A Law of Large Numbers for an Interacting Particle System with Confining Potential

Mean field limits for non-Markovian interacting particles: convergence to equilibrium, GENERIC formalism, asymptotic limits and phase transitions

In this paper, we study the mean field limit of interacting particles with memory that are governed by a system of interacting non-Markovian Langevin equations. Under the assumption of quasi-Markovianity (i.e. that the memory in the system can be described using a finite number of auxiliary processes), we pass to the mean field limit to obtain the corresponding McKean-Vlasov equation in an extended phase space. We obtain the fundamental solution (Green's function) for this equation, for the case of a quadratic confining potential and a quadratic (Curie-Weiss) interaction. Furthermore, for nonc…

Toward the Fourier law for a weakly interacting anharmonic crystal

For a system of weakly interacting anharmonic oscillators, perturbed by an energy preserving stochastic dynamics, we prove an autonomous (stochastic) evolution for the energies at large time scale (with respect to the coupling parameter). It turn out that this macroscopic evolution is given by the so called conservative (non-gradient) Ginzburg-Landau system of stochastic differential equations. The proof exploits hypocoercivity and hypoellipticity properties of the uncoupled dynamics.

动力学相互作用扩散的大数定律

A law of large numbers for kinetic interacting diffusions

该研究基于动力学非线性偏微分方程最新进展，在有限时间范围和仅初始数据弱收敛的一般假设下，证明了受独立布朗力驱动的相互作用动力学粒子系统经验分布依概率收敛到非线性福克-普朗克偏微分方程。

具有幂律长程跃迁的相互作用随机粒子的局域化

Localization of interacting random particles with power-law long-range hopping

该研究针对具有幂律长程跃迁的相互作用随机粒子体系，采用格林函数多尺度分析方法，证明了强无序条件下全能量范围的幂律局域化特性。

Non-interacting fermions at finite temperature in a $d$-dimensional trap: universal correlations

We study a system of $N$ non-interacting spin-less fermions trapped in a confining potential, in arbitrary dimensions $d$ and arbitrary temperature $T$. The presence of the trap introduces an edge where the average density of fermions vanishes. Far from the edge, near the center of the trap (the so called "bulk regime"), physical properties of the fermions have traditionally been understood using the Local Density Approximation. However, this approximation drastically fails near the edge where the density vanishes. In this paper we show that, even near the edge, novel universal properties emer…

Arcsine Law as the Classical Limit for interacting Fock spaces

In the present paper we discuss how to generalize ``Quantum-Classical Correspondence'' by means of the notion of interacting Fock spaces, which associates algebraic probability theory and the theory of orthogonal polynomials of probability measures. As an application we show that the Arcsine Law is ``Classical Limit'' for interacting Fock spaces corresponnding to certain kind of symmetric probability measures such as q-Gaussians. We also discuss the case of the exponential distribution as a simple example of asymmetric probability measures.