  Beltrami fields occur as stationary solutions of the Euler equations of fluid
flow and as force free magnetic fields in magnetohydrodynamics. In this paper
we discuss the role of Beltrami fields when considered as operators acting on a
Hamiltonian function to generate particle dynamics. Beltrami operators, which
include Poisson operators as a special subclass, arise in the description of
topologically constrained diffusion in non-Hamiltonian systems. Extending
previous results, we show that random motion generated by a Beltrami operator
satisfies an H-theorem, leading to a generalized Boltzmann distribution on the
coordinate system where the Beltrami condition holds. When the Beltrami
condition is violated, random fluctuations do not work anymore to homogenize
the particle distribution in the coordinate system where they are applied. The
resulting distribution becomes heterogeneous. The heterogeneity depends on the
`field charge' measuring the departure of the operator from a Beltrami field.
Examples of both Beltrami and non-Beltrami operators in three real dimensions
together with the corresponding equilibrium distribution functions are given.
