Upload
others
View
1
Download
0
Embed Size (px)
Citation preview
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
Phys 112 (S2006) 9 Kinetic theory 2 B. Sadoulet
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
"
#
Phys 112 (S2006) 9 Kinetic theory 3 B. Sadoulet
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"
Phys 112 (S2006) 9 Kinetic theory 4 B. Sadoulet
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
$µ
Phys 112 (S2006) 9 Kinetic theory 5 B. Sadoulet
How do systems come in equilibrium?Energy transfer
If 2 gases at different temperatures are put in contact, moleculesof the hotter gas have in average higher energy and transfer netenergy to the lower temperature gas => temperature equilibrium.
Energy transport by diffusion. Not instantaneous!=> thermal conductivityHeat transfer equation
Similarly, transfer between wall of gas enclosure and gas.=>e.g., black body radiation: equilibrium between walls and photons
inside cavity
Momentum transferif shear between 2 fluid volumes
related to viscosity (see later)
Particle transferIf a gas system 1 is put in contact with another gas system 2 where
the concentration of gas molecules is lower , the higher density insystem 1 will favor diffusion of molecules to system 2 =>concentration equilibrium
=> Diffusion equation
Phys 112 (S2006) 9 Kinetic theory 6 B. Sadoulet
Scattering Mean Free PathInteraction cross section
Consider a beam of particles incident on a targetProbability of interaction in a slab of thickness dz =
Cross section : dimension = areaExample: hard spheres
Mean free pathProbability of interaction in interval dz
Survival probabilityParticle enters medium at z=0. l is the attenuation length
! n dz
!
! = "d2
dl =
1
!n
dz
dz
l
N z + dz( ) = N z( ) 1!dz
l
"#$
%&'
( 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
Phys 112 (S2006) 9 Kinetic theory 7 B. Sadoulet
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
Phys 112 (S2006) 9 Kinetic theory 8 B. Sadoulet
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
Phys 112 (S2006) 9 Kinetic theory 9 B. Sadoulet
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
Phys 112 (S2006) 9 Kinetic theory 10 B. Sadoulet
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 $d! S
S
"
! J !d! S
S
" =
! # !
V
"! J d3x
!n
!t+
! " #! J = 0
! J = !D
! " n
!n
!t= D"
2n
1
c2
!2A
!t2= "
2A
Phys 112 (S2006) 9 Kinetic theory 11 B. Sadoulet
Drift in an Electric FieldCharged Particles (electrons, holes, ions)Drift Velocity
Consider an electric field along the z axis. In addition to its random velocity, eachparticle will acquire a net velocity in z direction from acceleration betweencollisions
It is advantageous therefore to work with the time δt before the next collisioninstead of s.
If the collision time τc=l/v is constant, the probability of collision between δt and δt+dδt is
Mean displacement between collisions:
Each particle undergoes v/l collisions per unit time =>
=> Averaging on random velocities
Note: this strictly applies to case where collision time τc=l/v is constant. Otherwise
E
�
!v z =qE
m!t
�
!"z = v cos#"t +1
2
qE
m"t2
�
!zcoll.
= e"!t# c$ 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
!"# $#d"t
#c
=acceleration x τc
Phys 112 (S2006) 9 Kinetic theory 12 B. Sadoulet
Drift in an Electric Field (2)Mobility
Constant τc=l/v
Ohm’s lawConsider a wire of length L and sectional area A. If the wire is thin enough
! w = ˜ µ
! E
˜ µ =q
m!c
E =V
L where V is the applied voltage
I = nAqw = nAq
2
m! c
V
L"V =
L
A
1
nq2
m! c
conductivity #c
!"#
IL
A
Phys 112 (S2006) 9 Kinetic theory 13 B. Sadoulet
Einstein RelationConstant collision time
Consider the ratio
⇒for constant τc=l/v
General caseIf the field is low enough for the particle to remain in thermal
equilibrium at temperature τ = kBT , it can be easily shown byintegrating by part the integral giving <lv> that
We then still have
qD
!µ=m
! c
lv
3=
m
! c
!cv2
3
for constant !c = l/v" #$ %$
=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
Phys 112 (S2006) 9 Kinetic theory 14 B. Sadoulet
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
Phys 112 (S2006) 9 Kinetic theory 15 B. Sadoulet
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
Phys 112 (S2006) 9 Kinetic theory 16 B. Sadoulet
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
"
# $
%
& '
Phys 112 (S2006) 9 Kinetic theory 17 B. Sadoulet
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
Phys 112 (S2006) 9 Kinetic theory 18 B. Sadoulet
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
Phys 112 (S2006) 9 Kinetic theory 19 B. Sadoulet
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( )