Oxford Lectures 2025

From Werner KRAUTH

My 2025 Public Lectures at the University of Oxford (UK), entitled Algorithms and computations in theoretical physics, run from 21 January 2025 through 11 March 2025.

Lecture 1: 21 January 2025

We start our parallel exploration of physics and of computing with the concept of sampling, the process of producing examples (“samples”) of a probability distribution. In week 1, we consider “direct” sampling (the examples are obtained directly) and, among the many connections to physics, will come across the Maxwell distribution. In 1859, it marked the beginning of the field of statistical physics.

Here are the notes for the first lecture (21 January 2025).

Lecture 2: 28 January 2025

We start with a Special Topic A, where we supplement the first Lecture with two all-important special topics that explore the meaning of convergence in statistics, and the fundamental usefulness of statistical reasoning. We can discuss them here, in the direct-sampling framework, but they are more generally relevant. The strong law of large numbers, for example, will turn into the famous ergodic theorem for Markov chains.

Here are the notes for Special Topic A (28 January 2025).

Then, in this second lecture proper, we consider Markov-chain sampling, from the adults' game on the Monte Carlo heliport to modern ideas on non-reversibility. We discuss a lot of theory, but also six ten-line pseudo-code algorithms, none of them approximate, and all of them as intricate as they are short.

Here are the notes for Lecture 2.

Lecture 3: 04 February 2025

In this third lecture, we consider Markov-chain sampling in an abstract setting in one dimension. We discuss some theory, but also seven ten-line pseudo-code algorithms, none of them approximate, and all of them as intricate as they are short. At the end, we discuss the foundations of statistical mechanics, as seen in a one- dimensional example.

Here are the notes for Lecture 3.

• Here is the Metropolis algorithm for a single particle in an anharmonic potential
• Here is the factorized Metropolis algorithm for a single particle in an anharmonic potential
• Here is the Zig-zag algorithm for a single particle in an anharmonic potential

Lecture 4: 17 February 2025

In this fourth lecture, we treat classical many-particle systems that interact with hard-sphere potentials, a case of importance for the foundations of statistical mechanics but also for its applications. We move from Newtonian mechanics to Boltzmann mechanics and from classical mechanics to statistical mechanics, in a way that is again entirely example-based. We then treat the case of one-dimensional hard spheres and discover the Asakura--Oosawa interaction, the fifth force in nature. We will end with a discussion of the double nature of pressure: kinematic and thermodynamic.

Here are the notes for Lecture 4.

• Here is the event-driven algorithm for four hard disks in a square box, in a few dozen lines of code.

Lecture 5: 18 February 2025

In this fifth lecture, we treat classical many-particle systems that interact with general interactions, in the example of the harmonic chain.

Lecture notes for lecture 5 are forthcoming, but most of the material can be found here.

• Here is the Lévy-construction algorithm for the harmonic chain, in nine lines of code. It directly samples configurations.
• Here is the Metropolis algorithm for the harmonic chain, in two dozen lines of code. It samples configurations using the algorithm developed by Metropolis et al. in (1953).
• Here is the Hamiltonian Monte Carlo algorithm for the harmonic chain, in three dozen lines of code. It implements the algorithm developed by Duane et al. (1988).

Lecture 6: 25 February 2025

This sixth lecture is the first of two lectures on quantum statistical mechanics. Starting from the quantum harmonic oscillator, we introduce to density matrices and the Feynman path integral. The computational techniques that we concentrate on during these two weeks have widespread use in statistics and physics, and the fundamental matrix-squaring property of density matrices corresponds to the convolution of probability densities. The Lévy algorithm for the sampling of path integrals, on the other hand, is mathematically equivalent to what we discussed, in a totally different context, in Lecture 5.

Here are the notes for Lecture 6.

Lecture 7: 04 March 2025

This is the second of two lectures on computational quantum statistical mechanics. Building on the quantum harmonic oscillator, the density matrices and the Feynman path integral introduced in the sixth lecture, we will discuss a direct-sampling quantum Monte Carlo algorithm for bosons that illustrates the phenomenon of Bose–Einstein condensation. Before doing that, however, we must discuss quantum statistics of indistinguishable particles and set up a naive program against which to check our state-of-the-art simulation code.

Here are the notes for Lecture 7.

• Here is Two-cycles.py, an intricate seven-line algorithm for sampling random permutations... but only those without cycles longer than 2!
• Here is Direct-bosons.py, a three-dozen-line direct-sampling algorithm for ideal bosons in a three-dimensional harmonic trap.

Lecture 8: 11 March 2025

The final lecture in the set is on the classical Ising model. We go from basic (and not so basic) enumerations to the high-temperature expansion of the Ising model, touch on its exact solution by Kac and Ward (1952) and then treat sampling methods: Metropolis, heatbath, and perfect sampling, to finish with the Wolff (1989) cluster algorithm.

Here are the notes for Lecture 8.

• Here is Enumerate-ising.py that uses the Gray code enumeration to compute the density of states of the Ising model.
• Here is Markov-ising.py that uses the Metropolis algorithm to sample the Boltzmann distribution of the Ising model.
• Here is Heatbath-ising.py that uses the Heatbath algorithm for the above sampling problem.
• Here is CLuster-ising.py that uses the Wolff cluster algorithm, again for sampling the Boltzmann distribution of the Ising model.