New Horizons in Multidimensional Diffusion: The Lorentz Gas and the Riemann Hypothesis
New Horizons in Multidimensional Diffusion: The Lorentz Gas and the Riemann Hypothesis
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多维扩散的新视野：洛伦兹气体和黎曼假设
DOI：
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发表时间：
2011
期刊：
影响因子：
--
通讯作者：
C. Dettmann
中科院分区：
文献类型：
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作者：
C. Dettmann
洛伦兹气体是一个台球模型，涉及点粒子在凸散射体的周期性阵列中确定性扩散。在二维有限水平情况下，所有轨迹都涉及与散射体的碰撞，按通常的扩散因子 $sqrt{t}$ 缩放的位移呈正态分布，如 Bunimovich 和 Sinai 在 1981 年所示。在无限水平情况下，运动是超扩散的，但是当按 $sqrt {tln t}$ 缩放时，会恢复正态分布，并有其方差的显式公式。在这里，我们探索任意维度的无限视界情况，给出均方位移的明确公式，论证它与极限分布的方差不同，在小散射体极限下与黎曼假设联系起来，并为临界维度 d=6 提供证据，超过该维度相关性衰减表现出分数幂。该结果以许多猜想为条件，并通过多达十个维度的数值模拟得到证实。
The Lorentz gas is a billiard model involving a point particle diffusing deterministically in a periodic array of convex scatterers. In the two dimensional finite horizon case, in which all trajectories involve collisions with the scatterers, displacements scaled by the usual diffusive factor $sqrt{t}$ are normally distributed, as shown by Bunimovich and Sinai in 1981. In the infinite horizon case, motion is superdiffusive, however the normal distribution is recovered when scaling by $sqrt {tln t}$, with an explicit formula for its variance. Here we explore the infinite horizon case in arbitrary dimensions, giving explicit formulas for the mean square displacement, arguing that it differs from the variance of the limiting distribution, making connections with the Riemann Hypothesis in the small scatterer limit, and providing evidence for a critical dimension d=6 beyond which correlation decay exhibits fractional powers. The results are conditional on a number of conjectures, and are corroborated by numerical simulations in up to ten dimensions.
影响因子：
1.8
作者：
I. Melbourne
通讯作者：
I. Melbourne