At this point one may note that the 2nd and 4th derivatives of the imaginary potential at the origin are crucial ingredients for the determination of the asymptotic value of $\langle p^2 \rangle$. For the HTL potential defined in eq. (B.1), the 2nd derivative at the origin is divergent and the integral adopted in the definition (B.2) of $\phi$ needs to be regulated. Adopting a UV cut-off $\Lambda \gg 1$, one gets

$$ \phi(\tilde{r}; \Lambda) = 2 \int_0^\Lambda dz \frac{z}{(z^2+1)^2} \left(1 - \frac{\sin(\tilde{r}z)}{\tilde{r}z}\right) = \frac{c_2(\Lambda)}{3!} \tilde{r}^2 - \frac{c_4(\Lambda)}{5!} \tilde{r}^4 + \dots, \quad (G.5) $$

where $\tilde{r} = rm_D$, $z = q/m_D$ and $\Lambda = q_{max}/m_D$. Then, recalling that $W = \alpha\phi$ (see eq. (B.1)) we obtain the derivatives of $W$ w.r.t. $r$:

$$ \mathcal{H}(0) = W''(0) = \frac{\alpha T m_D^2}{3} c_2(\Lambda) \quad \text{and} \quad W^{(4)}(0) = -\frac{\alpha T m_D^4}{10} c_4(\Lambda) \quad (G.6) $$

where

$$ c_2(\Lambda) = 2 \int_0^\Lambda dz \frac{z^3}{(z^2+1)^2} \approx \ln(\Lambda^2) \quad \text{and} \quad c_4(\Lambda) = 2 \int_0^\Lambda dz \frac{z^5}{(z^2+1)^2} \approx \Lambda^2, $$

leading to

$$ \mathcal{H}(0) = \frac{2\alpha T m_D^2}{3} \ln \frac{q_{\max}}{m_D}, \quad \text{and} \quad W^{(4)}(0) = -\frac{\alpha T m_D^2 q_{\max}^2}{10}. \quad (G.7) $$

Due to the quartic dependence of $W^{(4)}(0)$ on $q_{max}$, the choice of the value of $q_{max}$ can affect significantly the value of $\langle p^2 \rangle$ at equilibrium. Let us notice that this is true even in the weak coupling case as the $\frac{\tilde{W}^{(4)}(0)}{W''(0)}$ ratio that appears to be responsible for $\gamma \neq 0$ does not depend on the coupling between the heat bath and the subsystem.

Besides the regularization issues just discussed, there is another potential source of deviation between the QME steady state and the canonical Maxwell-Boltzmann law: a potential mismatch between the fluctuation and the dissipation terms (respectively denoted $\mathcal{L}_2$ and $\mathcal{L}_3$ in our work). Quite generally, the structures of $\mathcal{L}_2 \propto W$ and $\mathcal{L}_{3,\text{lin}} \propto \nabla W \cdot \nabla$ — where the {cst, lin, quad} subscript refers to the number of derivatives acting on $\mathcal{D}$, as already introduced in the construction of table 1 — conspire to reach an exact Maxwell-Boltzmann law in the semi-classical approximation. However, adding a $\mathcal{L}_{3,\text{cst}}$ contribution is equivalent to shifting the imaginary potential in $\mathcal{L}_2$ from $W \to W + \frac{\Delta W}{4MT}$. It is to be viewed effectively as a fluctuation term. If such shift is not “transferred” to the terms $\propto \nabla W$ in the $\mathcal{L}_{3,\text{lin}}$ structure, this may induce deviations with respect to the Maxwell-Boltzmann law. This source of deviation is absent in the present work because of our restriction to the “minimal set” of superoperators. However, it may be at the origin of the discrepancies $\propto \frac{T}{M}$ that are for instance observed in appendix C of [35] (more specifically eq. (C2) in [35] illustrates the imbalance between the $\mathcal{L}_{\text{cst}}$ and $\mathcal{L}_{\text{lin}}$ structures, respectively based on $F_1$ and $\vec{F}_2$).

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