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Ch. 10. Statistical PhysicsCh. 10. Statistical Physics

이 병 호서울대 전기공학부byoungho@snu.ac.kr

Seoul National University NRL HoloTech

Statistical PhysicsStatistical 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

fermions (spin=1/2, 3/2,…: 예: electrons)

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p.335a

Seoul National University NRL HoloTech

Statistical PhysicsStatistical Physics

확률

경우의수

Most probable configuration

예: Random walk problem

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MaxwellMaxwell

James Clerk Maxwell(1831-1879)

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BoltzmannBoltzmann

Ludwig Boltzmann(1844-1908)

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Fig. 10-1, p.337

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Seoul National University NRL HoloTech

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 ConfigurationMost Probable Configuration

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( )

( )

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 ConfigurationMost Probable Configuration

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0 0

exp

( ) ( ) ( )

( ) ( ) ( )

iMB

B

i i MB MB

i MB

Ef Ak T

n g f n E dE g E f E dENN n n E dE g E f E dEV

∞ ∞

⎛ ⎞= −⎜ ⎟

⎝ ⎠= → =

= → = =∑ ∫ ∫

Maxwell-Boltzmann DistributionMaxwell-Boltzmann Distribution

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Fig. 10-2, p.339

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Fig. 10-4, p.341

Maxwell’s Speed DistributionMaxwell’s Speed Distribution

3/ 2 224( ) exp

2 2B B

N m mvn v dv v dvV k T k Tπ

π⎛ ⎞ ⎛ ⎞

= −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

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Fig. 10-5, p.341

2

2

2

22

0

12

( ) ( ) ( ) ( ) exp2

( ) ( ) 4

( ) ( ) 4 exp2

( )

MBB

B

E mv

mvn E dE g E f E dE g E A dEk T

g E dE f v dv C v dv

mvn E dE n v dv A v dvk T

N n v dvV

π

π

∞

=

⎛ ⎞= = −⎜ ⎟

⎝ ⎠= =

⎛ ⎞= = −⎜ ⎟

⎝ ⎠

= ∫

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Maxwell’s Speed DistributionMaxwell’s Speed Distribution

0

2

( ) 8/

3

3

B

B

Brms

vn v dv k TvN V mk Tvmk Tvm

π

∞

= =

=

=

∫

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Equipartition of EnergyEquipartition 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 Thas an average energy of kBT/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.

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노벨상과 노벨상의 사이노벨상과 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.