Anomalous Diffusion in Deterministic Systems

确定性系统中的反常扩散

• 批准号: EP/F031807/1
• 负责人: Ian Melbourne
• 金额: $ 37.95万
• 依托单位: University of Surrey
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2008
• 资助国家: 英国
• 起止时间: 2008 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FF031807%2F1
• 关键词: Anomalous; Diffusion; Deterministic; Systems

Certain gas models (planar periodic Lorentz flows) were introduced as deterministic models for Brownian motion. A statistical law called the almost sure invariance principle (ASIP) makes precise the connection with Brownian motion. Recent work established the ASIP for the Lorentz flow.Brownian motion has a characteristic growth rate of order sqrt t (square root of t ) where t denotes time. Currently, there is a great deal of interest in anomalous diffusion with a different growth rate. The aim of this project is the systematic study of anomalous diffusion (both subdiffusive and superdiffusive) in deterministic systems. In the superdiffusive (fast) case, this means proving the ASIP with Brownian motion replaced by a Levy process. In the subdiffusive (slow) case, we will focus on Sinai diffusion where the growth rate is of order (log t)^2.The results will apply to infinite horizon Lorentz gases and Bunimovich stadia. In addition, there are applications in dynamical systems with symmetry, including spiral waves in excitable media, and travelling waves.

某些气体模型（平面周期洛伦兹流）被引入作为布朗运动的确定性模型。一个被称为几乎必然不变性原理（ASIP）的统计定律使布朗运动的联系更加精确。最近的工作建立了Lorentz流的ASIP，布朗运动具有sqrt t（t的平方根）阶的特征增长率，其中t表示时间。目前，人们对具有不同增长率的反常扩散有很大的兴趣。这个项目的目的是系统地研究确定性系统中的反常扩散（亚扩散和超扩散）。在超扩散（快速）的情况下，这意味着用Levy过程代替布朗运动来证明ASIP。在亚扩散（慢扩散）情形中，我们将重点讨论增长率为（log t）^2量级的西奈扩散。结果将适用于无限视界洛伦兹气体和布尼莫维奇视距。此外，还有对称性动力系统的应用，包括可激发介质中的螺旋波和行波。

项目成果

Decay of correlations for flows with unbounded roof function, including the infinite horizon planar periodic Lorentz gas

具有无界屋顶函数的流动的相关性衰减，包括无限地平线平面周期性洛伦兹气体

• 作者: Ian Melbourne (Author)

Statistical Properties and Decay of Correlations for Interval Maps with Critical Points and Singularities

具有临界点和奇点的区间图的统计特性和相关性衰减

• DOI: 10.1007/s00220-013-1709-y
• 发表时间: 2013
• 期刊: Communications in Mathematical Physics
• 影响因子: 2.4
• 作者: Luzzatto S

Central limit theorems and suppression of anomalous diffusion for systems with symmetry

对称系统的中心极限定理和反常扩散的抑制

• DOI: 10.1088/0951-7715/29/10/2941
• 发表时间: 2016
• 期刊: Nonlinearity
• 影响因子: 1.7
• 作者: Gottwald G

Decay of correlations for nonuniformly expanding systems with general return times

具有一般返回时间的非均匀扩展系统的相关性衰减

• DOI: 10.48550/arxiv.1108.5437
• 发表时间: 2011
• 作者: Melbourne I

ON YOUNG TOWERS ASSOCIATED WITH INFINITE MEASURE PRESERVING TRANSFORMATIONS

论与无限措施保持转变相关的年轻塔

• DOI: 10.1142/s0219493709002816
• 发表时间: 2011
• 期刊: Stochastics and Dynamics
• 影响因子: 1.1
• 作者: BRUIN H