State equations and order fluctuations in 1D and 3D oscillators chains

One-dimensional chains have been investigated a lot during the last few decades, and many anomalies were found. In these particular systems, the validity of local thermodynamic equilibrium (LTE) is not ensured and this leads to not well-defined thermodynamic quantities. In this thesis project, we analyse anharmonic oscillator chains in a non-equilibrium condition, due to a temperature gradient. They can move in an one-dimensional and three-dimensional space, and their ends are coupled with Nosé-Hoover thermostats set at different temperatures. We consider two different kinds of nearest-neighbours anharmonic potentials: β-Fermi-Pasta-Ulam-Tsingou (FPUT) potential and an elastic interaction with a quadratic repulsion term. We study numerically the validity of a linear relation between density of particles and kinetic temperature profiles. This state equation was only verified for one-dimensional chains that ensure the particles ordering. In this work, we analyse chains with interactions that both can and cannot ensure this condition. Moreover, we generalise this relation for three-dimensional chains and we check its validity for our systems. For these chains, we also investigate the validity of LTE through numerical analysis of the position fluctuations behaviour as a function of the particles number, trying to understand the asymptotic growth order.