Byeong-Joo Lee POSTECH - MSE calphad@postech.ac.kr

1. Microstructure Evolution • Statistical • Thermodynamics Byeong-Joo Lee POSTECH - MSE calphad@postech.ac.kr

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

3. 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 n = 1, 2, 3, … 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 ▷ Microstate: each of the possible states for a macrostate. (n1, n2, …, nk)로 정의되는 하나의 macrostate를 만들기 위해, 있을 수 있는 수많은 경우의 수 하나하나를 microstate라 한다. ▷ Ensemble: mental collection of macrostates 어떠한 시스템에 가능한 (quantum mechanically accessible 한) macrostate (하나하나가 (n1, n2, …, nk)로 정의되는)의 mental collection을 ensemble이라 한다. ▷ Each microstate is equally probability. 같은 energy level에서 모든 microstate의 실현 확률은 동등하다. ▷ Ensemble average =time average

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– Boltzmann

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 ▷ Consider an Isolated System composed of two part in Thermal contact. @ equilibrium in Classical Thermodynamics: maximum entropy (S) in Statistical mechanics: maximum probability (Ω) ▷ There exists a monotonic relation between S and Ω →

12. Calculation of Macroscopic Properties from Partition Function

13. Ideal Mono-Atomic Gas

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

15. Entropy – S = kln 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 & 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 & 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 & Debye Model for Heat Capacity – Einstein All the 3N equivalent harmonic oscillators have the same frequency vE Defining Einstein characteristic temperature

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

22. Einstein & Debye Model for Heat Capacity – Comparison

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

24. Einstein & 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.