Jan. 6, 8, 2003
Lecture 1, 2: Basic thermodynamics and Kinetic theory of
the ideal gas
Ref: GNS, Ch.1
Statistical
mechanics is using a handful of quantities to describe a large number of
particles. Its macroscopic counterpart
is thermodynamics. We will review the essential concepts and techniques of
thermodynamics in the first few lectures.
Key words
Thermodynamic equilibrium
Isolated, closed, open system
Homogeneous-heterogeneous
State quantities-extensive and intensive
Equation of state
Temperature
Zeroth law of thermodynamics
Kinetic theory of the idea gas
Mexwell’s velocity distribution
Points of discussion
1.
Why thermodynamics is different from mechanics,
electrodynamics and quantum mechanics?
2.
How to define and measure temperature
?
3.
The difference between equilibrium state and
stationary state
4.
Average kinetic energy of the ideal gas is
proportional to kT
Homework (Due 1/17)
1. What would be the Maxwell’s velocity distribution for a
2D gas system?
Jan. 10, 13, 17, 2003
Lecture 3 4, 5: Equation of state and virial
expansion
Ref: GNS, Ch.1
We
will discuss the equation of state of real gases and their virial
expansion forms.
Key words
Pressure
Work and heat
Chemical potential
Specific heat
Virial expansion
Gas constant
Van der Waals equation of state
Coefficient of expansion
Isothermal compressibility
Points of discussion
1. What is the work
of adding one particle to a system?
2. What is the
relation between gas constant R and Boltzmann’s
constant K?
3. Why Cp/Cv?
Homework (Due 1/17)
2. Determine the first expansion coefficient B’ in another
type of virial expansion
for the van der Waals
equation of state.
No class on 1/15
(Wed)
Jan. 22, 24, 2003
Lecture 6, 7 : Entropy and the
second law
Ref: GNS, Ch.2
Entropy
is a state function but the change of entropy expressed as 
is only true for
reversible processes.
Key words
Internal energy
Reversible and irreversible work
Heat engine
Carnot cycle
Efficiency of a thermal engine
Points of discussion
1. First law of
thermodynamics
2. Internal energy is
a state function
3. Specific heat of
an ideal gas
4. Carnot theorem
5. Define entropy via Carnot cycle
6. Entropy of an ideal gas
Homework (Due 1/24)
3. Calculate
the work and heat of each step in a Carnot cycle.
What is the total change of W, Q, U and entropy of the universe after one cycle.
4. (a) One kg of water at 273 is brought into contact with a
heat reservoir at 373K. When the water has reached 373K, what is the entropy
change of the water of the heat reservoir, and of the surrounding? (b) If the
water had been heated from 273 to 373K by first bringing it into contact with a
reservoir at 323K and then with areservoir at 373K, what would have been the entropy change
of the universe?
Jan. 24, 27, 2003
Lecture 7, 8: Fundamental equations and equation of state
Ref: GNS, Ch.2
Callen,
1-9, 2-1, 2, 3
Knowledge
of all equations of state of a system=knowledge of the fundamental equation.
Key words
Fundamental equation
Equation of state
Extensive and intensive variables
Energy and entropy representation
Euler’s equation
Gibbs-Duhem relation
Points of discussion
1. What are the
properties of the fundamental equation?
2. Intensive
variables are homogeneous equation of zeroth order
3. Energy and entropy
representation of the fundamental equation
4. Gibbs-Duhem relation presents relationships among the intensive
variables
5. The fundamental
equation of an ideal gas system
6. Entropy of mixing
7. Principle of
maximum entropy
Homework (Due
5. (a) Show that the
fundamental equation 
where R, v, q are positive constants, satisfies all
postulates.
(b) Derive three
equations of state from this fundamental equation and show that the equations
of states are homogeneous zero order.
(c) Find m as a function of T, V, N.
(d) Show by a
diagram the dependence of pressure on volume for fixed temperature for this
system. Draw two such isotherms, corresponding to two values of the
temperature, and indicate which isotherm corresponds to the higher temperature.
(e) Using the
explicit expressions of T, p, m to
verify the Euler relation.
Lecture 9: The statistics of two-state systems
Ref: GNS, Ch.2
Kittel,
Ch.1
Since entropy is an extensive quantity and it is related to
the number of microscopic states that are compatible with a given macroscopic
state. Entropy can be defined by klnW where W
is the number of microscopic states.
Key words
Phase space
H-theorem
Gaussian distribution
Most probable distribution
Fluctuation
Points of discussion
1. Gibbs paradox
2. Why air molecules
in a room will not concentrate in a small bottle by themselves?
3. What is the
distribution function for a two-state system?
4. What is the
average number of particles in one state and its fluctuation?
5. Gaussian
distribution becomes very sharp, when the number of particles is Avogadro.
6. Two spin systems in thermal contact will
result in the equality of fractional spin excesses.
7. How small is the
deviation from the most probable configuration?
8. Entropy always
increases when two systems are brought into thermal contact.
Homework (Due
- Let g(N,s) be the multiplicity function of a binary system of N particles with two spin states, and 2s is the spin excess. Show that in the limit of s/N<<1 and N>>1,
Jan. 31,
Lecture 10, 11: Free energy and enthalpy
Ref: GNS, Ch.4
Callen,
Ch.5-6
Other than the internal energy there are three more
potentials, obtained from Legendre transformation,
have been developed in the history to describe the evolution of thermodynamic
systems.
Key words
Lagendre transformation
Helmholtz potential = free energy
= Helmholtz free energy
Enthalpy
Exothermal and endothermal
Energy minimum principle
Entropy maximum principle
Points of discussion
1. Geometric
construction of Legendre transformation
2. The difference
between internal energy U and free energy F
3. For isothermal
systems the free energy is analogous to the entropy for isolated systems
4. For isobaric,
adiabatic systems, equilibrium=minimum enthalpy
February 5, 7, 2003
Lecture 12, 13: Gibbs potential and the grand potential
Ref: GNS, Ch.4
The grand potential will be used extensively in statistical
mechanics.
Key words
Free enthalpy = Gibbs potential
Exergonic and endergonic
Grand potential
Particle bath
Maxwell relations
Points of discussion
1. G=mN
2. In an isothermal,
isobaric system, equilibrium=minimum free enthalpy
3. With a heat bath
and a particle bath, equilibrium= minimum grand potential
4. König-Born construction of Maxwell relations
5. TdS and energy equations are
useful to calculate heat and energy change.
Homework (Due 2/14)
7. Using Maxwell relations and the first law to
show that 
where the isobaric
expansion coefficient 
and the isothermal compressibility 
. From this relation, you can convince yourself
that 
and the equality only
holds at T=0.
Feb. 10, 12 2003
Lecture 14, 15: First order phase transition-a
thermodynamic view
Ref: GNS, Ch.3
Callen, Ch.9
Zemansky and Dittman, “Heat and
thermodynamics”, Ch. 10-11
Key words
First and second energy equation
First and second TdS equation
Gibbs’ phase rule
Clausius-Clapeyron equation
Maxwell construction
Critical point
Superheated liquid and supercooled
gas
Points of discussion
1. What is the
internal energy expression of a van der Waals gas?
2. What is the
entropy of a van der Waals gas?
3. Why in a pure
material, there can be at most a triple point?
4. The instability of
van der Waals equation indicates gas-liquid phase
transition
5. At critical point, the density of liquid and vapor are
identical.
6. At critical point, PV/NkT=3/8
for any a and b of the van der Waals
equation of state.
Homework (Due 2/24)
8. Express the internal energy from a Berthelot equation of state 
.
Feb. 14, 2003
Lecture 16: Critical phenomena
Ref: Callen,
Ch.10
Critical phenomena is one of the
most intriguing and mathematically challenging field in theoretical
physics. This lecture only provides you
a very superficial introduction to this rich field
Key words
Order parameter
Critical exponents
Critical opalescence
Points of discussion
1. At critical point, the latent heat is 0 and
the first order liquid-vapor phase transition becomes second order.
2. By using reduced pressure, temperature, and
volume, the van der Waals equation becomes universal.
3. Can there be a critical point in the fusion
line?
4. The critical exponents are not totally
independent. They satisfy many equalities.
5. Landau theory on critical phenomena
Homework (Due 2/24)
9. Show that
the critical exponents obtained from van der Waals
equation are d=3, b=1/2, g=1.
Feb. 18, 19, 2003
Lecture 17, 18 : Phase space and
microscopic states
Ref: GNS, Ch.5
Pathria,
Ch.2
Count the number of microscopic states which are compatible
with a macroscopic state is the first step in statistical mechanics
Key words
Phase space
N-sphere
Quantum concentration
Points of discussion
1. The phase space of classical systems
2. How to calculate the volume and surface of a N-sphere?
3. How to determine entropy and all
thermodynamic functions from the knowledge of phase space?
4. Gibbs correction factor
5. Physical significance of the unit of a phase
space cell
6. Particle in
a box quantum mechanically
7. What
determines the classical and quantum regime in statistical mechanics?
Homework (Due
10. Pathria, 2.5
11. Pathria, 2.8
Feb. 21, 24, 26, 2003
Lecture 19, 20, 21: Ensemble theory
Ref: GNS, Ch.5
Pathria,
Ch.2
Ensemble average=Time average
Key words
Representative point
Hamiltonian equations
Liouville’s theorem
Microcanonical ensemble
Equal a priori probability
Points of discussion
1. The unit cell in a
N-dimensional phase space is hn.
2. The observed value of a physical quantity is
determined by an ensemble average.
3. The time dependence of an ensemble average is
governed by Liouville’s theorem.
4. The ensemble average of a physical
quantity=time average of a physical quantity
5. The density function is a constant for microcanonical ensemble.
6. The most probable
distribution for an isolated system is equal a priori distribution
7. Thermal dynamics quantities of N
distinguishable harmonic oscillators
8. Thermal dynamics quantities of ultrarelativistic gases
9. Dimension issues
Homework (Due
12. What would
be the difference if you consider quantum harmonic oscillators or quantum ideal
gases in Ex. 6.2 and 6.3 of GNS?
Mid-term exam will be on
No class on 3/3, 5, 7.
Feb. 28, Mar. 17, 21,
2003
Lectures 22, 23,
24: Canonical ensemble theory (I)
Ref. Ch.7 of GNS
Ch.3 of Pathria
Canonical ensemble can describe most of the physical systems
in contact with heat baths.
Key words
Lagrange multiplier
Partition function
Fluctuation
Single particle state
Points of discussion
1. Reservoir
dictates accessible states of a system
2. One can
obtain the Boltzmann factor from either the reservoir
or ensemble approach
3. 
in the canonical ensemble is analogous to the 
in a microcanonical ensemble
4.
Thermodynamics of ideal gas and classical harmonic oscillator
5. Connection
between microcanonical and canonical ensembles
6. The
N-particle state and single particle state in a system
Homework (Due
13. Calculate
all thermodynamic functions of quantum harmonic oscillators by canonical
ensemble
March 26, 31, 2003
Lectures 25, 26:
Canonical ensemble theory (II)
Ref. Ch.7 of GNS
Ch.3 of Pathria
There are several ways to calculate partition function in
the canonical ensemble framework. Make
sure you understand the underlying conceptual differences.
Key words
Density of states
Single particle state
Equipartion theorem
Virial theorem
Points of discussion
1. The
difference of using single particle state to calculate partition function vs.
phase space approach
2. What is the
physical significance of equipartition theorem and Clausius’ virial theorem?
3. How to
obtain the density of states from a given partition function
4. Velocity
distribution of ideal gas particles
5. 1-D ideal
gas systems
6. Calculate
the number of microscopic states in a different way (from the phase space
volume)
7. Ideal gas
under gravity
Homework (Due
14. (a) Show that the density of orbitals
of a free electron in one dimension is 
, where L is the length of the line. (b) Show that in two dimensions, for a square
of area A, 
independent of e.
No class on 3/28.
April 2, 4, 2003
Lectures 27, 28:
Magnetic systems
Ref. Ch.8 of GNS
We will discuss thermodynamics of paramagnetism
and two-state systems
Key words
Paramagnetism
Bohr mageton
Lande g-factor
Langevin function
Brillouin function
Points of discussion
1. Particles in
a gravitational field
2. Another way
to get average energy of a system of harmonic oscillators
3. Magnetic
systems in classical and atomic treatment
Homework (Due
15. If the energy eigenvalues of
an s-dimensional oscillator can be written as
; where n=0,1,2,3,…
Evaluate the partition function and S, U, m of a
system of N oscillators, and compare your results with a corresponding system
of sN oscillators. Show, in particular, that the chemical
potential 
April 7, 9, 2003
Lectures 29, 30:
Grand Canonical Ensemble
Ref.
In a realistic world the number of particles in a state may
not be fixed, but is a function of temperature, e.g., electron distribution in
a solid. To describe such systems we can
use grand canonical ensemble where the number of particles in a “system” can be
interchanged among all the “mental copies”.
Key words
Grand partition function
Grand potential function
Ξ (T,p,N)
Fugacity
Points of discussion
1. Grand
partition function can be expressed as a sum of partition function weighted by
chemical potentials
2. Calculate
any averages of physical quantities in the grand canonical ensemble
3. Ideal gas in
grand canonical ensemble
4. Review of
all the partition function
5. Fluctuations
in the grand canonical ensemble
6. Vapor
pressure of a solid-vapor equilibrium system
7. Temperature
dependence of the partition function dictates the thermodynamics
Homework (Due
16. For a solid composed of N atoms, one can approximate the
thermal excitation of the atoms by small oscillations. The total degree of freedom is thus
equivalent to 3N 1-D oscillators. Let the frequency distribution of the
oscillators be g(w) and
for 
= 0 
where the Debye
frequency 
is defined by 
.
Calculate the Cv
with this model and find at high temperatures, Cv
approaches 3Nk (Dulong-Petit law), while at low
temperatures Cv behaves as T3.
April 11, 14, 2003
Lectures 31, 32:
Quantum Systems in Grand Canonical Ensemble
Ref. Ch.12 of GNS
We will discuss the occupation numbers of quantum states in
Bose-Einstein and Fermi-Dirac statistics and their
classical limit, Maxwell-Boltzmann statistics.
Key words
Occupation numbers
Bose-Einstein
Fermi-Dirac
Poisson statistics
Points of discussion
1. Grand
partition and thermodynamics functions in BE, FD and MB statistics
2. How many
particles will be any quantum mechanical state?
Homework (Due
18. Pathria p.102 problem: 4.3
19. Pathria p. 103 problem: 4.8
April 14, 16, 18, 2003
Lectures 32, 33,
34: Ideal Bose Systems
Ref. Ch.13 of GNS
When 
, quantum statistics is required. Bose systems have its characteristics in
average occupation number of each quantum state. Systems of photons and phonons
are two important applications. BEC happens in the momentum space, but is
manifested as superfluidity in coordinate space.
Key words
Bose-Einstein condensation
, and its derivative
Critical temperature of BEC
Planck spectrum
Points of discussion
1. How to
define classical and quantum regimes?
2. The ground
state in the summation must be treated separately.
3. What causes
BEC?
4. BEC can be
viewed as a phase transition in PV diagram, but it occurs even without interparticle interaction.
5.Why the chemical potential of photon and phonon are zero?
6. Heat
capacity has a kink indicating a second order phase transition.
7. Photon and
phonons
Homework (suggested)
20. Pathria p.190 problem: 7.6
21. Pathria p.190 problem: 7.13
22. Pathria p. 190 problem: 7.14
April 18, 21, 23,
2001
Lectures 40, 41,
42: Ideal Fermi Systems
Ref. Ch.14 of GNS
Exclusion principle dictates the occupation number of each
state to be no more than one. Thus fermi gas does not
condense. Instead, there can be a very large positive chemical potential for
the system.
Key words
, and its derivative
chemical potential
Fermi energy
Landau diamagnetism
De Haas-Van Alphen effect
Points of discussion
1. Chemical
potential in the fermi system can be greater than 0,
so the fugacity function can be greater than 1.
2. Pressure of
the non-relativistic fermi gas at classical limit is
2/3 of the energy density while the relativistic gas approaches 1/3.
3. Entropy is 0
at T=0
4. The heat
capacity behaves as T3 when T approaches 0.
5. The Fermi
energy (T=0) for electrons in Cu can be 7ev and that for a nucleon in neutron
stars can be 27Mev (but the neutron star is very cold!)
Homework (suggested)
23. Calculate
the numerical values of the Fermi energy for conducting electrons in Ag and He3
(density is 63Å3 per atom)