcos"

#

$ %

&

' ( 1+

1

no

dno

dzscos"

#

$ %

&

' (

!zbetween collisions

= scos"#ds

lo

e

$ slo 1 + s $

s2

2lo

%

& '

(

) *

1

no

dno

dzcos"

%

& '

(

) * d cos"

2

d+2,

s

�

sdn

n0dz


Diffusion: Concentration Gradienttaking into account that

we get

Each particle undergoes v/l scatters per unit time. Hence, the meantransport velocity along the concentration gradient is

Averaging on velocity distribution

sm

0

!

" e

#s

lods

lo

= m!lom

!zbetween collisions

=1

32lo

2" 3lo

2( )1

no

dno

dz= "

lo2

3

1

no

dno

dz

d !z

dt= w

z= "

lov

3

1

no

dno

dz= "D

1

no

dno

dz

dz

dt= !z

between collisions

v

lo

= "lov

3

1

no

dno

dz


Diffusive transportTransport

=> Particle flux or more generally

Fisk’s law: Opposite to gradientThis is one example of transport: In addition to random velocity v there is a coherent transport

(or drift) velocity w and net fluxes of particlesSimilar transport of charged particles explains electric mobility , of energy explains heat

conduction, of momentum explains viscosity.

Conservation of the number of particlesConsider a volume V. The decrease of the number of particles inside volume has to be equal to

the total particle flux through the surface.

but by the Divergence Theorem

This has to be fulfilled whatever the volume. Hence the particle conservation equation:

Diffusion equationReplacing J by its value

we get≠ wave equation

Jz = nowz = !Ddn

dz

! J = !D

! " n

n d! S

!n

!tV

" d3x = #

! J $d! S

S

"

! J !d! S

S

" =

! # !

V

"! J d3x

!n

!t+

! " #! J = 0

! J = !D

! " n

!n

!t= D"

2n

1

c2

!2A

!t2= "

2A


Drift in an Electric FieldCharged Particles (electrons, holes, ions)Drift Velocity

Consider an electric field along the z axis. In addition to its random velocity, eachparticle will acquire a net velocity in z direction from acceleration betweencollisions

It is advantageous therefore to work with the time δt before the next collisioninstead of s.

If the collision time τc=l/v is constant, the probability of collision between δt and δt+dδt is

Mean displacement between collisions:

Each particle undergoes v/l collisions per unit time =>

=> Averaging on random velocities

Note: this strictly applies to case where collision time τc=l/v is constant. Otherwise

E

�

!v z =qE

m!t

�

!"z = v cos#"t +1

2

qE

m"t2

�

!zcoll.

= e"!t# c$ d!t

#c

d cos%2

d&2'

v cos%!t +1

2

qE

m!t 2(

)

*

+ =qE

m# c

2=qE

mv 2l2

dz

dt= !z

between collisions"v

l=qE

m

l

v

wz =qE

m

l

v=qE

m! c

�

wz

=qE

m

2

3

l

v+

1

3

!l

!v

"

#

$

%

&

' =qE

m(

c eff

e

!"t / #c

survival probablility

!"# $#d"t

#c

=acceleration x τc


Drift in an Electric Field (2)Mobility

Constant τc=l/v

Ohm’s lawConsider a wire of length L and sectional area A. If the wire is thin enough

! w = ˜ µ

! E

˜ µ =q

m!c

E =V

L where V is the applied voltage

I = nAqw = nAq

2

m! c

V

L"V =

L

A

1

nq2

m! c

conductivity #c

!"#

IL

A


Einstein RelationConstant collision time

Consider the ratio

⇒for constant τc=l/v

General caseIf the field is low enough for the particle to remain in thermal

equilibrium at temperature τ = kBT , it can be easily shown byintegrating by part the integral giving <lv> that

We then still have

qD

!µ=m

! c

lv

3=

m

! c

!cv2

3

for constant !c = l/v" #$ %$

=2 1 / 2mv

2

3= ! = k

BT

�

D =!!

c

m=kBT ˜ µ

q

�

D =!

m

2

3

l

v+

1

3

dl

dv

"

# $

%

& ' =

!!c eff

m=! ˜ µ

q

�

D =!!

c eff

m=kBT ˜ µ

q


Constancy of the total chemical potentialBalance between Electric Potential and Density Gradient

Consider a charged particle in an electric field along Oz=> For constant τc=l/v, this induces a driftvelocity

which will increase the concentration along wzAny density gradient will induce a diffusion such that

Inversely a gradient of charged particles will induce an electric field which will create a drift velocityThese two contributions will balance when these two velocities are opposite

=> at equilibrium

integrating to have the potential

we get

Remembering that the internal chemical potential is

we conclude that the total chemical potential is constant

�

wz

= ˜ µ E

�

wz

= !D1

n

dn

dz= !

" ˜ µ

q

1

n

dn

dz

�

˜ µ E = !˜ µ

q

1

n

dn

dz

�

qE !"1

n

dn

dz= 0

V(z) = ! Edzzo

z

"qV(z) + ! log

n z( )

n1

"

# $

%

& ' = constant

µint z( ) = ! logn z( )

nQ

"

# $

%

& '

qV (z) + µint z( ) = constant


Energy and Momentum TransferAverage on random directions of scatter. Energy transfer 1->2 Momentum transfer

Consequence : heat conduction + viscosity

�

!E =1

2m

! v '2

2

"! v 2

2

2=1

2m

! v 1

2

"! v 2

2

2

�

!! p = m

! v '2"! v 2

2= m

! v 1"! v 2

2


Thermal ConductivityConsider a medium in local thermal equilibrium but with a thermal gradient along

z. Diffusion will transport energy from hotter region to cooler regions:Consider a particle 1 which just has been scattered: its initial velocity is v1 and angle θ,ϕ. At the

next collision with particle 2 after path s, it will transfer in average

if particle 1comes from region of temperature T1 and particle 2 comes from a region oftemperature T2.

The mean energy transport along z per collision is

Taking into account the total number of collisions per unit timewe obtain the energy flux along z (averaged over v) is

thermal conductance where C is the heat capacityper unit volume

< Average energy transfer ! "z >=3

2kB

T1 # T2

2( )$ scos% e

#s

l ds

l

d cos%2

d&2'

with T1! T

2= !

"T

"z

#z = !"T

"z

scos$

!<Average energy transfer "#z >= -3

2kB

$T$z

% scos&( )2 e

's

l ds

l

d cos&2

d(2)

= -3

2kB

$T$z

l2

3v

l

JQz = !3

2nkB

"T

"z

lv

3 or ! J Q = !#

! $ T

! =3

2nk

B

lv

3= C

lv

3= CD

1

2m

v1

2 ! v2

2

2

"

#

$ $

%

&

' '

=3

2kB

T1! T

2

2

"

# $

%

& '


Thermal Conductivity (2)Heat equation

The local increase of temperature with time isBy same argument of energy conservation

=>

This is the diffusion equation again!

!T

!t=1

C"#

2T = D#

2T

!T

!t=1

C

!u

!t

!u

!t+

! " #! J Q = 0

C!T = !u where u is the energy density


Relation to Brownian motionExample of current fluctuation across a resistor

Let us consider a charge moving between 2 plates whose voltagesdiffer by V .

The conservation of energy implies

If the charge is moving randomly

Power spectrumHowever for calculation of noise through a circuit which has some

frequency dependence, we need to compute the noise as a functionof the frequency.

Called the power spectrum or the spectral density of the noise.

--------

+++++++

+

Electron transfer

qEdx = Vdq or

dq

dt= q

E

V

dx

dt=q

Lv x

L

E

!i =

q

Lv x and !i = 0 !i2

=q

L

"#$

%&'

2

v x

2 at a given time t

< !i2

"( ) > dv

Note: v is the velocity ! " the frequency


Johnson NoiseAs a model of a resistor we consider the same system as last slide with

N electrons

One electron moving a length s along the directionq produces a square pulse of length

and amplitude

Fourier transformFor each flight path between interactions

where the factor 2 comes from the fact for the spectral density wecombine positive and negative frequencies ≠ Fourier transform

--------

+++++++

+

Electron transfer

L

E

area A

N = nLA

�

!t =s

v

!i =qv

Lcos"

!f v( ) = e!2i"#t

!$

$

% f t( )dt = e!2i"# (t+& t /2) sin "#&t( )

"#&i ' e!2i"# (t+& t /2)&t&i for small #

The modulus of !f v( ) at low frequency is &t&i and its phase is random

�

!i "( )int

= 0

!i2"( )

int

= 2 ˜ f v( )2

int

= 2 !t2!i

2

int

t t+δτ!if t( )

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