ELEMENTS OF STATISTICAL THERMODYNAMICS AND QUANTUM THEORY

1. ELEMENTS OF STATISTICAL THERMODYNAMICS AND QUANTUM THEORY • STATISTICAL MECHANICS OF INDEPENDENT PARTICLES • ▪ Macrostates versus Microstates • ▪Phase Space • ▪Quantum Mechanics Considerations • ▪ Equilibrium Distributions for Different Statistics • THERMODYNAMIC RELATIONS • ▪Heat, Work and Entropy • ▪Lagrangian Multipliers • ▪Entropy at Absolute Zero Temperature • ▪ Macroscopic Properties in Terms of the Partition Function

2. IDEAL MOLECULAR GASES • ▪ Monatomic Ideal Gases • ▪Maxwell’s Velocity Distribution • ▪Diatomic and Polyatomic Ideal gases • STATISTICAL ENSEMBLES AND FLUCTUATIONS • BASIC QUANTUM MECHANICS • ▪ Schrödinger Equation • ▪A Particle in a Potential Well or a Box • ▪Atomic Emission and Bohr Radius • ▪ Harmonic Oscillator • EMISSION AND ABSORPTION OF PHOTONS BY MOLECULES OR ATOMS • ENERGY, MASS AND MOMENTUM IN TERMS OF RELATIVITY

3. 3N positions and 3N momenta • Molecular dynamics simulation computer simulation technique where the time evolution of a set of interacting atoms (N) is followed by integrating their equations of motion statistical mechanics method for 6N-dimensional phase space link between the microscopic behavior and static and dynamic properties averaging → thermodynamic properties transport properties

4. Thermodynamic properties Temperature: Internal Energy: Pressure:

5. Intermolecular potential models Ne, Ar, Kr, Xe : Lennard-Jones (12-6) Pair two-body potential, inert gas, Van der Waals bond Water : ST2, SPC/E, TIP4P, CC Pair two-body potential, polarization Si, C : SW, Tersoff, Simplified Brenner three-body potential, covalent bond Metals : EAM, FS, SC two-body potential, embeded atom, electron cloud Soft Sphere Model

6. Exponent for Repulsion • Little Theoretical Basis • Due to Mathematical Convenience • Exponent for Cohesion (Attraction) • van der Waals Force • Keesom Force + Debye Force + London Dispersion Force • Good for Closed Shell Systems (Ar, Kr) Lennard-Jones Potential

7. Statistical mechanics Equilibrium distribution of certain types of particles (molecules, electrons, photons, phonons) in the velocity space • Kinetic theory Nonequilibrium processes, microscopic description of transport phenomena • Statistical methods

8. Statistical Mechanics of Independent Particles • Classical vs. statistical thermodynamics • independent particles • their energies: independent of each other • total energy: sum of the energies of individual particles

9. . . . ei, Ni . . . e2, N2 e1, N1 e0, N0 • Macrostates versus Microstates V, N, U volume, V number of particles, N internal energy, U constraints

10. macrostate: corresponds with a given set of • numerical values of N1, N2, …, Ni, …, and thus • satisfies the two constraints specified by the number of particles in each of the energy levels of the system • microstate: specified by the number of particles in • each energy state • degeneracy: the number of quantum states for a • given energy level. • thermodynamic probability • Number of microstates for each macrostate: number of • ways in which we can choose Ni’s from N particles

11. excited states ground state Energy level → quantum states Degeneracy macrostate microstate : (1), (2 1 0), (0 1 0 1 0)

12. constraints Example : Consider 3 particles, labeled A, B, and C Equilibrium state

13. a six-dimensional space formed by three coordinates for the position and three coordinates of the momentum or velocity elemental volume dV in Cartesian position space y dy dz dx x z • Phase Space

14. rsinq elemental volume dV in spherical position space dq dAn q df f

15. element in Cartesian velocity (or momentum) space vy for momentum dvy dvz dvx vx vz

16. element in spherical velocity (or momentum) space vsinq dq dan q df f

17. concept: a trajectory to include not only positions but also particle momenta plotting the positions and momenta of N particles in a 6N-dimensional hyperspace • Phase space trajectories phase space: 3N-dimensional configuration space and 3N-dimensional momentum space At one instant, the positions and momenta of the entire N-particle system are presented by one point in this space.

18. displacement : To obtain with initial conditions Ex) One-Dimensional Harmonic Oscillator: isolated system r system potential energy x r0 In an isolated system,

19. 6 4 2 position x(t) 0 -2 -4 -6 0 1 2 time 6 4 2 momentum p(t) 0 -2 -4 -6 0 1 2 time

20. 6 4 2 momentum p(t) 0 -2 -4 -6 -6 -4 -2 0 2 4 6 position x(t) To determine the phase-space trajectory : ellipse

21. h: Planck’s constant, • Quantum Mechanics Considerations • energy of a photon: • speed of light: • rest energy of a particle: • photon momentum: • de Broglie wavelength and frequency for a particle moving with velocity v << c

22. Heisenberg uncertainty principle The position and momentum of a given particle cannot be measured simultaneously with arbitrary precision. • Pauli exclusion principle In quantum theory, independent particles of the same type are indistinguishable. For certain particles, such as electrons, each quantum state cannot be occupied by more than one particle.

23. Equilibrium Distributions for Different Statistics Goal : Find the occupation in each energy level when the thermodynamics probability is maximum (equilibrium point). • MB can be considered as the limiting case of BE or FD model.

24. The specification, at any one moment, that there are N0particles in energy level e0 with degeneracy g0 N1particles in energy level e1 with degeneracy g1 . . . . . . . . . Niparticles in energy level ei with degeneracy gi . . . . . . . . . in a container of volume V when the gas has a total number of particles N and an energy Uis a description of a macrostate of the gas. There are many quantum states corresponding to the same energy levels and that the degeneracy of each state level is much larger than the number of particles which would be found in any one level at any time.

25. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 A B C A B C B A C C B A A C B C B A Ex) Consider the Niindistinguishable particles in any of the gi quantum states associated with the energy level ei. Any one particle would have the same gichoices in occupying gi different quantum states. A second particle would have the same gi choices, and so on. Thus, total number of ways in which Nidistinguishable particles could be distributed among gi different quantum states would be 6 ways (3!) in which 3 distinguishable particles can occupy 3 given quantum states → divided by 3! for indistinguishable particles

26. The number of ways to distribute Niindistinguishable particles among gi quantum states The number of ways W in which this macrostate may be achieved is given by W : thermodynamic probability of the particular macrostate or the number of microstate If V, N, and U are kept constant, the equilibrium state of the gas corresponds to that macrostate in which W is a maximum.

27. . . . N0 N1 N2 level 2 level 1 level 0 • Maxwell-Boltzmann statistics Number of ways to arrange Ndistinguishable particles Number of ways to put Ndistinguishable particles on each energy level if there is no limit for the number of particles on each energy level Consider one ofN! ways of arranging N distinguishable particles Each arrangement of N0, N1, N2, … particles in each energy level should be considered as one case.

28. N0particles in energy level e0 with degeneracy g0 N1particles in energy level e1 with degeneracy g1 . . . . . . . . . Niparticles in energy level ei with degeneracy gi . . . . . . . . . Thus ways of putting Nidistinguishable particles into gi distinguishable degeneracy for each arrangement If degeneracy is included, Each of Nidifferent particles can be put on any gi. The first particle can be placed in gi ways, the second particle can also be placed in gi ways, and so on.

29. partition . . . . . . . . . . . . . . . • Bose-Einstein statistics Number of ways to put Niindistinguishable particles on gidistinguishable quantum states on the ith energy level, if there is no limit on the number of particles in any quantum states Consider placing Ni copies of the same book among gi shelves. Number of partitions: gi - 1 Consider the number of ways to arrange Ni books and gi – 1 partitions together. The problem is now to find the ways of arranging Ni+ (gi - 1) with gi - 1indistinguishable partitions.

30. ways of putting N indistinguishable objects into g distinguishable boxes

31. . . . g0 g1 g2 • Fermi-Dirac statistics Number of ways to put Niindistinguishable particles on a set of gi quantum states on the ith energy level. Each quantum state can be occupied by no more than one particle. (Ni < gi) Equivalent to find Number of ways to select Ni objects from a set of gi distinguishable objects On the ith energy level

32. Ex. 3-3 4 indistinguishable particles 2 energy levels 3 degeneracies Find: 1) thermodynamic probability of all arrangements a) BE b) FD 2) most probable arrangements

33. Most probable arrangement

34. known: unknown: • States of maximum thermodynamic probability Energy level Degeneracy

35. For a continuous function • Method of Lagrange multipliers A procedure for determining the maximum/minimum point in a continuous function subject to one or more constraints At the maximum/minimum point, If xi’s are independent, If xi’s are dependent and related by m (m < n) constraints, n - m independent variables

36. Ex) Positive values of x, y, z that maximize constraint: bj: Lagrangian multipliers n equations, n - m independent variables, mbi’s

38. For MB statistics with degeneracy lnx ln8 ln7 ln6 ln5 ln4 ln3 ln2 x ln1 0 1 2 3 4 5 6 7 8 • MB distribution maximum probability Stirling’s approximation: for x >> 1, Since x >> 1

39. Negative signs are chosen because a and b are generally nonnegative for molecular gases.

40. or • BE distribution • FD distribution Since dNi can be chosen arbitrary, MB distribution Similarly,

41. heat added Work done Thermodynamic Relations from the microscopic point of view • Heat and Work redistribution of particles among energy levels shift in the energy levels associated with volume change

42. Energy level Degeneracy Heat added to a system moves particles from lower to higher energy level. Work done on the system moves energy levels to higher values.

43. SA, WA SB, WB Two subsystems of an isolated system • Entropy The entropy of an isolated system increases when the system undergoes a spontaneous, irreversible process. At the conclusion of such a process, when equilibrium is reached, the entropy has the maximum value consistent with its energy and volume. The thermodynamic probability also increases and approaches a maximum as equilibrium is approached.

44. If we let Since entropy is an extensive variable, the total entropy of the composite system is The thermodynamic probability, however, is the product, The only function that satisfies this relation is the logarithm. k is turn out to be the Boltzmann constant kB

45. Lagrange Multipliers For MB statistics hold for all thee types of statistics In a reversible process in a closed system

46. Helmholtz function MB distribution: BE distribution: FD distribution:

47. Comparison of the distributions

48. MB curve: y = 1/ex MB curve lies between the BE and FD curves only valid for y << 1 (diluted gas region) lots of states are unoccupied at high temperatures (x → 0), Ni ≈ gi (y → 1) Since gi ↑with energy↑, occupation number↑ At low temperatures, the population of the lower states is favored. • BE curve: y = 1/(ex-1) As x → 0, y → ∞, and for large x, y ≈ e-x. The distribution is undefined for x < 0. Particles tend to condense in regions where ei is small, that is in the lower energy states (Bose condensation). • FD curve: y = 1/(ex+1) For x = 0, y = 1/2, and for large x, y ≈ e-x. As x → -∞, y → 1. At the lower levels with ei – m negative, the quantum states are nearly uniformly populated with one particles per states.

49. For BE statistics • Entropy at Absolute Zero Temperature • The 3rd law of thermodynamics at very law temperature (T → 0), b = 1/ kBT → ∞ Hence N = N0: all particles will be the lowest energy level (ground states). If g0 = 1, as it is the case for pure substance, then Bose-Einstein condensation

50. For MB statistics or partition function • Macroscopic Properties in Terms of Partition Function • Partition function Z from conservation of particles,