Announcements
⭐ Submission form for NB1 (due 22 Oct 2026) click here.
Week 1: 18-20 Aug (asynchronous)
□ Read on random variables, pdfs, characteristic functions, convolutions, joint pdfs, conditional pdfs.
Week 2: 25-27 Aug: Stochastic variables
References vK I, RL 4
□ Provide a scientific or mathematical definition of the probability of an event/experimental outcome.
□ Provide a technical description of a stochastic variable.
□ Give the conditions needed for a function to be a valid probability distribution function.
□ Demonstrate how the characteristic function is used as a generating function for the cumulants of a probability distribution.
□ Define the elements of the covariance matrix.
□ Give the central limit theorem and emphasize the conditions needed for it to hold.
W03 4 Sep: Averages
References vK I
! Moments and cumulants are expectation values of polynomial functions of a random variable. They provide measurable information about unknown probability distributions.
□ Give the characteristic function as an expectation value of a function of a stochastic variable.
□ Demonstrate how the characteristic function and its logarithm generate moments and cumulants (expectation value of special linear combinations of powers of the stochastic variable).
□ Discuss some interesting and useful situations where "special linear combinations of powers of a stochastic variable" are needed.
W04 8-10 Sep: Multivariate distributions
References vK I.3
□ Define the notation used for a multivariate PDF. Show how these are normalized.
□ Define the notation used for a reduced or marginal multivariate PDF as a partially summed multivariate PDF. Try doing the same for quantum density matrices with a partial trace and highlight the similarities.
□ Define the notation used for a multivariate conditional PDF. Write down Bayes' rule as an equation and in words.
W05 16-18 Sep: Functions of stochastic variables
References vK I.4, I.5
□ Show how the addition of two independent stochastic variables leads to a convolution of PDFs.
□ Show that the addition of two independent stochastic variables is simple in Fourier space. Discuss if this simplification can be done if the variables are dependent.
□ Discuss some applications why one would need transformations or functions of a stochastic variable like Y = f(X). Write down the characteristic function of Y and apply it to the addition of squares of a stochastic variable (e.g. Maxwell's speed distribution as an energy density, or the similar construction of a χ2 distribution for error analysis).
W06 22-24 Sep: Random walks and stochastic processes
References RL 5.C.2
□ Given a symmetric random walk (p = q = 1/2) with an exact number of steps N, give the probability that the walk has k steps to the right. Show how the binomial distribution arises in the random walk problem.
□ Give the Stirling approximation.
□ Discuss how the PDF of displacements in the random walk approaches a Gaussian distribution.
□ Define a stochastic process and transition probabilities.
□ Describe the master equation for asymmetric random walks.
□ Obtain the drift-diffusion equation from the master equation.
W07 29 Sep - 1 Oct: Exercises
Midterms (in class, coverage is week 01 to week 06): 08 Oct 2026 (Th). May use books, notes, no electronic devices.
W09 13-15 Oct: Equilibrium Statistical Mechanics I
Ref. Pathria 1-2
□ In a microcanonical ensemble, show that the energy of an N-atom ideal gas satisfies the equipartition theorem.
□ From the fundamental equation, derive the mechanical equation of state, as well as the heat capacities of the ideal gas.
⭐ Submission form for NB1 (due 22 Oct 2026) click here.
W10 20-22 Oct: Equilibrium Statistical Mechanics II
□ Derive the Boltzmann factor ni ∝ exp(-βEi) in a canonical ensemble with fixed particle number, fixed average weighted energy, and extremal Boltzmann entropy.
☆ Problem Set:
1A. Consider N distinguishable particles in a 3D infinite quantum well of volume V. Derive the ideal gas equation of state for this system.
1B. Derive the entropy of this gas of distinguishable particles and illustrate Gibbs paradox.
1C. Resolve Gibbs paradox by imposing indistinguishability and using Stirling's approximation in the thermodynamic limit.
Some class requirements:
1. □ Notebook entries
You must submit two notebook files (one at midterm, and another before the finals week). Each file must contain weekly notebook entries with a new section or page made for each week. Each entry should demonstrate that each box-bulleted □ outcome item in the syllabus has been achieved. For example, you can write a short summary or outline of the topic, or give a list of steps in the derivation of a core result, or provide a worked out representative example. I suggest that you monitor your progress by ticking each box □ in this course guide as you complete these notebook entries.
2. ☆ Problems
You must submit one problem set (before the finals week). Each set must contain a solution to a problem in this course pack, which is marked by a star ☆ in the course outline. You are encouraged to discuss these problems with your classmates and help each other, but you must work independently and write up your own reports and solutions. You can also ask me for help: I will assess your solutions, so it is best to ask me first. It is recommended to do the problems in parallel with the learning units, but you can work on them in any order and at your own pace.
For reference, past course guide is here: Physics 151
Faculty
- Francis Paraan