Diffusion Evolution of the distribution function Boltzmann equation velocity modulation

Transport propertiesq

( , )dm

j p r p' p'

v E

nej v ne E E

2meter /volt-second

j E

is the conductivity in mhos/meter or siemens/meter

Find the conductivity of a resistor

j E

0VI

area A

length L

conductivity

0VI

A L

0

IL

AV

Conductivity of a plasma

12 3

edv

m qEdt

cmax 0

e

qEv dt

m

ce

qE

m

average velocity ce

qE

2m c

e

qmobility

2m

e econductivity n q2

ec

e

n q

2m

Diffusion

R

C

I( z,t )

z

V( z,t )RI( z,t )

z

R

V( z,t )C

t

Diffusion2

2

V( z,t ) V( z,t )RC

z t

1D

RC

meter

ohms farads

2

meter

second

2

1

Dohms/meter farads/meter

2

z RClet

t

( ) ( , )U tV z t

2

2

( , ) ( , )V z t V z tRC

z t

2

2

1 ( , )V z t

RC z

22 2

2 2

1 ( , ) 1

2

V z t d U RC

RC z dRC t t

22

2

1 d U

zdRC t

2

z RC

t

( )( , )

UV z t

t

( , )V z t

t

3

2

1

2

U dU

d ttt

3 32 2

( , ) 1

2 2

V z t U dU z RC

dt tt t

2

z RC

t

( )( , )

UV z t

t

3 32 2

( , ) 1

2 2

V z t U dU z RC

dt tt t

32

( , ) 1

22

V z t U dU

d tt tt

( , )V z t

t

3

2

1

2

U dU

d ttt

22

2 32

1 1

22 2

d U RC U dU

d tdRC t t tt

2

2

1 ( , ) ( , )V z t V z t

RC z t

2

2

d Ud U

dd

2

2

d U dUU

dd

2

2

d Ud U

dd

constant - Ud U

Ud

dU

Ud

d

lncabin

cabincabin

2

ln( )2

U

2

2U e

2

2U e

2

z RC

t

( ) ( , )U tV z t

2

4( , )

RCz

toVV z t et

-5 0 50

0.2

0.4

0.6

0.8

1de

nsity

z

t = 1t = 2t = 3t = 4t = 5

2

0( , )4

RCzV z t V erf

t

2

0( , )4

RCzV z t V erfc

t

1erfc erf

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1de

nsity

z

t = 1t = 2t = 3t = 4t = 5

Diffusion

x

area A

2

2

n( x,t ) 1 n( x,t )

Dx t

Harmonic oscillatorhttp://www.kettering.edu/~drussell/Demos/SHO/mass.html

2202

d xx 0

dt

0 0x( t ) x cos t

dx( t )v( t )

dt 0 0 0x sin t

Maxwell Boltzmann distribution

2

Maxwell Boltzmann 3BB

1 pf exp

2m T2 m T

Boltzmann equationhttp://grus.berkeley.edu/~jrg/ay202/node32.html

Ludwig Eduard Boltzmann (1844-1906). The physicist whose greatest achievement was in the development of statistical mechanics, which explains and predicts how the properties of atoms determine the visible properties of matter such as viscosity, thermal conductivity, and diffusion. He is reputed to have smuggled wine into the Faculty Club during his 1904 visit to Berkeley--at that time Berkeley was a dry town

Boltzmann equation

is known at a time tf ( x,v,t ) dN( x,v,t )

f ( x x,v v,t t ) dN( x x,v v,t t ) f ( x v t ,v a t ,t t )

Boltzmann equation

f ( x,v,t )

?t

f f ( x,v,t )

xv

f

t

collisions

f f f dfFv mt x v dt

v Fm

f dv

v dt

f dx

x dtcollisions

Water bag distribution

2

Maxwell Boltzmann 3BB

1 pf exp

2m T2 m T

Boltzmann equation

collisions

f f f dfFv mt x v dt

f f

v 0t x

0f f ( x vt ) n fdv

Velocity modulationhttp://www2.slac.stanford.edu/vvc/accelerators/klystron.html

Velocity modulation

2e 0 1

1m v q V V sin t2

0 1e

2qv V V sin t

m

0 1

e 0

2qV Vv 1 sin t

m V

0 1

e 0

2qV Vv 1 sin t

m 2V

0V 1V

Velocity modulationhttp://www.google.com/search?hl=en&client=firefox-a&rls=org.mozilla%3Aen-

US%3Aofficial&hs=aBD&q=Applegate+diagram&btnG=Search

0 1

e 0

2qV Vv 1 sin t

m 2V

Phase space interpretation of velocity modulation

Velocity modulation animation

0 1

e 0

2qV Vv 1 sin t

m 2V

Velocity modulationhttp://www2.slac.stanford.edu/vvc/accelerators/klystron.html

Velocity modulationhttp://www.answers.com/topic/klystron

“bump on a tail distribution”

2

Maxwell Boltzmann 3BB

1 pf exp

2m T2 m T

Summary of physics principles

• Waves and particles are related – photo electric effect

Summary of physics principles

• Schrödinger equation gives us empty states and quantum numbers

• Fermi function tells us if the state is filled• Boltzmann equation describes how the

distribution of states evolves in space and time

• Velocity modulation helped win World War II and demonstrates the Boltzmann equation

Energy bands• With more than one atom, one has to

inquire about possible interaction between individual particles.

• Think of two race cars (or two witches) – the one behind uses less energy if it is following very closely behind the first one.

• Splitting of individual energy levels yielding a band.

2 21 1m v v mv2 2

mv v

Energy bands• Probability density functions from two

adjacent atoms in close juxtaposition causes interaction and splitting of the lowest state yielding a band.

separated close together

band

Energy bands• Valence band – the top most energy band

containing electrons

• Conduction band – the energy band just above the valence band

• Electrons in the conduction band can move from one location to another

• Why did the chicken cross the road? • Aristotle: It is the nature of chickens to cross roads. • Issac Newton: Chickens at rest tend to stay at rest,

chickens in motion tend to cross roads. • Albert Einstein: Whether the chicken crossed the road or

the road moved beneath the chicken depends on your frame of reference.

• Werner Heisenberg: We are not sure which side of the road the chicken was on, but it was moving very fast.

• Wolfgang Pauli: There already was a chicken on this side of the road.