  High-dimensional dynamical systems projected onto a reduced-order model cease
to be deterministic and are best described by probability distributions in
state space. Their equations of motion map onto an evolution operator with a
deterministic component describing the projected dynamics, and a stochastic one
from the neglected dimensions. It is shown that, for projections in which the
deterministic component is dominant, `physics-free' stochastic Markovian models
can be constructed that mimic many of the statistics of the real flow, even for
fairly crude operator approximations. Deterministic models converge to steady
states. This is related to general properties of Markov chains and illustrated
with data-driven models for a moderate-Reynolds number turbulent channel.
