Ch. 9. Statistical Mechanics Ch. 9. Statistical Mechanics
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
확률
경우의 수
Most probable configuration 예 : Random walk problem
Based on …
Based on …
James Clerk Maxwell (1831-1879)
Ludwig Boltzmann (1844-1908)
Maxwell and Boltzmann
Maxwell and Boltzmann
Example
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
1
i
i i i
i i i
i i
E i
B
Q f h
N N N
n n n n n
N N N
N E N e e
k T
β α
α β
α β
β
∂ + ∂ + ∂ =
∂ ∂ ∂
≈ − >>
∂ −
− + + =
∂
=
= −
Most Probable Configuration
Most Probable Configuration
( )
0 0
exp
( ) ( ) ( )
( ) ( ) ( )
( / )
i MB i
B
i i MB MB
i MB
f E 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
N V N
∞ ∞
⎛ ⎞
= ⎜ − ⎟
⎝ ⎠
= → =
= ∑ → = ∫ = ∫
경우에 따라서는 대신 사용
Maxwell-Boltzmann Distribution
Maxwell-Boltzmann Distribution
Example
[ ]
( )
2 2
2
3/ 2
3/ 2
0 0
( ) ( ) ( ) ( ) exp
1
2 2
( )
( ) ( ) 2
( ) exp
( ) exp
2
MB
B
B
B B
n E dE g E dE f E Ag E E dE k T E mv p
m g p dp Bp dp
g E dE g p dp m B EdE n E dE C E E dE
k T
E C
N n E dE C E dE k T
k T π
∞ ∞
⎛ ⎞
= = ⎜ − ⎟
⎝ ⎠
= =
=
= =
⎛ ⎞
= ⎜ − ⎟
⎝ ⎠
⎛ ⎞
= = ⎜ − ⎟ =
⎝ ⎠
∫ ∫
Ideal Gas
Ideal Gas
( )
3/ 2 3/ 20 0
( ) 2 exp
3 2 3 2
B B
B
B
N E
E En E dE E dE
k T k T Nk T
E k T
π π
∞ ∞
⎛ ⎞
= = ⎜ − ⎟
⎝ ⎠
=
=
∫ ∫
Ideal Gas Ideal Gas
Total energy of N gas molecules
Average molecular
energy
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
2
3/ 2 2
2
2
1 2
( ) 4 exp
2 2
3
B B
B rms
E mv
dE mvdv
m mv
n v dv N v dv
k T k T
v v k T
m
π π
=
=
⎛ ⎞ ⎛ ⎞
= ⎜ ⎟ ⎜ − ⎟
⎝ ⎠ ⎝ ⎠
= =
Maxwell’s Speed Distribution
Maxwell’s Speed Distribution
Maxwell’s Speed Distribution
Maxwell’s Speed Distribution
