Long time dynamics for interacting oscillators on graphs

稠密随机图上的弱相互作用振子

Weakly interacting oscillators on dense random graphs

本文研究图序列上弱相互作用粒子系统，证明当图序列收敛到图子时经验测度满足大数定律，其极限为图子加权的非线性 Fokker-Planck 方程解，并刻画了收敛到经典 McKean-Vlasov 极限的图序列。

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.

Euclidean Gibbs states of interacting quantum anharmonic oscillators

A rigorous description of the equilibrium thermodynamic properties of an infinite system of interacting $ν$-dimensional quantum anharmonic oscillators is given. The oscillators are indexed by the elements of a countable set $\mathbb{L}\subset \mathbb{R}^d$, possibly irregular; the anharmonic potentials vary from site to site. The description is based on the representation of the Gibbs states in terms of path measures -- the so called Euclidean Gibbs measures. It is proven that: (a) the set of such measures $\mathcal{G}^{\rm t}$ is non-void and compact; (b) every $μ\in \mathcal{G}^{\rm t}$ obey…

Oscillatory behavior in a model of non-Markovian mean field interacting spins

We analyze a non-Markovian mean field interacting spin system, related to the Curie--Weiss model. We relax the Markovianity assumption by replacing the memoryless distribution of the waiting times of a classical spin-flip dynamics with a distribution with memory. The resulting stochastic evolution for a single particle is a spin-valued renewal process, an example of two-state semi-Markov process. We associate to the individual dynamics an equivalent Markovian description, which is the subject of our analysis. We study a corresponding interacting particle system, where a mean field interaction …

高温区相互耦合振子异常行为的推导

Derivation of anomalous behavior from interacting oscillators in the high-temperature regime

本文在高温区研究具有体积和能量守恒的相互耦合振子模型，通过泰勒展开提取谐振主部，并证明两个涨落场分别收敛于 Ornstein-Uhlenbeck 过程或随机 Burgers 方程，以及偏斜 Lévy 过程驱动的分数阶扩散方程。