Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Emergence of a nonconstant entropy for a fast-slow Hamiltonian system in
  its second-order asymptotic expansion   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, $d \bar{S}_2 \neq 0$, effectively describing a non-adiabatic process.
Remarkably, this process satisfies on average (in the weak$^\ast$ limit) a
similar thermodynamic energy relation, i.e., $d \bar{E}^\perp_2 = F_0 d
\bar{y}_2 + T_0 d \bar{\bar{S}}_2$.

13
The categories of this paper are: quant-ph
20
The categories of this paper are: acc-phys
4
The categories of this paper are: cond-mat
1
The categories of this paper are: cs
6
The categories of this paper are: astro-ph
26
The categories of this paper are: bayes-an
34
The categories of this paper are: chem-ph
3
The categories of this paper are: physics
15
The categories of this paper are: nucl-ex
29
The categories of this paper are: funct-an
17
The categories of this paper are: q-fin
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The categories of this paper are: nlin
30
The categories of this paper are: q-alg
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The categories of this paper are: nucl-th
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The categories of this paper are: comp-gas
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The categories of this paper are: patt-sol