--- title: "One Box, Three Worlds: Yu Deng’s Fields Medal Work — Sandbox Physics" description: "From Newtonian hard spheres to the Boltzmann and fluid equations: a visual guide to the long-time bridge proved by Yu Deng, Zaher Hani, and Xiao Ma." canonical_url: "https://sandboxphysics.com/stories/yu-deng-fields-medal/" language: "en" page_type: "story" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/stories/yu-deng-fields-medal/" last_modified: "2026-10-22" --- 2026 Fields Medal · Yu Deng # One box.*Three worlds.* Every sphere obeys Newton’s laws. Why should the gas as a whole obey a different, statistical equation? Writing down both descriptions is easy. Proving that they remain one story after a long history of collisions is not. - **Areas cited**: PDE · probability · mathematical physics - **Reading time**: About 12 minutes - **Sources**: 8 items · graded by type 01 Microscopic **Track particles** Position, velocity, elastic impact 02 Mesoscopic **Track a distribution** Probability density over position and velocity 03 Macroscopic **Track a fluid** Density, temperature, bulk velocity The equation does not replace the particles. The proof shows that, in a dilute limit, both descriptions make the same statistical predictions. Reading route 01 / The problem ## The equation is old. The new achievement is a proof that two worlds really connect. Imagine a transparent box. Inside it there is no “temperature,” no “pressure,” and no gas treated as one object. There are only hard spheres: straight flights interrupted by elastic collisions. If every position and velocity were known, Newtonian mechanics would determine the next moment. That is not how we normally describe air. We use temperature and pressure, or a function$f(t,x,v)$that counts how much matter lies near position $x$ with velocity$v$ at time $t$. This is the language of the Boltzmann equation. One-sentence version Yu Deng, Zaher Hani, and Xiao Ma proved that, in an appropriate dilute hard-sphere limit, deterministic particle dynamics produces the long-time statistical evolution described by the Boltzmann equation, throughout the lifespan of the corresponding Boltzmann solution.[1] [2] The difficult word is “proved.” Physics had long relied on the bridge, and computation had long used the equation on the far side. The mathematical obstacle was to show why the statistical description remains valid after collisions create long, shared histories among particles. 02 / Three languages for one gas ## Pull the camera back, and the main character changes At the **microscopic scale**, every sphere moves freely between impacts, with momentum and kinetic energy redistributed at each elastic collision. The description is concrete but enormous. MICROSCOPIC $$ \dot x_i=v_i,\qquad \dot v_i=0 $$ Track each particle between impacts. KINETIC $$ \partial_t f+v\cdot\nabla_x f=Q(f,f) $$ Ask how many particles belong to each class. FLUID $$ \rho(t,x),\quad u(t,x),\quad T(t,x) $$ Density, bulk velocity, and temperature take over. At the **kinetic scale**, particle identity disappears. Free transport sits on the left of the Boltzmann equation; the collision operator$Q(f,f)$describes how impacts move particles from one velocity class to another. At the **macroscopic scale**, moments of the distribution produce fluid variables. A companion work by Deng, Hani, and Ma continues the chain to compressible Euler and incompressible Navier–Stokes–Fourier equations within the dilute hard-sphere framework and limiting regimes stated in the paper.[3] The particle world remembers everything. The statistical world deliberately forgets most details. **Why does prediction survive that loss?** 03 / The two words that blocked the road ## For a short time, particles are strangers. Later, they share a past. Oscar Lanford established the hard-sphere-to-Boltzmann bridge in 1975, but only for a sufficiently short interval. The new work extends the derivation to arbitrarily long intervals, provided the corresponding Boltzmann solution continues to exist.[2] [7] Waiting longer changes the proof because a collision creates both a new velocity and a relationship. A hits B, B later hits C, and C eventually meets A again. Treating A and C as independent samples can now double-count or miss correlations. t₀ t₁ t₂ t₃ The number of impacts is not the only difficulty. The same particles can meet again through different paths. Microscopic hard-sphere dynamics is reversible, while the Boltzmann equation carries a statistical direction of time. The theorem does not settle the arrow of time in one line. What it proves, under precise low-density and initial-data hypotheses, is that an effective statistical evolution can emerge from reversible dynamics and remain valid over long times. 04 / The technical heart of the proof ## Keep the memory: encode collision histories as “molecules,” then cut carefully One turning point began with a simple question: if a long interval is too difficult, can it be divided into short ones? Deng has recalled that the idea arrived in a Korean fried-chicken shop in Providence.[8]The answer required a new framework because every time layer creates more possible collision histories. 1. 01 ### Layer time Break a long interval into short windows while preserving the information passed between them. 2. 02 ### Track correlations with cumulants Particles do not stay independent, so track how strongly their histories are linked. 3. 03 ### Encode histories as CH molecules Translate complicated integrals into combinatorial diagrams whose loops and layers reveal dangerous histories. 4. 04 ### Cut and control the remainder Use a careful algorithm to decompose large structures into pieces that can be bounded. The original paper describes the core in the same order: propagate a long-time cumulant structure that retains full collision history, reduce the estimates to combinatorial properties of collision-history diagrams, and control them with an elaborate cutting algorithm.[2]“Cutting” here means splitting the estimate into pieces that can each be bounded; no cases are thrown away. 05 / The tool was first developed for waves ## Particles collide and waves resonate, but the mathematical obstruction has a familiar shape Before hard spheres, Deng and Hani studied wave turbulence. The microscopic side is a nonlinear Schrödinger equation tracking phases and interactions. The statistical side is a wave kinetic equation tracking average energy transfer among Fourier modes. PARTICLES **Hard-sphere impacts** Identifiable particles generate an expanding collision genealogy. $$ \text{Newton}\Longrightarrow\text{Boltzmann} $$ WAVES **Mode resonances** Deterministic waves generate increasingly complicated interaction trees. $$ \text{NLS}\Longrightarrow\text{WKE} $$ They first derived the wave kinetic equation at the kinetic timescale under a specified scaling law, then addressed a fuller range of scalings. The later work identifies arbitrarily large bad diagrams, uncovers systematic cancellations among them, and builds a robust algorithm for the remaining diagrams.[4] [5] Particles and waves are physically different, but both pose the same task: derive a statistical equation from deterministic microscopic dynamics. The long-time expansions and diagrammatic experience developed for waves became a template for the hard-sphere problem. 06 / Randomness is structure, not noise ## One wave can become unrecognizable while the ensemble law stays the same The official medal citation also highlights Deng’s probabilistic approaches to nonlinear Schrödinger dynamics. Here the randomness has a precise structure: the initial data are drawn from a well-defined statistical ensemble.[1] With Andrea Nahmod and Haitian Yue, Deng studied defocusing nonlinear Schrödinger equations on the two-dimensional torus. They proved almost-sure global well-posedness with respect to the associated Gibbs measure and obtained invariance of that measure under the dynamics, using new random averaging operators.[6] Do not stare at one frame Shuffle a deck thoroughly. Every card changes position, yet the rule for drawing a random card is unchanged. Gibbs invariance describes stability at the ensemble level, not a wave frozen in place. The common theme with kinetic theory is striking: individual trajectories may be extremely complicated, but if probabilistic structure propagates, collective laws can remain stable, rigorous, and predictive. 07 / Precise claims make the result stronger ## What the work establishes — and what it does not claim - **It establishes a long-time bridge.** In the paper’s dilute hard-sphere limit and regularity regime, the Boltzmann description remains valid throughout the lifespan of the corresponding solution. - **It reaches a central program within Hilbert’s sixth problem.** The companion paper connects Newtonian particles through Boltzmann kinetics to fluid equations in this setting. - **It is not a statement about every material, initial condition, or time.** Hard spheres, the dilute limit, initial statistical structure, and the lifespan of the Boltzmann solution are part of the theorem. - **It does not axiomatize all of physics at once.** The accurate claim concerns the Newton-to-fluid program through Boltzmann kinetic theory for a specific dilute hard-sphere gas. 08 / Now the simulation ## An animation cannot prove the theorem. It can make the proof’s hardest objects visible. The first interactive release puts the first two questions on one synchronized stage, in the order this article follows: Lab 01 · Live ### Two views of one gas Track hard spheres on the left and only their empirical distribution on the right. Change particle count and dilution to compare the descriptions. Lab 02 · Live ### Draw collision genealogies Select a particle and unfold its collision history. Watch an ordinary branch become a correlation loop after a recollision. Lab 03 · Coming next ### Cut long time by hand Compare a single expansion with a layered one, and see why short windows make each step simpler but the combinatorics harder. [Interactive Lab · Live **Enter “Hard Spheres to Boltzmann”**](https://sandboxphysics.com/labs/hard-spheres-to-boltzmann/) Understand what the picture means before making it move. Then a simulation becomes a microscope for the argument, not fireworks after the conclusion. SOURCES / CITATION POLICY ## Where each claim comes from The article relies first on the IMU’s official citation and the original research papers. News reports are used only for biographical details and the story of the research. Preprints are labeled as preprints. 1. [1] Official citation 2026 ### Fields Medal 2026 — Yu Deng: Short and Long Citation International Mathematical Union International Congress of Mathematicians 2026 The main source for what the medal citation covers. [Open source](https://www.mathunion.org/fileadmin/documents/2026-07/Yu_Deng_Citations.pdf) 2. [2] Original research · preprint 2024 ### Long time derivation of the Boltzmann equation from hard sphere dynamics Yu Deng, Zaher Hani, Xiao Ma arXiv:2408.07818; Annals of Mathematics, to appear The central long-time derivation from hard-sphere dynamics. [Open source](https://arxiv.org/abs/2408.07818) 3. [3] Original research · preprint 2025 ### Hilbert’s sixth problem: derivation of fluid equations via Boltzmann’s kinetic theory Yu Deng, Zaher Hani, Xiao Ma arXiv:2503.01800 Continues the bridge from Boltzmann to fluid equations in the dilute hard-sphere setting. [Open source](https://arxiv.org/abs/2503.01800) 4. [4] Original research · published 2023 ### Full derivation of the wave kinetic equation Yu Deng, Zaher Hani Inventiones Mathematicae 233, 543–724 A rigorous derivation of wave kinetics from nonlinear Schrödinger dynamics. [Open source](https://doi.org/10.1007/s00222-023-01189-2) 5. [5] Original research · preprint 2023 ### Derivation of the wave kinetic equation: full range of scaling laws Yu Deng, Zaher Hani arXiv:2301.07063 Source for bad diagrams, systematic cancellation, and the bounding algorithm. [Open source](https://arxiv.org/abs/2301.07063) 6. [6] Original research · published 2024 ### Invariant Gibbs measures and global strong solutions for nonlinear Schrödinger equations in dimension two Yu Deng, Andrea R. Nahmod, Haitian Yue Annals of Mathematics 200(2), 399–486 Primary source for Gibbs invariance and random averaging operators. [Open source](https://doi.org/10.4007/annals.2024.200.2.1) 7. [7] Proof guide · preprint 2026 ### Derivation of the Boltzmann equation from hard-sphere dynamics (after Y. Deng, Z. Hani, and X. Ma) Thierry Bodineau, Isabelle Gallagher, Laure Saint-Raymond, Sergio Simonella arXiv:2602.04407 An expert exposition of the proof architecture. [Open source](https://arxiv.org/abs/2602.04407) 8. [8] Institutional reporting 2026 ### UChicago Prof. Yu Deng receives Fields Medal, highest honor in mathematics University of Chicago News University of Chicago Background on the research process and the idea of cutting a long time into short intervals. [Open source](https://news.uchicago.edu/story/uchicago-prof-yu-deng-receives-fields-medal-highest-honor-mathematics) This first part tells the scientific story. Interactive models will come next and follow the same line of argument. [Back to every experiment](https://sandboxphysics.com/)