Quadrupolar gyration of a Brownian particle in a confining ring

We develop a minimal theoretical model that reveals a structured steady-state flux field with four alternating local circulation, a phenomenon we refer to as quadrupolar gyration. A passive Brownian particle is confined to move in a ring-shaped trap and driven far from equilibrium solely by anisotropic thermal fluctuations from two orthogonal heat baths held at different temperatures. By breaking detailed balance, this fundamental temperature anisotropy induces a robust nonequilibrium steady state characterized by probability currents of the particle's motion. Remarkably, these currents self-organize into a distinctive quadrupolar vortex pattern, providing a clear signature of emergent symmetry breaking, irreversible entropy production, and coherent motion in minimal passive systems. Our theoretical predictions, based on the narrow-ring approximation and the assumption of small thermal anisotropy, are validated by numerical simulations.