Validity of the Markovian approximation for macroscopic quantum dynamics

Determine whether, for a closed quantum system with Hamiltonian H and macro-space decomposition into Hν, the macroscopic weights pν(t)=||Pνψt||² agree with the time-t distribution of a time-homogeneous Markov jump process on the macro-state space with the proposed transition rates σνν′.

Background

The paper compares the unitary quantum evolution of the macro-space weights pν(t)=||Pνψt||² with a classical Markov jump process on the macro-state space. The proposed rates are defined from short-time transition probabilities involving the unitary operator e{-iHτ} and the macro-space projectors.

The Markovian approximation would yield a time-independent master equation for the macroscopic weights and would provide a route toward explaining entropy increase through predominantly right-moving transitions. The paper presents this as a conjecture associated with van Kampen and discussed by Strasberg et al., but does not establish it for the general quantum systems under consideration.

References

A related statement was proposed as a conjecture by van Kampen and recently argued for by Strasberg et al. . In the version relevant here, it says that $p_\nu(t)=|P_\nu\psi_t|2$ agrees with the distribution at time $t$ of a Markov jump process on $\mathscr{N}$ with ($t$-independent) jump rates

— On the Increase of Boltzmann Entropy in Closed Macroscopic Quantum Systems  (2610.18553 - Igelspacher et al., 13 Oct 2026) in Remark 'Markovian approximation', Section 2 (Strong Second Law subsection)

Eq. eq:hlocdetbal can be confirmed for certain random matrix models (see Remark~\ref{rmk:locdetWig}), but it remains open whether it is valid more generally.

eq:hlocdetbal:

σνν′σν′ν≈dνdν′.\frac{\sigma_{\nu\nu'}}{\sigma_{\nu'\nu}} \approx \frac{d_\nu}{d_{\nu'}}.

— On the Increase of Boltzmann Entropy in Closed Macroscopic Quantum Systems  (2610.18553 - Igelspacher et al., 13 Oct 2026) in Remark 'Markovian approximation', Section 2 (Strong Second Law subsection)