nearcommensurate

Nearcommensurate, often written as near-commensurate, describes a relationship between two periodicities that is close to a simple rational ratio but not exactly equal. In contrast, a strictly commensurate relationship has a rational ratio, while an incommensurate one has an irrational ratio. Near-commensurate therefore lies between these extremes: the components nearly repeat with a common period over long, but finite, lengths, producing a pattern that is ordered yet not perfectly periodic.

In condensed matter physics, near-commensurate structures frequently arise in modulated systems such as charge-density waves (CDWs). Nearly commensurate CDWs feature a wavevector that is close to a rational fraction of the underlying lattice vectors. This yields a patchwork of commensurate domains separated by discommensurations or domain walls. As external conditions like temperature or pressure change, the system may drift toward a strictly commensurate state or toward an incommensurate phase, with corresponding changes in electronic transport, optical response, and diffraction signatures.

In mathematical and dynamical contexts, near-commensurability describes approximate frequency ratios, where p ω1 ≈ q ω2 for small mismatch between integers p and q. Such near-resonant conditions can produce long-lived quasi-periodic behavior and range-limited frequency locking phenomena, often analyzed via concepts from Diophantine approximation and nonlinear dynamics.

Etymologically, the term emphasizes approximation to a commensurate relationship rather than exact equality. Nearcommensurate descriptions are used to characterize systems with long-range order that is nearly periodic, yet inherently inexact, requiring modulational or domain-wall descriptions.