---
title: "Humboldt University's Free Course Proves Physics and LLMs Share the Same Math"
subtitle: "Olaf Hohm releases master-level lecture notes tracing the shared mathematical language of spin glasses, Boltzmann machines, and modern LLMs"
url: "https://alphasignal.ai/news/humboldt-university-s-free-course-proves-physics-and-llms-share-the-same-math"
author: "AlphaSignal Newsroom"
author_url: "https://alphasignal.ai/editorial-team"
company: "AlphaSignal"
company_url: "https://alphasignal.ai/company/alphasignal"
published: "2026-05-07T15:08:49.000Z"
updated: "2026-06-15T01:07:24.742Z"
type: "Paper"
topics:
  - "Llms"
  - "Post Training"
subtopics:
  - "Alignment"
  - "Fine Tuning"
upvotes: 191
source_url: "https://arxiv.org/abs/2605.06394"
comments: 0
truncated: true
---

# Humboldt University's Free Course Proves Physics and LLMs Share the Same Math

> Olaf Hohm releases master-level lecture notes tracing the shared mathematical language of spin glasses, Boltzmann machines, and modern LLMs

Physics and machine learning have always shared a vocabulary. Terms like temperature, entropy, and energy function appear in both fields for a reason: the mathematical structures underlying them are genuinely the same. A new set of [lecture notes by Olaf Hohm](https://arxiv.org/abs/2605.06394) of Humboldt University Berlin makes this connection explicit and rigorous, building a self-contained path from classical statistical physics all the way to transformers and large language models.

The notes grew out of a master-level course on statistical physics. They are based on a master-level course at Humboldt University Berlin, covering about 30 to 40 percent of that course, focusing on the classical statistical mechanics of phase transitions and the renormalization group together with an introduction to neural networks and the basics underlying the modern AI revolution such as large language models. The stated goal is to make these ideas accessible to readers without a prior physics background, treating statistical physics as a branch of probability theory rather than as a separate discipline.

## The core thread: energy functions all the way down

The notes open with the Boltzmann-Gibbs distribution, the foundational probability distribution of statistical physics. From there, they introduce Ising spin models and their generalization to spin glasses, which are systems with disordered, randomly signed interactions between sites. The key insight that runs through the entire document is that neural networks are not just inspired by physics, they literally share the same energy function.

Hopfield networks and Boltzmann machines are governed by the same energy function as spin-glass models. In a Hopfield network, each neuron takes a value of +1 or -1 and is connected to every other neuron by a symmetric weight. The network evolves by updating each neuron to minimize a global energy function, and that energy function is exactly the spin-glass Hamiltonian, with neuron states playing the role of spins and connection weights playing the role of coupling constants.

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## Sources

- [Lecture Notes on Statistical Physics and Neural Networks (arXiv:2605.06394)](https://arxiv.org/abs/2605.06394)
- [Full HTML version of the lecture notes](https://arxiv.org/html/2605.06394)
- [Scaling Laws for Neural Language Models (Kaplan et al., arXiv:2001.08361)](https://arxiv.org/abs/2001.08361)