Protons travel along a one-dimensional backbone with tree-like side branches. The exact solution exists.
The asymmetric simple exclusion process (ASEP) is the canonical model of one-dimensional driven transport: particles hop along a lattice with a bias, obeying exclusion (no two particles on the same site). The steady state is exactly solvable via matrix product ansatz, and the model captures everything from ribosome traffic to vehicular flow.
Yukawa et al. (arXiv:2603.09694) extend the ASEP to a backbone lattice with tree-like network branches attached at regular intervals. The motivation is proton transport in solid oxide fuel cells: protons hop along oxygen networks where the backbone is a one-dimensional chain of oxygen sites and the branches are side networks of additional oxygen atoms.
The exact stationary-state solution survives the branching. Each tree branch acts as a particle reservoir whose effective injection/extraction rates are determined by the branch topology and the particle-hole symmetry of the ASEP on the branch. The steady-state current along the backbone depends on these effective rates, which in turn depend on the branch structure — depth, branching ratio, and hopping asymmetry within the branch.
The structural result: the branches don't just add capacity. They modify the phase diagram of the backbone. The boundary-induced phase transitions of the standard ASEP (low-density, high-density, maximal-current phases) shift and can merge or split depending on the branch geometry. A backbone that would be in the maximal-current phase without branches can be pushed into the low-density phase by branches that act as effective sinks.
The highway's capacity depends on its exits. The exits' effect depends on where they lead.
Yukawa et al., "Asymmetric simple exclusion process with tree-like network branches," arXiv:2603.09694 (2026).