Statistical Mechanics (mostly equilibrium)

Statistical Mechanics – MEGR 7117/8117

In-Class and Prep Notes, Spring 2023

0) Einstein’s complaint and course approach

1) Statistical mechanics overview; derivation Boltzmann bridge relation [connecting number of quantum energy states available to an N particle, fixed mass, nominally fixed energy system to the system’s macroscale entropy, S], pt-1

2) Physical meaning of fundamental (Tds) thermodynamic relation; derivation of Boltzmann bridge relation pt 2, including proof that Boltzmann constant is fixed (i.e., independent of the equilibrium thermodynamic system under study)

3) Fundamental Planck-Einstein-de Broglie (PEdB) postulates of quantum mechanics; using PEdB relations to connect guessed system Hamiltonian to an associated system Schrodinger equation

4) First example of guessing a system Hamiltonian and using PEdB relations (connecting wave-like and particle-like properties of elementary objects) to derive system Schrodinger equation

5) Statistically independent dynamics in composite systems (e.g., atoms and molecules); system and subsystem Hamiltonians and Schrodinger equations

6) Choosing between quantum and classical statistical mechanical models – scaling and rules of thumb; high temperature and low density (classical) limits of fermion and boson ideal (noninteracting) quantum (indistinguishable particle) gasses

7) Geometry-based approximate calculation of number of quantum states (Omega) available to an N-particle, insulated, fixed mass ideal gas

8) Using approximate Omega to back out macroscopic thermodynamic properties of insulated, fixed mass ideal gas

8a) Recipe for determining quantum (eigen) energies for molecules, assuming statistically independent translation, rotation, vibration, and electron (random) dynamics; demonstration that partition function for molecule equals the product of translation, rotation, vibration, and electronic partition functions

9) Derivation of W [= number of ways of observing N replica fixed mass systems (in a canonical ensemble of fixed mass, energy exchanging systems), where any and all subsets of N systems can be found in any available quantum energy state]; derivation of the canonical distribution [=probability of observing our actual system in any of (the known, calculated) available system quantum energy states

10) Finish derivation of canonical distribution; highlight bridge relation connecting (microscale) system partition function and (macroscale) Helmholtz free energy (applicable to fixed mass, energy exchanging, interacting and noninteracting, classical and quantum systems)

11) Non-interacting N particle systems: Expressing (total, quantum) system energy, E, as sum n_j epsilon_j, where epsilon_j is the j^th single particle quantum energy level and n_j is the number of particles (in the system of N particles) in state epsilon_j; derivation of bridge relation (for fixed mass, energy exchanging systems), connecting system partition function and Helmholtz free energy

12) Derivation of Bose-Einstein distribution for non-interacting (ideal, low temperature, high density) gas of bosons, giving the average number of particles in each single-particle quantum state (where the set of quantum energies is obtained by guessing/specifying a single particle Hamiltonian, followed by solution of the associated Schrodinger equation)

13) Derivation of Boltzmann distribution giving average number of particles (in N particle, noninteracting system) occupying each available single particle quantum state (applicable when temperature high enough, density low enough that quantum dynamics are weak/hidden on interparticle length scale or mean free path length scale); details on how/why Bode-Einstein and Fermi-Dirac distributions collapse to Boltzmann under these conditions

14) Derivation of Fermi-Dirac distribution, giving average number of (noninteracting) fermions occupying each available single particle quantum state; details on why the BE and FD distributions collapse to Boltzmann under these limits

15) Photons in a heated box (black body) – calculation of the average number of photons (per volume) within a given wave number interval – set-up for derivation of Planck’s law

16) Photons in a heated box: Derivation of Planck’s law (giving average photon (gas) energy as function of photon frequency); calculation of photon (gas) pressure

17) Road map for deciding between and using classical and quantum statistical mechanics models of non-interacting systems

18) Quantum and statistical modeling of monatomic ideal gas, accounting for statistically independent nuclear and electronic (random) dynamics

19) More noninteracting systems: Connecting system, particle, and particle subsystem partition functions; moving from system to particle to particle subsystem Schrodinger equations

20) Interacting (solid state) systems: Phonons (collective, N-particle wave modes) – system Hamiltonian and diagonalization to form a system of 3N uncoupled Hamiltonians

21) Phonons: Derivation of system partition function; binning of neighboring quantum energy states to create approximate degeneracy and transform (partition function) sum into an integral; derivation of degeneracy formula giving number of wave modes in (solid) volume V having wave numbers in interval [k, k+dk]

22) Phonons: Debye’s degeneracy model allowing approximate evaluation of phonon system partition function

Lecture Notes (ca. 2015)

1) Introduction; overview of macroscopic equilibrium thermo: lecture1-macrothermo-review

2) Derivation of Gibb’s relations, Maxwell relations, generalized du:  lecture1-notes

3) More macroscopic thermo:  lect2-macro-review-cont

4) Fundamental postulates, part 1: lecture2-part2-notes

5) Fundamental postulates, part 2: lecture2-aside-estimate-probability-1-ptcl-total-energy-5

6) Fundamental postulates, part 3:  lecture3-generallization-fund-postulates-n-indep-particle-systems-6

7) Boltzmann’s attempt to connect micro- and macroscale equilibrium thermo:  lecture2-part1-notes-7

8) Derivation of Boltzmann entropy relation (ca 2015): lec4-derivation-boltzmann-relation-8

9) Relation between k and universal gas constant; rough repeat of Reif’s degree-of-freedom argument: lec4-2015-connectktor-9

10) Rough rederivation of Reif’s DoF argument; rough alternate derivation of number of quantum states in an N-particle ideal gas: lec4-rough-2015-reifdofpluspathrianoofstatesigas-11

11) Quantum mechanics overview, part 1: lecture-4-quantum-mech-overview-part1-12

 12) Quantum mechanics overview, part 2 (rough): lecture-4-quantum-mech-overview-rough-version-part1-13

 13) Ensemble overview; QM recipe; when to choose quantum versus classical model: ensemble-method-overview-quantum-mech-recipe-thermal-wavelength-2015-14

14) Derivation of canonical distribution: canonical-distbn-derivation-2015-16

15)  Rederivation of canonical distribution; overview of phase space of N-particle system:   lec9-2015-phase-space-deriavtion1-canonical-density-fn-17

16) Improved derivation of canonical distribution: canon-distbn-good-dervn-version1-2015-18

17) Derivation of A=kTlnQ; G=μN; S=k ∑ plnp; S = k ln Ω: dervnaktlnq-gmun-sksumprlnpr-20

18) Degenerate states; example degeneracy estimate for single particle in aa box (not  in Pathria): dervn-canon-dist-degenerate-energy-levels-22

19) Rigorous derivation of microcanonical distribution (not in Pathria): rigorous-dervn-microcanonical-distribution-22

20) On the equivalence of state density, g(E), and Ω:  eqivalence-of-ge-omega-23

19) Rigorous derivation of microcanonical distribution (not in Pathria): rigorous-dervn-microcanonical-distribution-22

20) Example calculation of g(E) for classical ideal gas (Laplace transform inversion):  laplace-inversion-ex-ge-aside-quantum-conversion-factor-24

21) Example calculation of partition function, Q, for classical ideal gas:  example-calc-partition-fn-classical-ideal-gas-25

22) Derivation of Liouville theorem: dervn-continuum-cons-eqns-via-kirkwood-liouville-theorem-26

23) Interacting particle systems; molecular-scale conservation laws: interacting-classical-particle-systems-molecular-scale-conservation-laws-part1-27

24) Molecular-scale ensemble-averaged momentum equation: derivn-ensemble-avg-molecular-scale-momentum-eqn-28

25) Appendix 1: Quantum energy states of a free particle in a box: lecture-5-quantum-energy-lstates-free-ptcl-29

26) Appendix 2: Details underlying Pathria’s derivation of quantum phase space volume: lecture-11-dervn-of-no-quantum-states-per-unit-vol-phase-space-30

27) Appendix 3: Combinatorics warm-up:  combinatorics-1-lec5-2015-31

28) Appendix 4: X and P bases in quantum mechanics: lecture-18-pt1-x-and-p-bases-in-qm-32

Send email to rkeanini@charlotte.edu