When is a one-dimensional lattice small?

While the use of continuum approximations to describe systems consisting of many molecules is well established, it is not known how such approximations fail as the number of molecular components decreases. We study the one-dimensional Fermi-Pasta-Ulam chain in order to determine the critical value of the system size below which the system’s behavior deviates from the continuum limit, allowing us to delineate between “small” and “large” systems and define these terms precisely. For this system, the distinction between small and large is correlated with the appearance of an instability of the chain.