Growth Models and Hamilton-Jacobi PDEs
University of California
Abstract: Hamilton–Jacobi equation (HJE) is one of the most popular and studied PDE which enjoys vast applications in numerous areas of science. Originally HJEs were formulated in connection with the completely integrable Hamiltonian ODEs of celestial mechanics. They have also been used to study the evolution of the value functions in control and differential game theory. HJE associated with space-time stationary Hamiltonian functions are used to study turbulence in hydrodynamics. Several growth models in physics and biology are described by such HJEs and their viscous variants. In these models, a random interface separates regions associated with different phases and the interface can be locally approximated by the graph of a solution to a HJE. Naturally we would like to understand how the randomness affects the solutions and how the statistics of solutions are propagated with time. Lagrangian techniques in Aubry-Mather theory for action-minimizing trajectories, PDE techniques of weak KAM theory, and probabilistic methods related to first/last passage percolation problems have been employed to study long-time behavior of solutions. Most notably, a unique invariant measure has been constructed for any prescribed average velocity for some important examples of Hamiltonian functions. In these lectures I will give an overview of some of the existing results for the statistics of random solutions to HJEs. In particular, I will discuss a systematic approach for constructing Gibbsian solutions to Hamilton-Jacobi PDEs by exploring the Eularian description of the shock dynamics. Such Gibbsian solutions depend on kernels satisfying kinetic-like equations reminiscent of the Smoluchowski model for coagulating and fragmenting particles.
- Updated notes of the lecture will be made available at https://math.berkeley.edu/~rezakhan/KHJE_l'Aquila.pdf
Weakly interacting fermions: mean-field and semiclassical regimes
University of Basel
Abstract: The derivation of effective macroscopic theories approximating microscopic systems of interacting particles in some scaling limit is a major question in non-equilibrium statistical mechanics. In this course we will be concerned with the dynamics of systems made of many interacting fermions. We will focus on the mean-field regime, i.e. weakly interacting particles whose collective effect can be approximated by an averaged potential in convolution form, and review recent mean-field techniques based on second quantization approaches. As a first step we will obtain a reduced description given by the time-dependent Hartree-Fock equation. As a second step we will look at longer time scales, where a semiclassical description starts to be relevant, and approximate the many-body dynamics with the Vlasov equation, which describes the evolution of the effective probability density of particles on the one particle phase space. The structure of the initial data will play an important role at each step of the approximation.
- N. Benedikter, M. Porta, B. Schlein: Effective evolution equations from quantum dynamics. Springer Briefs in Mathematical Physics 7, 2016.
- C. Saffirio: The Vlasov equation as the mean-field and semiclassical limit of many interacting fermions. IAMP News Bulletin April 2022, http://www.iamp.org/bulletins/Bulletin-Apr2022-print.pdf.
Hydrodynamic scales of integrable many-particle systems
Technical University Munich
Abstract: While integrable many-particle systems are fine-tuned, microscopic models are very diverse. As to be discussed in the lectures, nevertheless the Euler type equations have always the same struscture. The Toda lattice will be used as a guiding example and further integrable models will be included. Covered are generalized Gibbs ensembles (GGE), random Lax matrix and its density of states, GGE averaged conserved fields and currents, and the resulting generalized hydrodynamics.
- H. Spohn, "Hydrodynamic scales of integrable many-particle systems", https://arxiv.org/abs/2301.08504.