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Analysis of pure nonlinear lattice with x^4 potential -A method of computational physics including computer algebra-

具有x^4势的纯非线性晶格的分析-包括计算机代数在内的计算物理的方法-

基本信息

批准号：
07640538
负责人：
SHIMOJI Sadao
金额：
$ 1.28万
依托单位：
Tokyo Engineering University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1995
资助国家：
日本
起止时间：
1995 至 1996
项目状态：
已结题

项目摘要

In order to investigate the common feature which may exist in the complex systems, such as 1/<bounded integral> fluctuation, the simplest model of nonlinear lattice is considered and analyzed : a purely nonlinear lattice with a single term of x^4 potential. Its solution obtained by numerical technique became chaotic as time elapses and the power spectrum of the solution has 1/<bounded integral> distribution. The trajectories of particles has equal sojourn probability in the 2N (N is the number of particles) phase spaceThis research project aims to clarify the dynamics of the lattice by using also the analytical technique. The model of the lattice is approximated by a quasi-linear wave equation, gamma_u=(gamma^2gamma_x)_x. The equation has analytical solutions for a special class of initial conditions. The solutions become generally multivalued in some region M after a time t_<v min>. The region M spreads with time. The analytical solutions and numerical solutions to the lattice agree in the whole region fot t<t_<v min> and they agree in the single valued region for t>t_<v min>. The energy integral extended to be applicable also to M is shown to be conserved in the whole region including M for all t. The solution to the lattice will become chaotic in M.The numerical analysis of the system of colliding particles in one dimensional space clarified that the velocity distribution tends to 1/upsiron distribution with time.

为了研究复杂系统中可能存在的共同特征，如1/&lt；有界积分&gt；涨落，考虑和分析了最简单的非线性晶格模型：具有x^4势的单项的纯非线性晶格。用数值方法得到的解随着时间的推移而变得混沌，并且解的功率谱具有1/&lt；有界积分&gt；分布。在2N(N是粒子数)相空间中，粒子的轨迹具有相等的逗留概率。本研究项目旨在通过使用分析技术来阐明晶格的动力学。晶格模型用拟线性波动方程Gamma_u=(Gamma^2 Gamma_x)_x来近似，对于一类特殊的初始条件，该方程有解析解。在时间t_t；vmin&gt；之后，解一般在某个区域M上变得多值。区域M随时间扩展。晶格的解析解和数值解在整个区域内是一致的，在单值区域内是一致的。推广到M的能量积分在包括所有t的M在内的整个区域是守恒的。晶格的解在M中将变得混沌。对一维空间中碰撞粒子系统的数值分析表明，速度分布随时间趋于1/usiron分布。