Diffusion processes relax toward equilibrium. Circulation — persistent probability current flowing in loops — is destroyed as the system equilibrates. This is among the most basic features of undriven diffusion: detailed balance forbids net circulation at steady state.
Namura and Nakao invert this. They design an optimal controller that forces a two-dimensional diffusion process to maintain a prescribed nonequilibrium circulation as its steady state, while simultaneously accelerating convergence to that circulating state.
The mechanism is spectral decomposition of the Fokker-Planck equation. The infinite-dimensional PDE control problem — how to shape a drift field to sustain circulation against diffusion — becomes finite and tractable when projected onto the eigenmodes of the Fokker-Planck operator. The objective functional penalizes both deviations from the desired probability density and deviations from the desired flux rotation, and the optimal control is found by manipulating the eigenmode amplitudes.
The key insight is that the eigenstructure of the Fokker-Planck operator — the same structure that describes relaxation to equilibrium — can be repurposed to sustain the very nonequilibrium state it would normally destroy. The same eigenmodes that encode exponential decay toward detailed balance can, under optimal forcing, encode exponential convergence toward a circulating state. The mathematical machinery is identical; only the target changes.
Numerical demonstrations show the controller achieving prescribed circulation patterns with rapid convergence. The computational cost stays low because the spectral reduction keeps the problem finite-dimensional.
Diffusion's eigenstructure does not inherently oppose circulation. Under spectrally decomposed control, the same framework that describes relaxation to equilibrium can be repurposed to sustain and accelerate convergence to prescribed nonequilibrium steady states.