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From statistical physics to complex systems
Can the whole not merely be the sum of its parts? How do collective patterns appear?
Could three molecules of water form ice?
Could higher-level abilities be created from interacting AI agents?
The notion of complexity pertains to systems in which somewhat unexpected properties emerge from the interplay of a sufficiently large number of entities — be they particles, living cells, artificial neurons, organisms, people, abtract agents... or even a mixture of some (or all!) of these.
Statistical physics has been the first branch of science to try and model in a mathematical manner such systems, focusing especially on the subtle and often elusive passage from the micro/individual level to the macro/collective level. This lecture course explores further how the mindset of statistical physics can provide fertile ground for the analysis and modelling of complexity — including across disciplinary boundaries.
Syllabus
I. Introduction [1h30]
1. What are complex systems?
→ Examples in nature, social sciences, etc.
2. Modelling complex systems
→ Statistical (descriptive) models & physical (generative) models
3. Narrative of the syllabus
II. Interactions, instabilities & collective effects [6h]
1. Curie-Weiss and the Random Ising Field Model paradigms
→ The Curie-Weiss paradigm: self-fulfilling prophecies, hysteresis
→ The Random Ising Field Model paradigm: crises & sudden opinion shifts
2. Flocking and collective motion
→ Vicsek model from microscopic to field theory for flocking
→ 1.0.1 statistical methods: maximum likelihood & baysian estimations
→ Generative model vs statistical model
3. Wealth exchanges 1h30 JRF
→ Kinetic exchange model
→ Altruistic vs selfish society
III. Agent-based models [4h30]
1. The Schelling model
→ Archetypal agent-based model in the social sciences
→ Phase diagram via analogy with spin systems & simulations
→ Markov chain description & analytical results
2. Grauwin et al. statphys formulation of the Schelling model
→ Local versus global detailed balance
→ Segregated steady state: "Maxwell construction" in social science context
3. “Agentivity” model
→ Mathematical framework
→ Agentic equilibrium
→ Examples & link with other equilibria
IV. Networks [4h30]
1. Formal introduction
→ Graph theory
→ Count-based measures: degree, centrality, betweenness, etc.
→ Random-walked based measures: Google PageRank, mixing index, etc.
2. Networks in practice
→ Preferential attachment for social networks
→ Planar networks for cities
→ Abstract networks: causal/Bayesian, ecosystems, art
3. Dynamical processes on networks
→ Temporal criticality: synchronization, failures
V. Heterogeneities [6h]
1. Inequalities
→ Thin tails/Fat tails, scale invariance
→ The « problem » with power-laws: concentration/localisation, generalised CLT
→ Example: Wealth condensation model
2. Multiplicative growth models for population
→ Uncorrelated growth, log-normal fluctuations
→ Sums of log-normals & condensation
→ Growth with redistribution/mixing
→ Continuous limit & the Hamilton-Jacobi-Bellman method
3. Measuring heterogeneities
→ Gini coefficient, segregation indices, etc.
→ Multiscalar measures, Random walk measures
→ Optimal transport measures
VI. Evolution & transmission [4h30]
1. Birth, death processes and branching processes
2. Tree models in biology and philology
→ Probability models for the tree(s) of life
→ Cultural transmission: the example of manuscripts
Practical
- Who?
Julien Randon-Furling, ENS Paris-Saclay (julien.randon-furling@ens-paris-saclay.fr)
Camille Scalliet, ENS, CNRS (camille.scalliet@phys.ens.fr)
- When?
Wednesdays 9:00 - 12:30.
Dates : 14, 21, 28 January, 4, 11, 18 February, 4, 11, 18 March
- Where?
Sorbonne Université Campus, room to be determined.
Prerequisites
The lecture course is designed to be accessible to a broad audience with a general training in mathematical sciences (physics, mathematics, economics, computational social sciences...) and a strong taste for modelling across disciplines. All open minds are welcome.
Exam
Exam Modalities: Scientific Flash Talk
Context: This course, "From Statistical Physics to Complex Systems," is designed to prepare you for research in fundamental and interdisciplinary physics. As you near the end of your M2 program and prepare for your internship and PhD, it is essential to develop key skills: synthesizing scientific content, presenting it clearly and concisely, and situating it within a broader framework.
Exam Format: The exam will take the form of a 5-minute Flash Talk (strictly timed), where you will present a scientific article of your choice related to the themes covered in this course. This format mirrors the expectations of scientific conferences, where the ability to communicate complex ideas effectively in a limited time is crucial. The audience will be composed of all other students taking the exam.
Pedagogical Objectives: This exercise aims to prepare you to: i) make a bibliographical search on a scientific topic and extract its essential elements. ii) Communicate your ideas clearly and persuasively, a critical skill in research. iii) Contextualize specific work within a broader framework, whether for a paper, thesis, or research project.
Schedule:
- Deadline to submit your slides: Monday 30th March 8am Paris time.
- Exam date: Wednesday April 1st, 9.00 to 12. 30. Everyone taking the exam should attend the whole session of flash talks.
On what aspects will I be judged?
Article Selection: The article must be scientifically rigorous and directly related to one or more concepts from the course (e.g., agent-based models, networks, phase transitions, inequalities, etc.). You must justify your choice by explaining why the article interested you and how it fits within the course framework.
Visual Support: A visual aid (slides) is mandatory. It should be clear, concise, and professional (avoid overloading with information).
Highlight the following:
- The research question of the article.
- The methods used.
- The main results and their significance.
- The strengths and limitations of the study.
Oral Presentation: Your presentation must be structured, fluid, and accessible to a general scientific audience (there should be physicists, mathematicians, and economists attending the lectures). You must explicitly connect the article to the course content.
Time Management: 5 minutes. Exceeding the time limit will result in a penalty.
Your final grade will be based on:
- Relevance of the article choice (25%).
- Quality of the visual support (25%): clarity, aesthetics, and effectiveness.
- Quality of the oral presentation (30%): structure, clarity, adherence to time, and ability to answer questions.
- Connection to the course (20%): explicit links to concepts, methods, or models covered in class.
Oh my god, I have never seen a flash talk in my life, should I still take the exam?
Yes! During the course, we will provide examples of Flash Talks (from conferences or former students) to guide you on format and expectations.