The classical statistical mechanics of Frenkel-Kontorova models

The scaling properties of the free energy, specific heat, and mean spacing are calculated for classical Frenkel-Kontorova models at low temperature, in three regimes: near the integrable limit, the anti-integrable limit, and the sliding-pinned transition (“transition by breaking of analyticity”). In particular, the renormalization scheme given in previous work for ground states of Frenkel-Kontorova models is extended to nonzero-temperature Gibbs states, and the hierarchical melting phenomenon of Vallet, Schilling, and Aubry is put on a rigorous footing.