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Emergence of a nonconstant entropy for a fast-slow Hamiltonian system in its second-order asymptotic expansion | | Matthias Klar
; Karsten Matthies
; Johannes Zimmer
; | | Date: |
21 Oct 2020 | | Abstract: | 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
$epsilon o0$ 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 $y_epsilon = y_0 +epsilon^2 (ar{y}_2
+[y_2]^epsilon)$. In the second part, we give the resulting asymptotic
expansion of the energy associated with the fast particle $E_epsilon^perp =
E_0^perp + epsilon [E_1^perp]^epsilon+ epsilon^2
(ar{E}_2^perp+[E_2^perp]^epsilon)$, a thermodynamic interpretation. In
particular, we note that to leading-order $epsilon o0$, the dynamics of the
fast particle can be identified as an emph{adiabatic} process with
emph{constant entropy}, $dS_0 =0$. This limit $epsilon o0$ 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, $d ar{S}_2
eq 0$, effectively describing a non-adiabatic process.
Remarkably, this process satisfies on average (in the weak$^ast$ limit) a
similar thermodynamic energy relation, i.e., $d ar{E}^perp_2 = F_0 d
ar{y}_2 + T_0 d ar{ar{S}}_2$. | | Source: | arXiv, 2010.10971 | | Services: | Forum | Review | PDF | Favorites | |
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