报告人：Jin Feng (Department of Mathematics, University of Kansas, USA)
时间：2023-07-21 9:00-10:00
地点：Room 1304, Sciences Building No. 1
Abstract:
One of the well known open problems at the interface of probability and mechanics is "how probability emerges, through first principle derivations of pure deterministic mechanical nature, in statistical and continuum mechanics?". To move this one step further, we pursue an understanding on hydrodynamic limit for interacting particles following deterministic Hamiltonian dynamics. Traditional approach on such a program faces many difficulties. One of them is about rigorous justification of canonical type ensembles. This is because relevant deterministic ergodic theory is still largely out of reach. Another huge barrier is on making sense of rigorous meaning of hyperbolic conservation laws. Such PDEs are needed to express F=ma and thermodynamic relations in the continuum.
To make possible relevant progress, we insist on using deterministic model as initial building blocks, but we lower expectations on other aspects of the issue so that we can introduce mathematical modifications to simplify. To be precise, first, we only study global in time action minimizing (instead of critical point) dynamics of particles, from a variational point of view. Second, we only consider situations where particle interactions are weak enough so that thermodynamics is mostly absent (isentropic case). Third, we are content with not seeing probability models explicitly in the end at this point, but only deterministic variational problems matching asymptotic (e.g. large deviations) of the probability models with some other hidden scaling parameters.
With the above reductions, we then explore a new formulation of the hydrodynamic limit issue as abstract multi-scale Hamilton-Jacobi theory in space of probability measures. Interestingly, there are enough mathematical tools available for us to offer a round-about way to avoid some hard technical issues and to make meaningful statements. The goal of this talks is to outline such a program.
This is work in progress with Toshio Mikami in Tsuda University, Japan.
