  We first discuss the geometrical construction and the main mathematical
features of the maximum-entropy-production/steepest-entropy-ascent nonlinear
evolution equation proposed long ago by this author in the framework of a fully
quantum theory of irreversibility and thermodynamics for a single isolated or
adiabatic particle, qubit, or qudit, and recently rediscovered by other
authors. The nonlinear equation generates a dynamical group, not just a
semigroup, providing a deterministic description of irreversible conservative
relaxation towards equilibrium from any non-equilibrium density operator. It
satisfies a very restrictive stability requirement equivalent to the
Hatsopoulos-Keenan statement of the second law of thermodynamics. We then
examine the form of the evolution equation we proposed to describe multipartite
isolated or adiabatic systems. This hinges on novel nonlinear projections
defining local operators that we interpret as ``local perceptions'' of the
overall system's energy and entropy. Each component particle contributes an
independent local tendency along the direction of steepest increase of the
locally perceived entropy at constant locally perceived energy. It conserves
both the locally-perceived energies and the overall energy, and meets strong
separability and non-signaling conditions, even though the local evolutions are
not independent of existing correlations. We finally show how the geometrical
construction can readily lead to other thermodynamically relevant models, such
as of the nonunitary isoentropic evolution needed for full extraction of a
system's adiabatic availability.
