The Momentum Autocorrelation Function in a Bernoulli Chain

This paper is devoted to the study of the statistical dynamics of the small amplitude coplanar vibrations of a compound pendulum consisting of N + 1 particles suspended in series by weightless strings in a gravitational field. All particles have the same mass m, except for the top particle whose mass is m(1 + 𝒬); and all strings are of equal length. The behavior of this system in the limit in which N → ∞ is of particular interest, because the maximum normal mode frequency is proportional to N½. In the limit N → ∞, asymptotic formulas with error estimates are obtained for the time dependence of the momentum autocorrelation function of: (1) the top particle when 𝒬 = 0; (2) the bottom particle when 𝒬 = 0; and (3) the top particle when N ≫ 𝒬 ≫ 1.