DEVELOPEMENT OF RENORMALIZATION GROUP METHODS AS TOOLS OF ANALYSIS AND THEIR APPLICATIONS TO DYNAMICAL SYSTEMS

作为分析工具的重正化群方法的发展及其在动态系统中的应用

• 批准号: 11640220
• 负责人: ITO Keiichi r.
• 金额: $ 1.6万
• 依托单位: SETSUNAN UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640220/
• 关键词: Renormalization Group; Recursion Formula; Scaling; O(N)Spin Model; Pauli-Fierz Model; Navier-Stokes Equation; Couette Flow; Hopf Bifurcation; Kolmogorov law; self-avoiding walk; connectivity constant; block spin変換; polymer展開; Kolmogorov理論; i

1. Ito and Tamura (Kanazawa Univ.) studied classical O(N) symmetric spin model by renormalization group (block spin transformation) method. In the first stage, they argued the integrability of the functional determinent det^<N/2>(1+2iG_ψ/√<N>) with respect to ψ, where ψ is the auxially field introcuced for Fourier Transformation. Using the technique called polymer (cluster) expansion, they showed that the inverse critical temperature β_c obeys the bound β_c>N log N in two dimensions, which implies the existence of strong deviation. (β_c〜N for the dimension more than or equal to 3.) It is believed that β_c=∞ in the present model. To establish this conjecture, they recursively apply the BST to the model to decompose the determinant into product of many determinants which comes from fluctuations of various distance scales. They showed that the main part of the recursion relations is quite simple, and reproduces the flow of the hiererchical approximation of Wilson-Dyson type. It remains to … More control small non-local terms of the recursion formulas to solve the problem completely.2. Ito and Hiroshima (Setsunan Univ.) investigate the Pauli-Fierz Model which is rgarded as a classical Quantum Electrodynamics (QED). Though QED is believed to be trivial if no momentum cutoff is introduced, the Pauli-Fierz model may not. They apply the renormalization group type argumemt to the Pauli-Fierz model (this idea is originally due to J.Froelich(ETH)). But their analysis remains to be seen.3. Teramoto considere Couette-Taylor problems of the perturbation to the Couette flow between two rotating cylinders, and shows that the stationary bifurcation or Hopf bifurcation occurs when the Taylor number increases. He proved it with the help of numerical analysis by computer.4. Teramoto and Ito investigated properties of turbulence, among them, the Kolmogorov law about the dissipation of energy and deviation from it. They tried to derive the deviation from the Navier-Stokes equation but they could not obtain concrete results this year.5. Shimada characterized all possible selfadjoint extensions of Aharonov-Bohm hamiltonian in terms of boundary conditions at origin which distinguish the operator treated by Aharonov and Bohm from other possible ones. He developed scattering theory for their operators and obtained the phase shift formula. Less

1. Ito 和 Tamura（金泽大学）通过重正化群（块自旋变换）方法研究了经典的 O(N) 对称自旋模型。在第一阶段，他们论证了函数行列式 det^<N/2>( 的可积性。 1+2iG_ψ/√<N>) 相对于 ψ，其中 ψ 是使用称为聚合物（簇）的技术引入的辅助场。展开式表明，逆临界温度β_c在二维上服从约束β_c>N log N，这意味着存在很强的偏差（对于大于或等于3的维度，β_c〜N）。为了建立这个猜想，他们将 BST 递归地应用于模型，将行列式分解为来自不同距离尺度波动的许多行列式的乘积。部分递归关系非常简单，并且再现了 Wilson-Dyson 类型的层次近似流程，仍然需要控制递归公式的小非局部项来完全解决问题。 2. Setsunan Univ.）研究了被视为经典量子电动力学（QED）的保利-菲尔兹模型，尽管 QED 如果没有动量则被认为是微不足道的。引入了截止，Pauli-Fierz模型可能没有。他们将重正化群类型论证应用于Pauli-Fierz模型（这个想法最初是由J.Froelich（ETH）提出的，但他们的分析还有待观察。3． Teramoto 考虑了两个旋转圆柱体之间的库埃特流扰动的库埃特-泰勒问题，并表明当泰勒数增加时会出现稳态分岔或 Hopf 分岔。借助计算机数值分析证明了这一点。 4. Teramoto 和 Ito 研究了湍流的性质，其中包括关于能量耗散的 Kolmogorov 定律及其偏差。他们试图推导出 Navier-Stokes 方程的偏差。今年他们未能获得具体结果。 5．玻姆从其他可能的算子中发展了散射理论，并得到了相移公式。

项目成果

K.R.Ito,H.Tamura: "N Dependence of Upper Bounds of Critical Temperatures of 2D O(N) Spin Models"Commun. Math. Phys.. 202. 127-168 (1999)

K.R.Ito, H.Tamura：“二维 O(N) 自旋模型临界温度上限的 N 依赖性”Commun。

K.R.Ito: "N Dependence of Critical Temperatures of Two-Dimensional O(N) Spin Models"Comm.Math.Phys.. 202. 127-168 (1999)

K.R.Ito：“二维 O(N) 自旋模型临界温度的 N 依赖性”Comm.Math.Phys.. 202. 127-168 (1999)

S.Shimada: "Scattering Theory for Aharanov-Bohm Hamiltonians"Proc. Of the 4th Workshop on Diff. Eqs.('99). (2000)

S.Shimada：“阿哈拉诺夫-博姆哈密顿量的散射理论”Proc。

S.Watarai,S.Miyashita et al.: "Entropy Effect on the Magnetization Process of Hexagonal XY-like Heisenberg Ferromagnets"Journal of Phys.Soc.Jpn.. 70. (2001)

S.Watarai，S.Miyashita 等：“熵对六方 XY 型海森堡铁磁体磁化过程的影响”Journal of Phys.Soc.Jpn.. 70. (2001)

S.Shimada: "Resolvent Convergence of Schroendinger Operators to Point Interactions"Journal of Math.Phys.. (2001)

S.Shimada：“薛定谔算子到点相互作用的求解收敛”数学物理杂志..（2001）