The Fermi-Pasta-Ulam problem(∗)

The Fermi-Pasta-Ulam model is a system of N+2 equal particles on a line with mutual interactions between adjacent particles, provided by a potential of the form V (r) = r/2 + αr/3 + βr/4; certain boundary conditions are also assigned, typically with the two extreme particles fixed. For α = β = 0 the system is a linear one, and by a familiar linear transformation it can be reduced to a system of N independent harmonic oscillators (called normal modes) with certain frequencies ωj = 2 sin[jπ/2(N + 1)] , j = 1, · · · , N . The total energy E then reduces to the sum E = j Ej of the N normal mode energies Ej , which are independent integrals of motion: Ej(t) = Ej(0). When the nonlinear interaction is active, the normal mode energies are no more integrals of motion, and a standard arguments of classical statistical mechanics suggests that their