Exact solutions in the FPU oscillator chain

The Fermi–Pasta–Ulam (FPU)-β system possesses low-dimensional solutions that are the exact solutions of reduced Hamiltonian systems. Identification of such low-dimensional solutions is important for a precise understanding of the energy exchange process among normal modes, especially the induction phenomenon. The reduced Hamiltonians can be systematically constructed by introducing in the mode number space the type I subsets defined by Poggi and Ruffo [Physica D 103 (1997) 251]. By a simple analysis of the selection rule for energy exchange, we present here general expressions for the type I subsets. The applicability of the expressions is demonstrated via a numerical experiment on the induction phenomenon.

Low-Dimensional Solutions in the Quartic Fermi–Pasta–Ulam System

Ergodicity and Time-Scales for Statistical Equilibrium in Classical Dynamical Systems

Reversible approach to statistical equilibrium in a nonlinear chain: an ensemble study

Nonlinear interaction between flexural modes is critical to heat conductivity and mechanical vibration of two-dimensional materials such as graphene. Much effort has been devoted to understand the underlying mechanism. In this paper, we examine solely the out-of-plane flexural modes and identify their energy flow pathway during thermalization process. The key is the development of a universal scheme that numerically characterizes the strength of nonlinear interactions between normal modes. In particular, for our square graphene system, the modes are grouped into four classes by their distinct symmetries. The couplings are significantly larger within a class than between classes. As a result, the equations for the normal modes in the same class as the initially excited one can be approximated by driven harmonic oscillators, therefore, they get energy almost instantaneously. Because of the hierarchical organization of the mode coupling, the energy distribution among the modes will arrive at a stable profile, where most of the energy is localized on a few modes, leading to the formation of "natural package" and metastable states. The dynamics for modes in other symmetry classes follows a Mathieu type of equation, thus, interclass energy flow, when the initial excitation energy is small, starts typically when there is a mode that lies in the unstable region in the parameter space of Mathieu equation. Due to strong coupling of the modes inside the class, the whole class will get energy and be lifted up by the unstable mode. This characterizes the energy flow pathway of the system. These results bring fundamental understandings to the Fermi-Pasta-Ulam problem in two-dimensional systems with complex potentials, and reveal clearly the physical picture of dynamical interactions between the flexural modes, which will be crucial to the understanding of their abnormal contribution to heat conduction and nonlinear mechanical vibrations.

Metastable states and energy flow pathway in square graphene resonators.

It is shown that the induction phenomenon in the exchange of energies of normal modes found by computer experiments is an inherent property of the anharmonic lattice vibration and its behavior obeys the time reversal even after the elapse of the induction period. The instability of the normal modes when one mode is excited initially is related to the induction phenomenon, and is discussed by the application of the instability region of the Mathieu function. The exponential orbital separation is also shown to take place after the elapse of the induction period. This guarantees the C-property of the anharmonic lattice vibration.

Naoteru Hirotomi

Papers

The Induction Phenomenon and Ergodicity in the Anharmonic Lattice Vibration