PHY 770 Spring 2014 -- Lecture 20 14/08/2014

PHY 770 -- Statistical Mechanics12:00* - 1:45 PM TR Olin 107

Instructor: Natalie Holzwarth (Olin 300)Course Webpage: http://www.wfu.edu/~natalie/s14phy770

Lecture 20

Chap. 9 – Transport coefficients The Boltzmann equation Treatment of the collision term Examples

*Partial make-up lecture -- early start time

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3 3

3 3

represents the number of particles in the6 dimensional

Define the distribution func

phase space a

tion ( , , ) :

( , , ) and at timebout .

( ,

, )

f t

f t dt

rd

f t

v

d rd v N

r vr v

r v

r v

The Boltzmann equation

( , , )r vcoll

f tm

ft t

Fv r v

PHY 770 Spring 2014 -- Lecture 20 64/08/2014

Analysis of collision term

coll added removed

ft

f ft t

Two-particle collision events

v1

v2’

v2

v1’

2 2 2 2

2 2 2 21 2 2 1 2 2

2

1 1 1 1

1

2

1

1 1

Conservation of momentum: ' '1 1 1 1Conservation of energy: ' '2 2 2 2

' '

m m m m

m v m v m v

g

m v

v v v v

v v v v

PHY 770 Spring 2014 -- Lecture 20 74/08/2014

Two-particle collision events

v1

v2’

v2

v1’2 2 2 2

2 2 2 21 2

1 1 1 1

1 1

1

2 1 2 2

2 21

' '1 1 1 1 ' '2 2 2 2

' '

m m m m

m v m v m m

g

v v

v v v v

v v v v

Assumptions• Only two-particle collisions considered• Only elastic collisions considered• Assume that force F does not effect collisions• Assume that distribution function f(r,v,t) is slowly varying

in r within collision volume• In a two-particle collision , the distribution functions for

the two particles are independent (uncorrelated)

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Analysis of collision term

Two-particle collision events

v1

v2’

v2

v1’ 3 31 2 1 2 1 2, ,' ' dv dv v v v v

Denotes the number of particles per unit time and per unit flux of particles 1 incident on particles 2 with initial velocities v1,v2 and final velocities v1’, v2’

Equivalent two-particle collision events

v1

v2’

v2

v1’

v1

v2’

v2

v1’v1

v2’

v2

v1’

1 2 1 2 1 2 1 2 1 2 1 2, , ,' , , ,' ' ' ' ' v v v v v v v v v v v v

PHY 770 Spring 2014 -- Lecture 20 94/08/2014

Analysis of collision term

bq

As discussed in Appendix E of your textbook, the scattering cross section describes the effective area cut out of the incident beam by the scattering process.

sin sin

lab

lab lab lab C

C

M

M

CM CM

d dbdb bdbd d d d

q q q q

Area of incident beam: 2 bdb

PHY 770 Spring 2014 -- Lecture 20 104/08/2014

Analysis of collision term

bq

Area of incident beam: 2 bdb

2 1

Volume of beam of particles 1 which will be scattered by particles 2per unit time: 2 bdbv v

coll added removed

ft

f ft t

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3 3 31 2 1 1 2 1 2 , 2 ' '', , ',

added

b db f t f t d v d vf d vt

r v r vv v

Analysis of collision term

3 3 31 2 1 1 2 1 2 2 , , , ,

removed

b db f t ff d v t d vt

d v r v r vv v

3 3 3 31 2 1 2

Since 2sin

Also note that ' '

CMCM CM

CM CM

CM CM

d dbdb b

d v

db dd d

d v d v d v

d

q q

32 2 2 2 , ', , ', , , , ,CM

CM

added removed

CM

f ft t

dd d vd

f t f t f t f t

r v r v r v r vv v

PHY 770 Spring 2014 -- Lecture 20 124/08/2014

( , , )r vcoll

f tm

ft t

Fv r v

Summary of results:

32 2 2 2 , ', , '

With:

, , , , ,CM

col

CM

l

MC

ft

dd d v

df t f t f t f t

v v r v r v r v r v

32 2 2

In the notation of your textbook:

( , ) , ', , ', , , , ,col

cml

CMf d d v gt

b g f t f t f t f t

r v r v r v r v

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Some rough estimates:

2 2

1 2 2

3 32 2 2

3

)/(2(m v )1 22 3 32 2

2

Number of collisions per second per unit volume for gas near equilibrium:

( ,

2

4

) ,

wh

,

ere 2

, ,CM cm

v kTmtot

tot

v b g fd d d v

m mn d d v e

n v v

t f t

vkT

v r v r v

v

v

v

C

2 when all particles have same mass kT mm

1 1Mean free path: 2 4 2

1 1Collision time: 4 2

tot

tot

n vn

nv v

C

PHY 770 Spring 2014 -- Lecture 20 144/08/2014

Properties of Bolzmann equation

( , , )r vcoll

f tm

ft t

Fv r v

Described the time evolution of the distribution of particles in six-dimensional phase space for a dilute gas. In the absence of external fields, the system should decay to equilibrium as shown by Boltzmann’s H Theorem:

3 3

3 3

Define: ( )

Can show that ( ) always decreases with collisi

( , , ) ln ( , , )

( , , ) ln ( , , ) 1

ons:

rH t d d

H t

d d

vf v t f v t

H f v tr v f v tt t

r r

r r

PHY 770 Spring 2014 -- Lecture 20 154/08/2014

Properties of Bolzmann’s H theorem continued (assume F=0)

3 3

3 3

( , , )

ln ( , , ) 1

ln ( , , ) 1

rcoll

coll

H fr v f tt t

fr v f t

d d

t

f t

d d

v r r v

r

v

v

3 3

32 2 2

ln ( , , ) 1

( , ) , ', , ', , , , ,CM cm

H r v f tt

d d

d d b g f t f t fv g t f t

r v

r v r v r v r v

2

3 32 2

32 2

:

ln ( , ,

Performing the same analysis but switching v

) 1

( , ) , ', , ', , , , , CM cm

H r v f tt

b g f t f t f t f t

d d

d d vg

r v

r v r v r v r

v

v

PHY 770 Spring 2014 -- Lecture 20 164/08/2014

Properties of Bolzmann’s H theorem continued (assume F=0)Adding the two expressions together and switching variables, we find:

3 3 3 22 2 2

2

( ,1 ' ' ln4 ' '

) CM cmffd d d v d g f f fH r v b g

tf

f f

Note that for all real positive , ln 0

Therefore 0

xy x y

Hy

t

x

This implies that at equilibrium:

2 2 0 and ' ' H ff f ft

PHY 770 Spring 2014 -- Lecture 20 174/08/2014

Conservation laws implied by Boltzmann equation

1 2 1 2

1 2 1

Consider some quantity (mass, momentum, etc) associated witha particle of velocity and position : ( , )If ( , ) is conserved during a collision ,

' ', ,

so that ( , ) ( , ) )'( ,

v r r vr v

r v r v r vv v v v

2

3

( , ), it is possible to show that:

( , ) 0

'

coll

fd vt

r v

r v

These results lead to identities involving the distribution function ( , , ) f tr v

PHY 770 Spring 2014 -- Lecture 20 184/08/2014

Approximate solutions to Boltzmann equation

23/

0

0 02

/(2 )

( , , ) ( , , ) 1 ( , , )For free particle moving in 3 dimensions in thermal equilibrium:

( , , ) ( )2

mv kT

k

f t f t h t

mfT

t f n e

r v r v r v

r v v

0

For 0 :

( , , )r rh

t tf t f h

F

v r v v v

32 2 2

0 3 02 2 2 2

( , ) , ', , ', , , , ,

( , ) , ', , ', , , ,

,

CM cm

CM c

col

m

l

b g f t f t f t ff d d v gt

f d d v g

t

b g h t h t h t h tf

r v r v r v r v

r v r v v r vv v r