  Cold ions in anisotropic harmonic potentials can form ion chains of various
sizes. Here, the density of ions is not uniform, thus the eigenmodes are not
phononic-like waves. We study chains of N>>1 ions and evaluate analytically the
long wavelength modes and the density of states in the short wavelength limit.
These results reproduce with good approximation the dynamics of chains
consisting of dozens of ions. Moreover, they allow to determine the critical
transverse frequency required for the stability of the linear structure, which
is found in agreement with results obtained by different theoretical methods
[D.H.E. Dubin, Phys. Rev. Lett. 71, 2753 (1993)] and by numerical simulations
[J.P. Schiffer, Phys. Rev. Lett. 70, 818 (1993)]. We introduce and explore the
thermodynamic limit for the ion chain. The thermodynamic functions are found to
exhibit deviations from extensivity.

Statistical Mechanics;Soft Condensed Matter;Atomic Physics;Quantum Physics