seoul national university - ch. 9. statistical mechanicsseoul national university seoul national...
TRANSCRIPT
Seoul National UniversitySeoul National UniversityCenter for Active Plasm onics
Application System s
Center for Active Plasm onics
Application System s
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
fermions (spin=1/2, 3/2,…: 예: electrons)
Seoul National UniversitySeoul National UniversityCenter for Active Plasm onics
Application System s
Center for Active Plasm onics
Application System s
Seoul National UniversitySeoul National UniversityCenter for Active Plasm onics
Application System s
Center for Active Plasm onics
Application System s
확률
경우의수
Most probable configuration
예: Random walk problem
Based on …Based on …
Seoul National UniversitySeoul National UniversityCenter for Active Plasm onics
Application System s
Center for Active Plasm onics
Application System s
James Clerk Maxwell(1831-1879)
Ludwig Boltzmann(1844-1908)
Maxwell and BoltzmannMaxwell and Boltzmann
Seoul National UniversitySeoul National UniversityCenter for Active Plasm onics
Application System s
Center for Active Plasm onics
Application System s
Example
Seoul National UniversitySeoul National UniversityCenter for Active Plasm onics
Application System s
Center for Active Plasm onics
Application System s
Seoul National UniversitySeoul National UniversityCenter for Active Plasm onics
Application System s
Center for Active Plasm onics
Application System s
1 2
1 1 2 2
1 21 2
1 2
!( , , , )! ! !
ln ( , , , ) entropy
n
n n
nn
n
N N N NU E N E N E N
NQ N N NN N N
Q N N N
= + + ⋅⋅⋅+= + + ⋅⋅⋅ +
⋅⋅⋅ =⋅⋅⋅
⋅ ⋅ ⋅
Most Probable Configuration Most Probable Configuration
Seoul National UniversitySeoul National UniversityCenter for Active Plasm onics
Application System s
Center for Active Plasm onics
Application System s
( )
( )
Lagrange's multiplier methodln 0
Stirling's approximationln ! ln for 1
ln0
1
i
i i i
i i ii
iE
i
B
Q f hN N N
n n n n n
N N NE
N
N e e
k T
βα
α β
α β
β
∂ ∂ ∂+ + =
∂ ∂ ∂
≈ − >>
∂ −− + + =
∂
=
= −
Most Probable Configuration Most Probable Configuration
Seoul National UniversitySeoul National UniversityCenter for Active Plasm onics
Application System s
Center for Active Plasm onics
Application System s
( )
0 0
exp
( ) ( ) ( )
( ) ( ) ( )
( / )
iMB i
B
i i MB MB
i MB
Ef E Ak T
n g f n E dE g E f E dENN n n E dE g E f E dEV
N V N
∞ ∞
⎛ ⎞= −⎜ ⎟
⎝ ⎠= → =
= → = =∑ ∫ ∫경우에따라서는 대신 사용
Maxwell-Boltzmann Distribution Maxwell-Boltzmann Distribution
Seoul National UniversitySeoul National UniversityCenter for Active Plasm onics
Application System s
Center for Active Plasm onics
Application System s
Example
Seoul National UniversitySeoul National UniversityCenter for Active Plasm onics
Application System s
Center for Active Plasm onics
Application System s
[ ]
( )
( )
22
2
3/ 2
3/ 2
0 0
3/ 2
( ) ( ) ( ) ( ) exp
12 2
( )
( ) ( ) 2
( ) exp
( ) exp2
2( ) exp
MBB
B
BB
BB
En E dE g E dE f E Ag E dEk T
pE mvm
g p dp Bp dp
g E dE g p dp m B EdE
En E dE C E dEk T
E CN n E dE C E dE k Tk T
N En E dE E dEk Tk T
π
ππ
∞ ∞
⎛ ⎞= = −⎜ ⎟
⎝ ⎠
= =
=
= =
⎛ ⎞= −⎜ ⎟
⎝ ⎠⎛ ⎞
= = − =⎜ ⎟⎝ ⎠⎛ ⎞
= −⎜ ⎟⎝ ⎠
∫ ∫
Ideal Gas Ideal Gas
Seoul National UniversitySeoul National UniversityCenter for Active Plasm onics
Application System s
Center for Active Plasm onics
Application System s
( )3/ 2
3/ 20 0
2( ) exp
3232
BB
B
B
N EE En E dE E dEk Tk T
Nk T
E k T
ππ
∞ ∞ ⎛ ⎞= = −⎜ ⎟
⎝ ⎠
=
=
∫ ∫
Ideal Gas Ideal Gas
Total energy of N gas molecules
Average molecular energy
Seoul National UniversitySeoul National UniversityCenter for Active Plasm onics
Application System s
Center for Active Plasm onics
Application System s
Equipartition of Energy Equipartition of Energy
2
2 2 2
1 32 21 1 1 12 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 kB T/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 subject to the equipartition of energy.
Seoul National UniversitySeoul National UniversityCenter for Active Plasm onics
Application System s
Center for Active Plasm onics
Application System s
2
3/ 2 22
2
12
( ) 4 exp2 2
3B B
Brms
E mv
dE mvdv
m mvn v dv N v dvk T k T
k Tv vm
ππ
=
=
⎛ ⎞ ⎛ ⎞= −⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
= =
Maxwell’s Speed Distribution Maxwell’s Speed Distribution
Seoul National UniversitySeoul National UniversityCenter for Active Plasm onics
Application System s
Center for Active Plasm onics
Application System s
그림
9.4 73K의 산소, 273K의 산소, 그리고
273K의
수소에서의
분자
속력분포.
Maxwell’s Speed Distribution Maxwell’s Speed Distribution