Heat flows from hot to cold, but mathematicians still cannot fully prove why this happens at the atomic scale. This project tackles that gap by building rigorous mathematical foundations for Fourier's law of heat conductivity—a rule engineers use daily but physicists have never fully derived from first principles. The research uses kinetic theory, the branch of physics that explains how large-scale properties like temperature emerge from countless microscopic particle collisions. By analysing equations such as the Boltzmann equation for gases and the wave kinetic equation for turbulence, the team aims to prove that heat conduction laws hold true even when systems are far from equilibrium. This is fundamental science: it addresses Hilbert's Sixth Problem, a century-old challenge to axiomatise physics mathematically. If successful, the work will not produce a new gadget or drug. Instead, it could sharpen climate models by putting their heat-transfer assumptions on solid mathematical ground, and help engineers design more efficient thermodynamic materials. The methods developed—linking stochastic particle systems to macroscopic equations—may also lead to better algorithms for coordinating autonomous vehicles or managing multi-agent systems. Past breakthroughs in kinetic theory, such as Cedric Villani's work on the Boltzmann equation, have already reshaped fields from plasma physics to data science.
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This proposal focuses on the analysis of partial differential equations and probability, motivated by the kinetic theory of particles, waves, or phonons in non-equilibrium or equilibrium statistical mechanics. The primary aim is to investigate the mathematical foundations of macroscopic physical laws, particularly the derivation and validation of Fourier's law of heat conductivity. By exploring its microscopic origins, the proposed research contributes to the Hilbert's Sixth Problem from various perspectives in kinetic theory. The first part of the proposal examines the derivation of Fourier's law of heat conductivity in microscopic models involving networks of interacting oscillators perturbed at the boundaries by heat baths. Significant results in this model have emerged from the stochastic analysis and statistical mechanics community, including contributions by Fields Medallist Martin Hairer. This project aims to characterize non-equilibrium steady states of this model, quantify scalings in terms of the number of particles, and finally validate the thermodynamic limit, through rigorous mathematical analysis and methods from spectral theory, optimal transport theory, functional inequalities, and probability. The second part investigates non-equilibrium steady states in the context of the kinetic theory of gases, focusing on the Boltzmann equation for dilute gases. This equation has been extensively studied, with breakthrough results by leading figures including Fields Medallists Cedric Villani and Pierre-Louis Lions. The goal is to study the Boltzmann equation under non-isothermal boundary conditions, which induce energy currents in the stationary state. Understanding them is critical towards validating Fourier's law macroscopically. These first two parts may have implications for environmental sciences by refining climate change prediction models through a better understanding of Fourier's law. Additionally, they could contribute to energy technologies and material sciences by aiding in the design of more efficient thermodynamic materials. The third part addresses Hilbert's Sixth Problem through the kinetic theories of waves and phonons. The main objects of study are the wave kinetic equation and the phonon Boltzmann equation, which arise in weak wave turbulence theory. The wave kinetic equation is the kinetic limit of several wave systems and nonlinear Schrödinger equations, while the phonon Boltzmann equation is the kinetic limit of anharmonic crystals, such as the historically significant Fermi-Pasta-Ulam-Tsingou (FPUT) oscillator chains, which were among the first problems to be simulated on early digital computers. We aim to understand the rich phenomenology of the long-time behaviour of solutions to these equations. Wave turbulence theory finds applications in various fields, including oceanography, nonlinear optics, plasma physics, astrophysics, condensed matter physics and acoustics. The fourth part proposes a novel consistency-stability approach to quantitatively derive the hydrodynamic limit from discrete stochastic particle systems to macroscopic partial differential equations. The main objectives include understanding the emergence of shocks in hyperbolic equations and providing a quantitative hydrodynamic limit for non-gradient systems. This quantitative understanding of macroscopic behavior emerging from microscopic interactions could lead to more efficient algorithms for multi-agent systems and autonomous decision-making.
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