From non-symmetric particle systems to non-linear PDEs on fractals

Hermitian, symmetric and symplectic random ensembles: PDEs for the distribution of the spectrum

Given the Hermitian, symmetric and symplectic ensembles, it is shown that the probability that the spectrum belongs to one or several intervals satisfies a nonlinear PDE. This is done for the three classical ensembles: Gaussian, Laguerre and Jacobi. For the Hermitian ensemble, the PDE (in the boundary points of the intervals) is related to the Toda lattice and the KP equation, whereas for the symmetric and symplectic ensembles the PDE is an inductive equation, related to the so-called Pfaff-KP equation and the Pfaff lattice. The method consists of inserting time-variables in the integral and s…

具有中心对称雅可比矩阵的一维自相似分形：渐近性与模数据

1D self-similar fractals with centro-symmetric Jacobians: asymptotics and modular data

本研究针对一维自相似分形图，借助中心对称雅可比矩阵、相互作用福克空间与T-代数构建量子中心极限，拓展了结果适用范围，并证实其结合方案自对偶，可导出模不变任意子系统。

Entanglement in permutation symmetric states, fractal dimensions, and geometric quantum mechanics

We study the von Neumann and Rényi bipartite entanglement entropies in the thermodynamic limit of many-body quantum states with spin-s sites, that possess full symmetry under exchange of sites. It turns out that there is essentially a one-to-one correspondence between such thermodynamic states and probability measures on CP^{2s}. Let a measure be supported on a set of possibly fractal real dimension d with respect to the Study-Fubini metric of CP^{2s}. Let m be the number of sites in a subsystem of the bipartition. We give evidence that in the limit where m goes to infinity, the entanglement e…

Toda lattice G-Strands

Hamilton's principle is used to extend for the Toda lattice ODEs to systems of PDEs called the Toda lattice strand equations (T-Strands). The T-Strands in the $n$-particle Toda case comprise $4n-2$ quadratically nonlinear PDEs in one space and one time variable. T-Strands form a symmetric hyperbolic Lie-Poisson Hamiltonian system of quadratically nonlinear PDEs with constant characteristic velocities. The travelling wave solutions for the two-particle T-Strand equations are solved geometrically, and their Lax pair is given to show how nonlinearity affects the solution. The three-particle T-Str…

非凸势场中的多物种麦克凯恩-弗拉索夫动力学

Multi-species McKean-Vlasov dynamics in non-convex landscapes

研究非凸势场中多物种随机相互作用粒子系统及其平均场麦克凯恩-弗拉索夫偏微分方程。通过一般假设建立多物种随机微分方程系统适定性等，针对二次相互作用势研究平稳解等性质，构造自由能泛函并证明解及其自由能向稳态收敛。