Statistical Physics Statistical Physics
Maxwell-Boltzmann Distribution
- Classical statistics: distinguishable particles
Bose-Einstein Distribution
- QM statistics: indistinguishable particles
bosons (integer spin:
예– photons)
Fermi-Dirac Distribution
- QM statistics: indistinguishable particles
p.335a
Statistical Physics Statistical Physics
확률
경우의 수
Most probable configuration
예: Random walk problem
Maxwell Maxwell
James Clerk Maxwell
(1831-1879)
Boltzmann Boltzmann
Ludwig Boltzmann
(1844-1908)
Fig. 10-1, p.337
1 2
1 1 2 2
1 2
1 2
1 2
( , , , ) !
! ! !
ln ( , , , ) entropy
n
n n
n
n n
N N N N
U E N E N E N Q N N N N
N N N Q N N N
= + + ⋅⋅⋅ +
= + + ⋅⋅⋅ +
⋅⋅⋅ =
⋅⋅⋅
⋅⋅⋅
Most Probable Configuration
Most Probable Configuration
( )
( )
Lagrange's multiplier method
ln 0
Stirling's approximation
ln ! ln for 1
ln 0
i
i i i
i i i
i i
E i
Q f h
N N N
n n n n n
N N N
N E N e e
α βα β
α β
∂ + ∂ + ∂ =
∂ ∂ ∂
≈ − >>
∂ −
− + + =
∂
=
Most Probable Configuration
Most Probable Configuration
0 0
exp
( ) ( ) ( )
( ) ( ) ( )
i MB
B
i i MB MB
i MB
f A E
k T
n g f n E dE g E f E dE
N n N n E dE g E f E dE
V
∞ ∞
⎛ ⎞
= ⎜ − ⎟
⎝ ⎠
= → =
= ∑ → = ∫ = ∫
Maxwell-Boltzmann Distribution
Maxwell-Boltzmann Distribution
Fig. 10-4, p.341
Maxwell’s Speed Distribution Maxwell’s Speed Distribution
3/ 2 2
4
2( ) exp
2
B2
BN m mv
n v dv v dv
V k T k T
π
π
⎛ ⎞ ⎛ ⎞
= ⎜ ⎟ ⎜ − ⎟
⎝ ⎠ ⎝ ⎠
2
2
2
2 2
1 2
( ) ( ) ( ) ( ) exp
2
( ) ( ) 4
( ) ( ) 4 exp
2
MB
B
E mv
n E dE g E f E dE g E A mv dE k T g E dE f v dv C v dv
n E dE n v dv A v mv dv k T π
π
=
⎛ ⎞
= = ⎜ − ⎟
⎝ ⎠
= =
⎛ ⎞
= = ⎜ − ⎟
⎝ ⎠
Maxwell’s Speed Distribution Maxwell’s Speed Distribution
0
2
( ) 8
/ 3
3
B
B
B rms
vn v dv k T
v N V m
v k T
m v k T
m
π
∞
= =
=
=
∫
Equipartition of Energy Equipartition of Energy
2
2 2 2
1 3
2 2
1 1 1 1
2 2 2 2
B
x y z B
m v K k T
m v m v m v k T
= =
= = =
A classical molecule in thermal equilibrium at temperature T has an average energy of k
BT/2 for each independent mode of motion or so-called degree of freedom.
Each variable that occurs squared in the formula for the
energy of a particular system represents a degree of freedom
노벨상과 노벨상의 사이 노벨상과 노벨상의 사이
Eugene Commins in 1993 ---
I have no definite philosophy of teaching, but after more than thirty years of experience I have learned a few things, most of them quite obvious.
The first and most important is that if one is to explain something clearly, one must first understand it.
The second is that students should be vigorously encouraged to play an active rather than passive role in their own education. Over the last thirty years I have observed among Berkeley students (especially
undergraduates) an unfortunate and increasing tendency toward passivity, the desire to be "entertained."
This may have something to do with the pervasiveness of television. It is regrettable that we occasionally pander to this tendency by rewarding teaching that is essentially nothing but show business. Perhaps we do this out of defensiveness to the repeated accusation that the Berkeley faculty ignores undergraduate teaching.
The last has to do with the training of graduate students in research. In my view, the most effective way to do this is by example, from day to day. I try to work together with my research students on perplexing questions and unsolved problems, and I do not like to ask them to undertake anything in the laboratory, however arduous, that I am not prepared to do myself. I like to think that in my laboratory they may learn general values, such as intellectual honesty, perseverance, and courage in the face of adversity, as well as specific technical and professional skills.
