The revision tutorial for the course is on Tuesday May 8, 11:00 in
Hall B in the Arts building (usual time and place for the
Tuesday lecture)
A schedule of content and exercises will appear gradually
below.
To give you the opportunity to test yourself on an exam, I am making
the
Spring 2010 exam available with solutions.
The exam is here.
A set of solutions to this exam is here.
Week 1:
Notes:
We introduced microstates and macrostates of a system as well as
equations of state. We introduced a simple models for a gas (the classical
ideal gas) and calculated its energy in terms of its temperature, using
the ideal gas law and direct kinetic arguments.
We discussed the equipartition theorem and some of its implications for
the energy of monatomic and diatomic ideal gases and solids.
This material is treated in sections 1.2 and 1.3 of the book by
Schroeder
Week 2:
Notes:
There was only one lecture this week. We discussed the equipartition
theorem and its applications in more detail and in particular the concept
of freezing out of degrees of freedom.
Week 3:
There were three lectures this week.
We discussed how energy may be exchanged in the form of work or heat and
introduced the first law of thermodynamics, which says that, in any
process, the change in the energy of the system is equal to the heat
absorbed, minus the work done by the system. We also discussed the
calculation of work done by gases due to changes in their volume and we
defined heat capacities (and corresponding specific heats) and calculated
them for constant volume and constant pressure processes in ideal gases
and solids. We also introduced the multiplicity and entropy of
macrostates and the second law of thermodynamics This material is covered in sections 1.4, 1.5, 1.6
and 2.1 of the book by Schroeder. Please make sure you read all of chapter
1 by Schroeder, especially if you did not take a thermodynamics course
before.
Exercises:
Assignment 1: please hand in the following exercises (from the book by
Schroeder):
1.12, 1.18, 1.24 and either 1.22 or 1.55 (choose one of the last two)
Other recommended exercises (not to be handed in):
1.17, 1.21, 1.26, 1.34, 1.44
1.22 was worked out in the tutorial by Ciarán and Mario
1.55 was worked out in the tutorial by Úna and Francis
Week 4:
Notes:
We again introduce the fundamental assumption
of statistical mechanics, the microscopic definition of the
entropy and the second law of thermodynamics.
We then calculate the multiplicities or entropies for macrostates of
the 2-state paramagnet and the Einstein solid and introduced Stirling's
approximation of the factorial in order to be able to study the results
in more detail. This material is covered in sections 2.1,2.2, 2.3, 2.4 and 2.6
of the book by Schroeder.
Exercises:
Assignment 2 can be found here.
Please hand this in no later than Tuesday February 28, 11:05.
This assignement will be worked out by Neal and Brendan.
Week 5:
Notes:
This week was all about convincing ourselves of the second law of
thermodynamics (large systems in equilibrium are found in maximum entropy
macrostates, up to small fluctuations).
We used Stirling's approximation to the
factorial to show that the probability of finding a given magnetisation
state in a 2-state paramagnet is given by a Gaussian function (to good
approximation). We argued that in large systems (many spins) such
fluctuations in the magnetisation are very small (proportional to the
square root of the number of spins, which is very small compared to the
number of spins in large systems). We then argued that the entropy is
extensive and that any large system in equilibrium should have an
approximately gaussian probability density for fluctuations of any
observable quantity, with a standard deviation proportional to the square
root of the number of components of the systems (spins/particles) for
extensive quantities. Similar material is covered in sections 2.4 and 3.3
of the book by Schroeder.
Exercises: Recommended exercises in the book by Schroeder:
2.17 and 2.19 (if you didn't do them yet), 2.18, 2.22, 2.26, 2.32,
2.34, 2.35, 2.42.
These exercises don't have to be handed in.
2.18 and 2.22 will be worked out in the tutorial by Brian and Megan
Week 6:
Notes:
We calculate the entropy of the ideal gas using coarse graining of
the classical phase space (the space of configurations of momenta and
positions of all particles of the gas). We introduced two elements from
quantum mechanics into the description of the gas. First of all the
natural volume of the coarse graining cell is inferred from the
uncertainty principle (introducing Planck's constant into the
description). Secondly, if the particles of the gas are indistinguishable,
we consider any classical states which are related by a permutation of the
positions and momenta of the particles as the same state. This introduces
a factor of (1/N!) into the multiplicity of the gas, where N is the
number of particles.
By a consideration of interacting systems we showed that the various
derivatives of the entropy characterize the equilibrium properties of a
system. In particular we see that the temperature is directly related to
the partial derivative of entropy with respect to the energy.
Exercises: Assignment 3 consists of exercises 2.26, 2.32, 3.5
and 3.8 in Schroeder.
Please hand these in no later than Tuesday March
13, 11:05.
Other recommended exercises: 2.31, 3.2, 3.3, 3.4
Week 7:
Notes:
We showed the relation between the entropy and macroscopic quantities
such as energy, temperature, volume and pressure, summarized in the
thermodynamic identity. We showed how the entropy can be found from
the heat capacity of the system and the various latent heats which occur
at phase transitions.
We also introduced the chemical potential, which appears in the
thermodynamic identity for systems whose number of particles can vary
As an application, we calculated the energy and magnetisation of a
two-state paramagnet
as a function of temperature (with T obtained from S), deriving Curie's
law, which states that energy and magnetisation are inversely
proportional to T at high temperatures, as
a special case.
Material on the correspondence between statistical physics and
thermodynamics is in various section of chapter 3, notably 3.2 and 3.4
(thermodynamic identity etc.).
For people who have taken thermodynamics, section 4.1 gives an
interesting new perspective on heat engines.
In fact, Quite a bit of the material covered in chapters 4 and 5 of
Schroeder's book was either covered earlier in
thermodynamics (MP460) or more in the spirit of that course. I won't
have time to treat it here. Please have
at least a read through this material. It is quite interesting and has many
examples that were not treated in much detail in thermodynamics, for
example a really interesting section on various cooling techniques, such
as magnetic cooling.
Exercises: Recommended exercises in the book by Schroeder:
3.11, 3.14, 3.23, 3.25, 3.32, 3.34, 3.39.
After week 7, we had a study week.
Week 8:
Notes:
We started studying systems in equilibrium with reservoirs of energy
and particles (reservoirs being large systems which stay at constant
temperature and chemical potential). We introduced and derived the
Boltzmann and Gibbs probability distributions and partition functions for
the canonical and grand canonical formalisms (see Schroeder
sections 6.1 and 7.1 for these derivations).
As a first example, we calculated the average energy of a two state paramagnet
at a given temperature
In the tutorial, we also made some remarks on the definition of
temperature. We noted in
particular that systems with a finite number of states such as a two
state paramagnet will have a "negative temperature" (negative derivative
of entropy by energy) at (very) high energy,
but that this is usually not an equilibrium property as such systems can
never be in equilibrium with a "normal" system at positive
temperature.
We discussed thermal averages and showed how to obtain the average energy
as a derivative of the canonical partition function
(see Schroeder section 6.2).
We also derived the factorization formulas for the partitions functions of
non-interacting systems (where the energies of the parts add up),
including including approximate factorization formulas for dilute systems of
identical particles (section 6.6).
Exercises: Assignment 4 consists of exercises 3.11, 3.25, 6.18
and 6.19 in Schroeder. Ex. 6.20 is also recommended.
Week 9:
Notes:
We derived the equipartition theorem for the average energy of
classical, quadratic degrees of freedom, using
the Boltzmann distribution (Schroeder section 6.3) and spent a bit of
time thinking about "freezing out" of quantum degrees of freedom at low
temperatures.
We then revisited the ideal gas, this time treating it fully
quantum mechanically as a system of non-interacting quantum particles in a box
(cf. Schroeder section 6.7).
We expressed our notion of a `dilute' gas more precisely in terms of the
quantum length or quantum volume (there should be a lot more volume
available per particle than the quantum volume).
We allowed the particles to have internal degrees of freedom, decoupled
from their translational degrees of freedom (so that the partition
function factorized into a translational and an internal part). As an
(important) example, we started treatment of the rotational degrees of
freedom of diatomic molecules (section 6.2).
Exercises: Recommended exercises in the book by Schroeder
are
6.20 partition functions for harmonic oscillators
6.26, 6.29 - on rotational degrees of freedom and `freezing out'
6.31, an equipartition result for linear degrees of freedom
6.32 particle in a 1-dimensional potential
6.22, on multi-state paramagnets
7.6, a similar exercise to 6.18 and 6.19, on the fluctuations of the
number of particles in a system in contact with a particle reservoir.
After week 9, we had our Easter break.
Week 10:
Notes:
We gave two derivations of the relation F=-kT log(Z) between the
canonical partition function Z and the Helmholtz free energy
F=U-TS (see
Schroeder section 6.5 for one of them). We noted that the
entropy, pressure and chemical potential can all be obtained as
partial derivatives of the free energy and we calculated these
quantities for the ideal gas (we reproduced the ideal gas law and
the Sackur-Tetrode formula for the entropy).
We then started on quantum statistics, noted that there are two kinds
of particles, bosons and fermions, distinguished by their behaviour under
exchanges (symmetric vs. antisymmetric)
In an ideal gas, the
difference between these becomes important when the temperature is such
that number of available states per particle is low, or equivalently when
the gas is not dilute in the sense mentioned above.
Exercises:
Assignment 5 consists of exercises 6.22,6.31 and 6.32
Other recommended exercises are as last week, and additionally,
6.39, on escape velocity of gas molecules.
6.41, derivation of the Maxwell-Boltzmann distribution for a 2-dimensional
gas.
6.43 alternative formula for the entropy
6.44 Free energy and chemical potential for a dilute gas of
non-interacting particles.
6.48, Sackur-Tetrode formula for a diatomic gas.
6.51, Semiclassical derivation of the translational partition function for
a gas of identical particles.
Week 11:
Notes:
Using the grand canonical formalism (section 7.1),
we derive the Fermi-Dirac and Bose-Einstein distributions for the
occupation numbers of states in a system of non-interacting fermions resp.
bosons (all this in section 7.2).
We introduce the density of states (section 7.3) and its use
in combination with the occupation number distribution functions.
We then start on degenerate Fermi gases (section 7.3),
showing that the Fermi energy and chemical potential are equal
at zero temperature and calculating the Fermi energy as well as the
average energy at zero temperature.
Exercises:
Assignment 6 consists of exercises 7.23 and 7.28 in Schroeder