  A class of fast-slow Hamiltonian systems with potential $U_\varepsilon$
describing the interaction of non-ergodic fast and slow degrees of freedom is
studied. The parameter $\varepsilon$ indicates the typical time-scale ratio of
the fast and slow degrees of freedom. It is known that the Hamiltonian system
converges for $\varepsilon\to0$ to a homogenised Hamiltonian system. In this
article, we study the situation where $\varepsilon$ 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.
