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Published online by Cambridge University Press: 05 June 2012
In 1955 Fermi, Pasta and Ulam (FPU) (Fermi et al., 1955) and Tsingou (see Douxois, 2008) undertook a numerical study of a one-dimensional anharmonic (nonlinear) lattice. They thought that due to the nonlinear coupling, any smooth initial state would eventually lead to an equipartition of energy, i.e., a smooth state would eventually lead to a state whose harmonics would have equal energies. In fact, they did not see this in their calculations. What they found is that the solution nearly recurred and the energy remained in the lower modes.
To quote them (Fermi et al., 1955):
The results of our computations show features which were, from beginning to end, surprising to us. Instead of a gradual, continuous flow of energy from the first mode to the higher modes, … the energy is exchanged, essentially, among only a few. … There seems to be little if any tendency toward equipartition of energy among all the degrees of freedom at a given time. In other words, the systems certainly do not show mixing.
Their model consisted of a nonlinear spring–mass system (see Figure 1.1) with the force law: F(Δ) = –k(Δ+α Δ2), where Δ is the displacement between the masses, k > 0 is constant, and α is the nonlinear coefficient.
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