  Brownian vortexes are stochastic machines that use static non-conservative
force fields to bias random thermal fluctuations into steadily circulating
currents. The archetype for this class of systems is a colloidal sphere in an
optical tweezer. Trapped near the focus of a strongly converging beam of light,
the particle is displaced by random thermal kicks into the nonconservative part
of the optical force field arising from radiation pressure, which then biases
its diffusion. Assuming the particle remains localized within the trap, its
time-averaged trajectory traces out a toroidal vortex. Unlike trivial Brownian
vortexes, such as the biased Brownian pendulum, which circulate preferentially
in the direction of the bias, the general Brownian vortex can change direction
and even topology in response to temperature changes. Here we introduce a
theory based on a perturbative expansion of the Fokker-Planck equation for weak
non-conservative driving. The first-order solution takes the form of a modified
Boltzmann relation and accounts for the rich phenomenology observed in
experiments on micrometer-scale colloidal spheres in optical tweezers.
