  This is a set of four lectures devoted to simple ideas about turbulent
transport, a ubiquitous non-equilibrium phenomenon. In the course similar to
that given by the author in 2006 in Warwick [45], we discuss lessons which have
been learned from naive models of turbulent mixing that employ simple random
velocity ensembles and study related stochastic processes. In the first
lecture, after a brief reminder of the turbulence phenomenology, we describe
the intimate relation between the passive advection of particles and fields by
hydrodynamical flows. The second lecture is devoted to some useful tools of the
multiplicative ergodic theory for random dynamical systems. In the third
lecture, we apply these tools to the example of particle flows in the Kraichnan
ensemble of smooth velocities that mimics turbulence at intermediate Reynolds
numbers. In the fourth lecture, we extends the discussion of particle flows to
the case of non-smooth Kraichnan velocities that model fully developed
turbulence. We stress the unconventional aspects of particle flows that appear
in this regime and lead to phase transitions in the presence of
compressibility. The intermittency of scalar fields advected by fully turbulent
velocities and the scenario linking it to hidden statistical conservation laws
of multi-particle flows are briefly explained.
