Heat, Wave, and Schrödinger Dynamics of Gibbs Measures

Gibbs measures, first introduced in statistical mechanics, describe the statistical equilibria of complex physical systems. One can visualize a Gibbs measure by imagining water waves rippling across a lake on a windy day. The waves are constantly in motion, and yet the overall appearance of the lake's surface remains roughly unchanged over time. This is because, although the system is dynamic, certain statistical properties like the number of waves, average height, and average velocity remain constant. In this way, the surface of the lake is not in equilibrium in the traditional sense, but rather in statistical equilibrium. Beyond water waves, Gibbs measures and their dynamics play a central role in several areas of physics, including condensed matter physics, plasma physics, and Euclidean quantum field theory. In addition to being of physical significance, Gibbs measures and their dynamics pose numerous mathematical challenges. The central mathematical challenge lies in understanding how the randomness introduced by the Gibbs measure evolves under the dynamics. In the setting of water waves described above, for example, one may need to understand the likelihood of a large wave being formed by several smaller waves. This project advances our mathematical understanding of physical phenomena and helps train both undergraduate and graduate students, thereby strengthening the STEM workforce in the United States. The principal investigator (PI) and his collaborators pursue three projects at the interface of partial differential equations and probability theory, which aim to solve major open problems concerning Gibbs measures and their dynamics. In the first project, the PI studies the cubic nonlinear Schrödinger equation in three dimensions and plans to prove the invariance of the corresponding Gibbs measure. This is the first invariance result for a Gibbs measure that is critical with respect to probabilistic scaling. In the second project, the PI studies the well-posedness of the stochastic wave maps equation. Due to the geometric nature of the wave maps equation, this project not only combines techniques from partial differential equations and probability theory, but also differential geometry. In the third project, the PI plans to establish the global well-posedness of the stochastic Yang-Mills-Higgs heat flow in two dimensions. This result can be used to rigorously construct the two-dimensional Yang-Mills-Higgs measures, which play a central role in Euclidean quantum field theory. The three research projects described above forge new connections between partial differential equations, probability theory, and differential geometry. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.