In a system of heterogeneous components characterized by some continuous trait, is there a distribution of that trait that optimizes relaxation, i.e. the rate of return to equilibrium? And what about intrinsically non-equilibrium systems?
This project starts with a kinetic theory approach to polydisperse gases (particles of different sizes, or forms, or masses - here mostly the latter), in the tracer limit then for other settings, with an intriguing result: the optimum is not a continuous distribution but a discrete mixture of very few distinct species.
This property holds for dissipative gases, where there is no thermodynamic equilibrium, but optimal relaxation retains its meaning as describing the fastest process of energy redistribution.
For future work, I believe there may be some value to applying this idea to very different microscopic interactions - e.g. ecological or economic systems. In both cases, the collapse of a continuum of heterogeneous components into a few discrete classes would be a very interesting phenomenon to explain from optimality.