26 - 28 June 2024 4th BMS-BGSMath Junior Meeting @ZIB
26 June - 11 July 2024 “Interacting particle systems, mean field limits, phase transitions and fluctuations” @FU/ZIB
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26 June - 11 July 2024 “Interacting particle systems, mean field limits, phase transitions and fluctuations”
Our MATH+ Distinguished Visiting Fellow Grigorios A. Pavliotis (Imperial College London) will give a lecture series on “Interacting particle systems, mean field limits, phase transitions and fluctuations” in June and July.Abstract: In these lectures we will present a very brief introduction to the study of weakly interacting diffusions and of their mean field limit. We will introduce the models that we will study and we will discuss about different methods for proving propagation of chaos. Then, we will study phase transitions, i.e. nonuniqueness of stationary states for the mean field dynamics. We will also comment briefly on the properties of fluctuations around the mean field dynamics. The lectures is targeted towards MSc and PhD students.
Prerequisites include: stochastic differential equations, basic functional analysis, some familiarity with PDEs and statistical mechanics.
Contents:
Lecture 1: Introduction and examples: Ornstein-Uhlenbeck interacting particles, the Desai-Zwanzig model, the noisy Kuramoto (XY, O(2), Brownian mean field) model.
Lecture 2: Propagation of chaos (1): uniform propagation of chaos via coupling methods in the convex case.
Lecture 3: Propagation of chaos (2): entropic methods, break-down of uniform propagation of chaos in the non-convex case and phase transitions.
Lecture 4: Phase transitions and fluctuations: The free energy formulation, integral equation for the invariant measure(s) of the mean field dynamics, phase transitions for the McKean-Vlasov PDE on the torus. Gaussian fluctuations in the absence of phase transitions.
Assessment: If students want to take the short course for credit, the assessment will consist of assigning students a relevant paper asking them write a short report.
Dates and location – all lectures start at 2 pm (14:00)
Wednesday, 26 June – Seminar room 031 in Arnimallee 6, Pi-building
Tuesday, 02 July – Lecture Hall at ZIB
Wednesday, 09 July – Lecture Hall at Arnimallee 3
Tuesday, 11 July – Seminar room 032 in Arnimallee 6, Pi-building
Bibliography
Dawson, D.A. Critical dynamics and fluctuations for a mean-field model of cooperative behavior. J Stat Phys 31, 29–85 (1983).
Malrieu, F. https://mathscinet-ams-org.iclibezp1.cc.ic.ac.uk/mathscinet/2006/mathscinet/search/author.html?mrauthid=672787
Logarithmic Sobolev inequalities for some nonlinear PDE's. Stochastic Process. Appl. (2001), no. 1, 109–132
Chavanis, PH. The Brownian mean field model. Eur. Phys. J. B 87, 120 (2014).
Carrillo, J.A., Gvalani, R.S., Pavliotis, G.A., Shlichting, A.. Long-Time Behaviour and Phase Transitions for the Mckean–Vlasov Equation on the Torus. Arch Rational Mech Anal 235, 635–690 (2020).
Delgadino, M.G., Gvalani, R.S. & Pavliotis, G.A. On the Diffusive-Mean Field Limit for Weakly Interacting Diffusions Exhibiting Phase Transitions. Arch Rational Mech Anal 241, 91–148 (2021).
Delgadino, M.G., Gvalani, R.S., Pavliotis, G.A., Smith, S.A.. Phase Transitions, Logarithmic Sobolev Inequalities, and Uniform-in-Time Propagation of Chaos for Weakly Interacting Diffusions. Commun. Math. Phys. 401, 275–323 (2023).
Summer semester 2024
10 - 12 April 2024 5th Graduate Student Meeting in Applied Algebra and Combinatorics @FU
17 May 2024 Knot Theory Workshop @Potsdam
