Schedule and Homework for Statistical Mechanics (MP461)

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