Spectral analysis and phase transitions for long-range interactions in harmonic chains of oscillators

We consider chains of N harmonic oscillators in two dimensions coupled to a Langevin heat reservoir at fixed temperature, a classical model for heat conduction introduced by Lebowitz, Lieb, and Rieder (1967). We extend our previous results (Becker and Menegaki, 2021) significantly by providing a full spectral description of the full Fokker-Planck operator, also allowing for the presence of a constant external magnetic field for charged oscillators. We then study oscillator chains with additional next-to-nearest-neighbor interactions and find that the spectral gap undergoes a phase transition if the next-to-nearest-neighbor interactions are sufficiently strong and may even cease to exist for oscillator chains of finite length.