Exact, numerical, and mean field behavior of a dimerizing lattice in one dimension

The thermodynamics and dynamics of a one-dimensional dimer-forming anharmonic model is studied in the classical limit. This model mimics the behavior of materials with a Peierls instability. Specific heat C(T), correlation length l_c(T), and order parameter are calculated in three ways: (a) by mean-field (MF) approximation, (b) by numerical molecular dynamics (MD) simulation, and (c) by an exact transfer matrix method. The neighbor distribution function F(x) and vibrational density-of-states D(ω) (Fourier transform of velocity-velocity correlation function) are found numerically. MF theory fails completely to describe the destruction of long-range order by fluctuations, but nevertheless, the MF answers for C(T), l_c(T), F(x), and D(ω) give helpful interpretations of the exact behavior.