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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)





확률

경우의수

Most probable configuration

예: Random walk problem

Based on …Based on …





James Clerk Maxwell(1831-1879)

Ludwig Boltzmann(1844-1908)

Maxwell and BoltzmannMaxwell and Boltzmann





Example





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





( )

( )

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





( )

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





Example





[ ]

( )

( )

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





( )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





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.





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





그림

9.4 73K의 산소, 273K의 산소, 그리고

273K의

수소에서의

분자

속력분포.

Maxwell’s Speed Distribution Maxwell’s Speed Distribution

Seoul National University - Ch. 9. Statistical MechanicsSeoul National University Seoul National...

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