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. STATISTICAL ENSEMBLES AND FLUCTUATIONS BASIC QUANTUM MECHANICS
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. 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
3N positions
and 3N momenta
 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
Soft Sphere Model
 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

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

7. Statistical methods
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

8. Statistical Mechanics of Independent Particles
Classical vs. statistical thermodynamics
classical
macroscopic
statistical
microscopic
matter is continuous
matter is particulate
phenomenological approach
sum of molecules & atoms
equilibrium states
equilibrium distribution
Classical thermodynamics was worked out in essentially its present form. It is a phenomenological theory, describing the macroscopic properties of matter, most of which are amenable to direct measurement. No attempt is made to explain underlying causes or to provide a mechanistic description.
Statistical methods can obtain microscopic description that are related to macroscopic behaviors. The central idea is the probability density function applied to a large collection of identical particles. Statistical mechanics aims at finding the equilibrium distribution of certain types of particles.
Our object is understand basic concept~~ / and I will show you three equil~ / Finally, we can find equation of microscopic ~
independent particles
their energies: independent of each other
total energy: sum of the energies of individual particles

9. Macrostates versus Microstates.
e0, N0
e1, N1
e2, N2
ei, Ni
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
This is summary of the concept of statistical thermodynamics.
The number of microstates leading to a given macrostates is called the thermodynamic probability. It is the number of ways in which a given configuration can be achieved.
We will be interested in the number of microstates but not in their detailed specification.
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. Energy level excited states → quantum states ground state Degeneracy
In quantum theory, to each energy level there corresponds one or more quantum states described by a wave function.
When there are several quantum states that have the same energy, the states are said to be degenerate. The number of quantum states for a given energy level is called degeneracy.
The quantum state associated with the lowest energy level is called the ground state of the system; those that correspond to higher energies are called excited states.
And we can express statistical state of the system with concept of macrostate and microstate
The macrostate, configuration of the system, is specified by the number of particles in each of the energy levels. In this case, ~
The microstate is specified by the number of particles in each energy state. In general, there will be many different microstates corresponding to a given macrostate.
The number of microstates leading to a given macrostates is called the thermodynamic probability. It is the number of ways in which a given configuration can be achieved.
We will be interested in the number of microstates but not in their detailed specification.
→ quantum states
ground state
Degeneracy
macrostate
microstate :
(1), (2 1 0), ( )

12. Example : Consider 3 particles, labeled A, B, and C
constraints
Consider 3 particles, labeled A, B, and C, distributed among four energy levels, 0 1e 2e 3e such that the total energy is 3e.
The occupation numbers are No particles with energy 0, N1 with energy e. The total # of microstates is 10 and the number of possible macrostates is 3. For the most probable macrostates, the number of available microstates is 6.
While the numbers here are small, it is evident that the most “disordered” macrostates is the states of highest probability.
For very large number of particles, this state will be sharply defined and will be the observed equilibrium state of the system.
Equilibrium state

13. Phase Space
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 dx dy dz x y z

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

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

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

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

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

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

21. Quantum Mechanics Considerations
energy of a photon:
h: Planck’s constant,
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).
Model
Particle Properties
Case
Maxwell-Boltzmann
distinguishable
unlimited particles per quantum state
ideal gas molecules
Bose-Einstein
indistinguishable
photon, phonon
Fermi-Dirac
indistinguishable, identical
one particle per quantum state (Pauli exclusion principle)
electron, protons
Our goal is to find the occupation number of each energy level when the thermodynamic probability is a maximum.
The characteristics of various types of particles can be described by different distribution. I will show you four models.
Boltzmann distribution is model considered distinguishable, noninteracting particles.
MB distribution is the case of Boltzmann distribution for indistinguishable particles. The thermodynamic relations of molecular gases can be understood from MB model. And it also can be considered as the limiting case of BE or FD model.
FD distribution can be used to model the electron gas or protons. Fermions, particles followed FD distribution is indistinguishable and obey Pauli Exclusion principle.
BE distribution is important for the study of photons, phonons in solids, and atoms at low temperature. Boson, particles followed BE distribution is indistinguishable and any number can occupy a given quantum states.
MB can be considered as the limiting case of BE or FD model.

24. 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.
The specification, at any one moment, that there are
N0 particles in energy level e0 with degeneracy g0
N1 particles in energy level e1 with degeneracy g1
Ni particles 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 U is a description of a macrostate of the gas.

25. Ex) Consider the Ni indistinguishable particles in any of the gi quantum states associated with the energy level ei.
Any one particle would have the same gi choices 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 Ni distinguishable particles could be distributed among gi different quantum states would be A B C 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
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 Ni indistinguishable 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. . . . Maxwell-Boltzmann statistics
Number of ways to arrange N distinguishable particles
Number of ways to put N distinguishable particles on each energy level if there is no limit for the number of particles on each energy level
Consider one of N! ways of arranging N distinguishable particles
level 0 N0 N1 N2
. . .
level 1
level 2
Each arrangement of N0, N1, N2, … particles in each energy
level should be considered as one case.

28. . If degeneracy is included,
N0 particles in energy level e0 with degeneracy g0
N1 particles in energy level e1 with degeneracy g1
Ni particles in energy level ei with degeneracy gi .
Each of Ni different 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.
Thus
ways of putting Ni distinguishable particles into gi distinguishable degeneracy for each arrangement

29. . . . . . . . . . . . . . . . Bose-Einstein statistics
Number of ways to put Ni indistinguishable particles on gi distinguishable 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.
partition
. . . .
. .
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 - 1 indistinguishable partitions.

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

31. . . . Fermi-Dirac statistics
Number of ways to put Ni indistinguishable 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) g0 g1 g2
. . .
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. States of maximum thermodynamic probability
Energy level
We seek the distributions among energy levels Ni for an equilibrium state of the system. That is, we want to determine the number of particles Ni with energy ei for all n energy levels of the system, subject to the restrictive conditions : conservation of particles and conservation of energy.
we can get the value of energy level and degeneracy from 슈뢰딩거 equation.
For the second energy level, the situation is the same, except that there are only N-N1 particles remaining to deal with.
known:
unknown:
Degeneracy

35. Method of Lagrange multipliers
A procedure for determining the maximum/minimum point in a continuous function subject to one or more constraints
For a continuous function
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. bj: Lagrangian multipliers
n equations, n - m independent variables, m bi’s
constraint:
Ex) Positive values of x, y, z that maximize

38. lnx x MB distribution For MB statistics with degeneracy
maximum probability
Stirling’s approximation:
Same analogy, taking the logarithm of W_MB, and simplify equation using stirling’s formula. we obtain the equation.
In working out the derivatives, add these two Lagrange multiplier, we obtain same equation of ~~ x
lnx
ln1
ln2
ln3
ln4
ln5
ln6
ln7
ln8 1 2 3 4 5 6 7 8
for x >> 1,
Since x >> 1

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

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

41. Thermodynamic Relations
from the microscopic point of view
Heat and Work
redistribution of particles among energy levels
Now, we get ready to understand thermodynamic properties and relations from the microscopic point of view.
We will deal with the concept of Heat and work, entropy and third law of thermodynamics and so on..
The partition function is key to the evaluation of thermodynamic properties.
Let’s start with the concept of heat and work.
The statistical expression for internal energy is , taking the differential, we have .
Where εj is some function of an extensive property X (εj = εj(x)), Such as the volume,
Let y is ~
The classical analog to differential equation of internal energy is
For these two dU equatons, consider two states with X the same, that is dX=0
shift in the energy levels associated with volume change
heat added
Work done

42. Work done on the system moves energy levels to higher values.
Degeneracy
Energy level
That is
The first equation states that Heat transfer is energy resulting in a net redistribution of particles among the available energy levels, involving no work.
In this figure, heat added to a system shifts particles from lower to higher energy levels.
The second part of equation can be interpreted in terms of an adiabatic reversible process (that is no heat flow) in which work is done on the system. An increase in the system’s internal energy could therefore be brought about by a decrease in volume with an associated increase in the e j ‘s . The energy levels are shifted to higher values with no redistribution of the particles among the levels.
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. 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.
Two subsystems of an isolated system
SA, WA
SB, WB

44. Since entropy is an extensive variable, the total entropy of the composite system is
The thermodynamic probability, however, is the product,
If we let
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
The method of Lagrange multipliers is a procedure for determining the maximum/minimum point in a continuous function subject to one or more constraints.
Lagrange multiplier a and b are related to the physical properties of the assembly.
We consider b first. If we multiply derivative equation by Nj and sum over i, this two terms appear in the expression for log w
Of that equation. Making the substitution, we get
In a reversible process in a closed system

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

47. Comparison of the distributions
The distribution functions for identical indistinguishable particles can be represented by the single equation, where~
For diluted gas, MB distribution is an approximation to the FD and BE distributions, as we have seen.
Plots of y=~ and x=~ are shown in this figure.

48. MB curve: y = 1/ex BE curve: y = 1/(ex -1) FD curve: y = 1/(ex +1)
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. Entropy at Absolute Zero Temperature
The 3rd law of thermodynamics
at very law temperature (T → 0), b = 1/ kBT → ∞
For BE statistics
Then, let’s prove that these entropy equation satisfy the 3rd law of thermodynamics.
At very law temperature, the value of beta goes to infinity. Since ~
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. Macroscopic Properties in Terms of Partition Function
Partition function Z
For MB statistics
from conservation of particles,
Partition function is the most important parameter in the statistical thermodynamics. Because it makes that statistical parameters can connect classical thermodynamic parameters easily.
substitute the value of Lagrange multiplier b only,
And find ~
In this equation of MB distribution function, we define partition function Z is. ~
Partition function depends on the temperature and on the parameters that determine the energy levels and quantum states, degeneracy. or
partition function

51. Internal energy U

52. Entropy Helmholtz free energy Chemical potential Gibbs free energy
In spite of the physical meaning is not immediately clear, The partition function is important quantity in statistical thermodynamics.
Because of the partition function Z allows the calculation of macroscopic thermodynamic properties form the microscopic representation.
There are different types of partition functions. For MB distribution. The partition function Z is defined as ~
Using this definition, we can calculate thermodynamic properties.
Let us calculate first S for MB distribution.
Substitute the distribution function of equation in S=k log W,
and also substitute partition function Z. we already knows that sigma Ne is internal energy U.
Simplify equations, we obtain the equation of entropy with respect to partition function.
Gibbs free energy
Enthalpy
Pressure