1 PHY 770 Statistical Mechanics 1200 145 P M TR Olin 107 Instructor Natalie Holzwarth Olin 300 Course Webpage httpwwwwfuedunatalies14phy770 Lecture 19 Chap 9 Transport coefficients ID: 643977
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PHY 770 -- Statistical Mechanics12:00* - 1:45 PM TR Olin 107Instructor: Natalie Holzwarth (Olin 300)Course Webpage: http://www.wfu.edu/~natalie/s14phy770
Lecture 19Chap. 9 – Transport coefficients“Elementary” transport theoryThe Boltzmann equation
*
Partial make-up lecture -- early start time Slide2
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What is transport theory?Mathematical description of the averaged motion of particles or other variables through a host medium.Examples of transport parametersThermal conductivityElectrical conductivityDiffusion coefficientsViscositySlide4
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Simple transport theoryBase system – low density gas near thermal equilibrium;assume that the interaction energy is negligible compared to the kinetic energy of the particles.Slide5
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Estimation of mean free path for hard spheresAA
A
A
B
B
Collision radius
d
AB
A
BSlide6
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Estimation of mean free path for hard spheresSlide7
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d
AA
l
When there is only one type of particle (
A
):Slide8
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Self-diffusion
z
q
f
Assume that, in addition to geometric factors, the particle will reach the detector only if it does not have a collision.
rSlide9
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Self-diffusion
z
q
f
rSlide10
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The Boltzmann Equation(Additional reference: Statistical Mechanics, Kerson Huang)Assume a dilute gas of N particles of mass m in a box of volume V. In order to justify a classical treatment:Slide11
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The Boltzmann equation – continuedIn absence of collisions, the distribution of particles remains constant as
v
r
t
t+
d
tSlide12
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The Boltzmann equation – continuedSlide13
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The Boltzmann equation – continuedSlide14
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Digression on two particle scattering theory (see Appendix E)
r
1
r
2
z
x
ySlide15
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Review of scattering analysis from classical mechanics class:Slide16
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Scattering theory:Slide17
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Figure from Marion &
Thorton
, Classical Dynamics
bSlide18
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Differential cross sectionSlide19
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Note: The following is in the center of mass frame of reference.In laboratory frame: In center-of-mass frame:
V
1
m
1
m
target
v
1
m
origin
v
CM
r
Also note: We are assuming that the interaction between particle and target
V(r)
conserves energy and angular momentum.Slide20
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E
r
min
In center of mass reference frame:Slide22
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r
min
r
(f)
fSlide28
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Evaluation of scattering expression:Slide29
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Relationship between scattering angle q and impact parameter b for interaction potential V(r):Slide30
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Hard sphere scatteringSlide31
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The results above were derived in the center of mass reference frame; relationship between normal laboratory reference and center of mass: Laboratory reference frame: Before After
u
1
u
2
=0
v
1
v
2
y
z
m
1
m
2
Center of mass reference frame:
Before After
U
1
U
2
V
1
V
2
q
m
1
m
2
qSlide32
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Relationship between center of mass and laboratory frames of reference
V
1
V
CM
v
1
y
q
U
1
u
1
V
CMSlide33
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Relationship between center of mass and laboratory frames of reference -- continuedSlide34
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V
1
V
CM
v
1
y
q
Relationship between center of mass and laboratory frames of reference
For elastic
scatteringSlide35
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Digression – elastic scattering
Also note:Slide36
v
1
V
1
VCM
y
q
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Relationship between center of mass and laboratory frames of reference – continued (elastic scattering)Slide37
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Differential cross sections in different reference framesSlide38
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Differential cross sections in different reference frames – continued:Slide39
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Example: suppose m1 = m2Slide40
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Back to Boltzmann equation: If we can assume that the collisions are due to binary interactions, such that particles 1 and 2 interact: