Professor Celia Reina has contributed to an article in pre-print.
Abstract: A class of fast-slow Hamiltonian systems with potential Uε describing the interaction of non-ergodic fast and slow degrees of freedom is studied. The parameter ε indicates the typical time-scale ratio of the fast and slow degrees of freedom. It is known that the Hamiltonian system converges for ε→0 to a homogenised Hamiltonian system. In this article, we study the situation where ε is small but positive. In the first part, we rigorously derive the second-order corrections to the homogenised (slow) degrees of freedom. They can be decomposed into explicitly given terms that oscillate rapidly around zero and terms that trace the average motion of the corrections, which are given as the solution to an inhomogeneous linear system of differential equations. In the second part, we analyse the energy of the fast degrees of freedom expanded to second-order from a thermodynamic point of view. In particular, we define and expand to second-order a temperature, an entropy and external forces and show that they satisfy to leading-order as well as on average to second-order thermodynamic energy relations akin to the first and second law of thermodynamics. In a third part, we analyse for a specific fast-slow Hamiltonian system the second-order asymptotic expansion of the slow degrees of freedom from a numerical point of view. In particular, we discuss their approximation quality for short and long time frames and compare their total computation time with that of the solution to the original fast-slow Hamiltonian system of similar accuracy.