When Is a One-Dimensional Lattice Small?

Abstract

Dynamical systems may be either discrete or continuous, though it is often convenient to blur this distinction in order to formulate a continuum model of a large discrete system. For example, in systems in which macroscopic behavior alone is of interest, a continuum approach is justified because of the vast number of molecules present. But just how many molecules are necessary for a system to be accorded a continuum description? Due to recent advances in interrogating, fabricating, and controlling small collections of molecules [1], it now seems appropriate to consider the manner in which discrete physical systems acquire a continuum description. Such a study is not limited, of course, to molecular systems. As a preliminary question, we may ask, “How can the terms ‘small’ and ‘large’ be made precise?” To formulate this question in a physical context, we consider a model dynamical problem originally investigated by Fermi, Pasta, and Ulam (FPU) [2]. The model consists of a one-dimensional chain of point masses that interact with their nearest neighbors via nonlinear forces. FPU were surprised to find that even in the presence of nonlinearity the dynamics of the system was almost periodic, with a period known as the recurrence time. Further investigations of the lattice dynamics led to the discovery that the recurrence time has a power-law dependence on system parameters [3] (see also [4,5]). This can be explained in the continuum limit by soliton solutions of the Korteweg-de Vries (KdV) equation [6–9]. In this model, derived by letting the number of lattice points tend to infinity while keeping the total length of the lattice fixed [10], solitons flow around the lattice independently; the recurrences are due to the periodic return of these solitons to the initial configuration. The well-established scaling of recurrence time in the continuum limit makes the FPU lattice an ideal system to study the transition from a discrete lattice to a continuum system. We focus here on the behavior of the lattice as it passes through the twilight between “small” and “large” by examining the FPU recurrences as the system size changes. For large systems, when the dynamics is well modeled by the continuum KdV equation, we find that the recurrence can be characterized by a single parameter that measures the strength of the nonlinearity of the system. As the system size shrinks, however, a second parameter is required: the size of the system itself. The transition between these two regimes is well-defined and occurs at a critical value of the system size that appears to coincide with the onset of an instability in the dynamics. We are therefore able to formulate a precise criterion for where the transition between “small” and “large” lattices occurs. The FPU model consists of a one-dimensional lattice of point masses coupled by a linear force augmented by a quadratic nonlinearity and is described by the Hamiltonian