Asymptotic behaviour of a network of oscillators coupled to thermostats of finite energy

We study the asymptotic behaviour of a finite network of oscillators (harmonic or anharmonic) coupled to a number of deterministic Lagrangian thermostats of finite energy. In particular, we consider a chain of oscillators interacting with two thermostats situated at the boundary of the chain. Under appropriate assumptions we prove that the vector $(p,q)$ of moments and coordinates of the oscillators in the network satisfies $(p,q)(t)\to (0,q_c)$ when $t\to\infty$, where $q_c$ is a critical point of some effective potential, so that the oscillators just stop. Moreover, we argue that the energy transport in the system stops as well without reaching the thermal equilibrium. This result is in contrast to the situation when the energies of the thermostats are infinite, studied for a similar system in [14] and subsequent works, where the convergence to a non-trivial limiting regime was established. The proof is based on a method developed in [22], where it was observed that the thermostats produce some effective dissipation despite the Lagrangian nature of the system.