Thermalisation is diffusion, but in Hilbert space rather than real space.
When a small system couples to a large bath, it eventually reaches thermal equilibrium. The standard framework for understanding this — Fermi's golden rule, master equations, Eigenstate Thermalization Hypothesis — starts from specific assumptions about the nature of the coupling and the statistics of the bath. These frameworks work, but they obscure the geometric picture: what is thermalisation doing in the space of quantum states?
Lunkin (arXiv:2603.12234) models the interaction matrix elements as independent random variables in the non-interacting energy basis and derives a diffusion propagator that describes how a thermometer's state spreads through Hilbert space as it equilibrates with a many-body bath. The thermalisation timescale is set by the inverse of the typical level broadening — how much the interaction smears the originally sharp energy levels.
The sharp result: the rate at which you thermalise is determined by how wide your energy levels become when you turn on the interaction. Narrow broadenings mean slow diffusion through Hilbert space. Wide broadenings mean fast. The distribution of these broadenings — which can be heavy-tailed, as in systems with Lévy-distributed couplings — controls not just the rate but the character of the approach to equilibrium.
The framework extends beyond Markovian dynamics. Lunkin derives a non-Markovian generalization of global balance, capturing memory effects where the system's past trajectory through Hilbert space influences its future exploration. In systems with heavy-tailed broadening distributions, these memory effects dominate: the diffusion is anomalous, and simple exponential relaxation gives way to power-law tails.
What makes this more than a mathematical restatement: the diffusion picture predicts thermalisation timescales from a single statistical property of the interaction (the broadening distribution), without needing to diagonalize the full Hamiltonian or invoke ETH. The clock is the broadening. Everything else follows.
Lunkin, "Thermalisation as Diffusion in Hilbert Space," arXiv:2603.12234 (2026).