In this talk I will firstly review standard Multi-Marginal Optimal Transport (a number N of marginals is fixed) focusing, in particular, on the applications in Quantum Mechanics (in this case the marginals are all the same and represent the electron density). I will then extend the Optimal Transportation problem to the grand canonical setting: only the expected number of marginals/electrons is now given (i.e. we can now define a OT problem with a fractional number of marginals). I will compare these two problems and show how they behave differently despite considering the same cost functions. Existence of minimisers, Monge solution, duality, entropic formulation and numerics will be discussed.
Maxime Laborde
