A system of ordinary differential equations describing the interaction of a fast and a slow particle is studied, where the interaction potential Uε depends on a small parameter ε. The parameter ε can be interpreted as the mass ratio of the two particles. For positive ε, the equations of motion are Hamiltonian. It is known [6] that the homogenised limit ε → 0 results again in a Hamiltonian system with homogenised potential U hom . In this article, we are interested in the situation where ε is small but positive. In the first part of this work, we rigorously derive the second-order correction to the homogenised degrees of freedom, notably for the slow particle yε = y0 + ε 2 (ȳ2 + [y2] ε ). In the second part, we give the resulting asymptotic expansion of the energy associated with the fast particle), a thermodynamic interpretation. In particular, we note that to leading-order ε → 0, the dynamics of the fast particle can be identified as an adiabatic process with constant entropy, dS0 = 0. This limit ε → 0 is characterised by an energy relation that describes equilibrium thermodynamic processes, dE ⊥ 0 = F0dy0 + T0dS0 = F0dy0, where T0 and F0 are the leading-order temperature and external force terms respectively. In contrast, we find that to second-order ε 2 , a non-constant entropy emerges, d S2 = 0, effectively describing a non-adiabatic process. Remarkably, this process satisfies on average (in the weak * limit) a similar thermodynamic energy relation, i.e., d Ē⊥ 2 = F0dȳ2 + T0d S2.

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