1. Statistical Thermodynamics
Byeong-Joo Lee
POSTECH - MSE

2. 1. Stirling’s approximation
Warming Up – Mathematical Skills
1. Stirling’s approximation
2. Evaluation of the Integral
3. Lagrangian Undetermined Multiplier Method

3. Macroscopic vs. Microscopic
Basic Concept of Statistical Mechanics – Macro vs. Micro
View Point
Macroscopic vs. Microscopic
State
Macrostate vs. Microstate

4. Particle in a Box – Microstates of a Particle
for 66 : 8,1,1 7,4,1, 5,5,4

5. System with particles – Microstates of a System

6. Macrostate / Energy Levels / Microstates –

7. Scope and Fundamental Assumptions of Statistical Mechanics
▷(n1, n2, …, nk)로 정의되는 하나의 macrostate를 만들기 위해,
있을 수 있는 수많은 경우의 수 하나하나를 microstate라 한다.
▷ 어떠한 시스템에 가능한 (quantum mechanically accessible 한)
macrostate (하나하나가 (n1, n2, …, nk)로 정의되는)의 mental
collection을 ensemble이라 한다.
▷ 같은 energy level에서 모든 microstate의 실현 확률은 동등하다.
▷ Ensemble average는 time average와 같다.

8. Number of ways of distribution : in k cells with gi and Ei
▷ Distinguishable without Pauli exclusion principle
▷ Indistinguishable without Pauli exclusion principle
for gi with ni
▷ Indistinguishable with Pauli exclusion principle
for gi with ni

9. Evaluation of the Most Probable Macrostate – Boltzman

10. → Evaluation of the Most Probable Macrostate – B-E & F-D
Bose-Einstein Distribution →
Fermi-Dirac Distribution

11. Definition of Entropy and Significance of β
▷ Thermal contact 상태에 있는 두 부분으로 이루어진 Isolated System을 고려.
이에 대한 평형 조건은
Classical Thermodynamics에서는 maximum entropy (S)
Statistical mechanics에서는 maximum probability (Ω)
▷ S와 Ω는 monotonic relation을 가지며 →

12. Calculation of Macroscopic Properties from the Partition Function

13. Ideal Mono-Atomic Gas

14. Ideal Mono-Atomic Gas – Evaluation of k
for 1 mol of gas

15. Entropy – S = k ln W

16. Equipartition Theorem
The average energy of a particle per independent component of motion is
translational kinetic energy :
rotational kinetic energy :
vibrational energy :
kinetic energy for each independent component of motion has a form of

17. Equipartition Theorem
The average energy of a particle per independent component of motion is
※ for a monoatomic ideal gas :
for diatomic gases :
for polyatomic molecules which are soft and vibrate easily
with many frequencies, say, q:
※ for liquids and solids, the equipartition principle does not work

18. Einstein and Debye Model for Heat Capacity – Background & Concept
3N independent (weakly interacting) but distinguishable
simple harmonic oscillators.
for N simple harmonic vibrators
average energy per vibrator

19. Einstein and Debye Model for Heat Capacity – number density
Let dNv be the number of oscillators whose frequency lies
between v and v + dv
where g(v), the number of vibrators per unit frequency band,
satisfy the condition
The energy of N particles of the crystal

20. Einstein and Debye Model for Heat Capacity – Einstein
All the 3N equivalent harmonic oscillators have the same frequency vE
Defining Einstein characteristic temperature

21. Einstein and Debye Model for Heat Capacity – Debye
A crystal is a continuous medium supporting standing longitudinal
and transverse waves
set

22. Einstein and Debye Model for Heat Capacity – Comparison

23. Einstein and Debye Model for Heat Capacity – More about Debye
Behavior of
at T → ∞ and T → 0
at T → ∞
→ x2
at T → 0
: Debye’s T3 law

24. Einstein and Debye Model for Heat Capacity – More about Cp
for T << TF

25. Statistical Interpretation of Entropy – Numerical Example
A rigid container is divided into two compartments of equal volume by a partition. One compartment contains 1 mole of ideal gas A at 1 atm, and the other compartment contains 1 mole of ideal gas B at 1 atm.
(a) Calculate the entropy increase in the container if the partition between the two compartments is removed.
(b) If the first compartment had contained 2 moles of ideal gas A, what would have been the entropy increase due to gas mixing when the partition was removed?
(c) Calculate the corresponding entropy changes in each of the above two situations if both compartments had contained ideal gas A.