The foundations of weak turbulence theory is explored through its application to the (alpha) Fermi-Pasta-Ulam (FPU) model, a simple weakly nonlinear dispersive system. A direct application of the standard kinetic equations would miss interesting dynamics of the energy transfer process starting from a large-scale excitation. This failure is traced to an enforcement of the exact resonance conditi...

We study instabilities and relaxation to equilibrium in a long-range extension of the Fermi-Pasta-Ulam-Tsingou (FPU) oscillator chain by exciting initially the lowest Fourier mode. Localization in mode space is stronger for the long-range FPU model. This allows us to uncover the sporadic nature of instabilities, i.e., by varying initially the excitation amplitude of the lowest mode, which is th...

We consider the FPU model with nonlinearity starting with terms of order n ≥ 3. We compute the resonant normal form in the region where only one low frequency modes is excited and deduce rigorous results on the correspondence between the dynamics of the normal form and that of the complete system. As n varies, we give a criterion in order to deduce whether the FPU phenomenon (formation of a met...

A numerical and analytical study of the relaxation to equilibrium of both the Fermi-Pasta-Ulam (FPU) α-model and the integrable Toda model, when the fundamental mode is initially excited, is reported. We show that the dynamics of both systems is almost identical on the short term, when the energies of the initially unexcited modes grow in geometric progression with time, through a secular avala...

We demonstrate that in the atomic-scale limit the thermal conductance K of the Fermi-Pasta-Ulam (FPU) model and its variants strongly deviates from the mesoscopic behavior due to the relevance of contact resistance. As a result, atomic chains follow log K=nu log T, where the power-law coefficient nu is exactly two times larger than the mesoscopic value. We smoothly interconnect the atomic and m...

It is shown numerically that for Fermi-Pasta-Ulam (FPU) chains with alternating masses and heat baths at slightly different temperatures at the ends, the local temperature (LT) on small scales behaves paradoxically in steady state. This expands the long established problem of equilibration of FPU chains. A well-behaved LT appears to be achieved for equal mass chains; the thermal conductivity is...

The S/390@ floating-point unit (FPU) on the fourth-generation (G4) CMOS microprocessor chip has been implemented in a CMOS technology with a 0.20-pm effective channel length and has been demonstrated at more than 400 MHz. The microprocessor chip is 17.35 by 17.30 mm in size, and one copy of the FPU including the dataflow and control flow but not including the FPR register file is 5.3 by 4.7 mm ...

We use reduced precision checking (RPC) to detect errors in floating point arithmetic. Prior work explored RPC for addition and multiplication. In this work, we extend RPC to a complete floating point unit (FPU), including division and square root, and we present precise analyses of the errors undetectable with RPC that show bounds that are smaller than prior work. We implement RPC for a comple...