1996 Fiscal Year Final Research Report Summary
Analysis of pure nonlinear lattice with x^4 potential -A method of computational physics including computer algebra-
| Project/Area Number |
07640538
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
物理学一般
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| Research Institution | Tokyo Engineering University |
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Principal Investigator |
SHIMOJI Sadao Tokyo Engineering University Department of Engineering Science Professor, 工学部, 教授 (50216123)
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| Co-Investigator(Kenkyū-buntansha) |
KAWAI Toshio Keio University Department of Physics Professor, 理工学部, 教授 (30146724)
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| Project Period (FY) |
1995 – 1996
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| Keywords | nonlinear lattice / quasilinear wave equation / multivalued solution / chaotic behavior |
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| Research Abstract |
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 space
This 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.
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