  In the last decade, minimal kinetic models, and primarily the Lattice
Boltmann equation, have met with significant success in the simulation of
complex hydrodynamic phenomena, ranging from slow flows in grossly irregular
geometries to fully developed turbulence, to flows with dynamic phase
transitions. Besides their practical value as efficient computational tools for
the dynamics complex systems, these minimal models may also represent a new
conceptual paradigm in modern computational statistical mechanics: instead of
proceeding bottom-up from the underlying microdynamic systems, these minimal
kinetic models are built top-down starting from the macroscopic target
equations. This procedure can provide dramatic advantages, provided the
essential physics is not lost along the way. For dissipative systems, one such
essential requirement is the compliance with the Second Law of thermodynamics.
In this Colloquium, we present a chronological survey of the main ideas behind
the Lattice Boltzmann method, with special focus on the role played by the $H$
theorem in enforcing compliance of the method with macroscopic evolutionary
constraints (Second Law) as well as in serving as a numerically stable
computational tool for fluid flows and other dissipative systems out of
equilibrium.
