Emergence of a nonconstant entropy for a fast-slow Hamiltonian system in its second-order asymptotic expansion
Matthias Klar, Karsten Matthies, Johannes Zimmer · arXiv (Cornell University) · 2020
A system of ordinary differential equations describing the interaction of a fast and a slow particle is studied, where the interaction potential $U_\epsilon$ depends on a small parameter $\epsilon$. The parameter $\epsilon$ can be interpreted as the mass ratio of the two particles. For positive $\epsilon$, the equations of motion are Hamiltonian. It is known (Bornemann, Springer Lecture Notes in Mathematics 1687) that the homogenised limit $\epsilon\to0$ results again in a Hamiltonian system with homogenised potential $U_{\hom}$. In this article, we are interested in the situation where $\epsilon$ 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 $y_\epsilon = y_0 +\epsilon^2 (\bar{y}_2 +[y_2]^\epsilon)$. In the second part, we give the resulting asymptotic expansion of the energy associated with the fast particle $E_\epsilon^\perp = E_0^\perp + \epsilon [E_1^\perp]^\epsilon+ \epsilon^2 (\bar{E}_2^\perp+[E_2^\perp]^\epsilon)$, a thermodynamic interpretation. In particular, we note that to leading-order $\epsilon\to0$, the dynamics of the fast particle can be identified as an \emph{adiabatic} process with \emph{constant entropy}, $dS_0 =0$. This limit $\epsilon\to0$ is characterised by an energy relation that describes equilibrium thermodynamic processes, $d E_0^\perp = F_0 d y_0 + T_0 dS_0 = F_0 d y_0$, where $T_0$ an $F_0$ are the leading-order temperature and external force terms respectively. In contrast, we find that to second-order $\epsilon^2$, a \emph{non-constant entropy} emerges, $d \bar{S}_2 eq 0$, effectively describing a non-adiabatic process. Remarkably, this process satisfies on average (in the weak$^\ast$ limit) a similar thermodynamic energy relation, i.e., $d \bar{E}^\perp_2 = F_0 d \bar{y}_2 + T_0 d \bar{\bar{S}}_2$.