Topological Path-Connectivity and Diffusion

We propose that diffusion in bounded regions is governed primarily by the topological condition of path-connectedness. Modeling particle motion by Brownian paths in a region Ω containing a receptor B under rather general conditions, we prove that if Ω is connected a particle starting anywhere in Ω hits B with probability of one (p = 1), regardless of geometric deformations or the specific diffusion–drift coefficients; if Ω is disconnected and the start point and B lie in different components, the particle never arrives. Monte Carlo simulations in 2D domains corroborate this: smoothly deforming a domain of fixed area leaves mean first-passage times essentially unchanged, whereas extreme deformation produces sharp increases in arrival time and, once occluded, complete transport failure. The possibility of a phase transition characterized by a topological invariant, connectedness, is discussed, in which connected domains guarantee eventual particle arrival at the receptor (p = 1), while broken connectivity makes arrival impossible (p = 0), so that the loss or restoration of connectedness implies a phase transition between 0 and 1. Generalizing the description of diffusion as a process governed by the Poisson equation provides a complementary perspective on the roles of deformation and connectivity. This formulation enables spatially resolved statements about variations in the mean arrival time and provides an intuitive, visually interpretable description through contour maps that explicitly reveal where and how path-connectivity is broken. Gas-phase (ammonia–litmus hemisphere), liquid-phase (Hele–Shaw dye) and solid-phase (proton diffusion in correlated oxides) experiments further confirm contour-level simultaneity and isotropic, diffusion-limited spreading. Overall, our findings elucidate a fundamental link between the topological invariant connectedness and probabilistic phenomena in complex systems, with potential applications in materials physics, biophysics and neurobiology.