  It is always some constraint that yields any nontrivial structure from
statistical averages. As epitomized by the Boltzmann distribution, the energy
conservation is often the principal constraint acting on mechanical systems.
Here, we investigate a different type: the topological constraint imposed on
`space'. Such constraint emerges from the null space of the Poisson operator
linking energy gradient to phase space velocity, and appears as an adiabatic
invariant altering the preserved phase space volume at the core of statistical
mechanics. The correct measure of entropy, built on the distorted invariant
measure, behaves consistently with the second law of thermodynamics. The
opposite behavior (decreasing entropy and negative entropy production) arises
in arbitrary coordinates. An ensamble of rotating rigid bodies is worked out.
The theory is then applied to up-hill diffusion in a magnetosphere.
