Monge-Amp\`ere gravitating fluids. Least action principles and particle systems

Abstract

The Monge-Amp\`ere gravitation theory (MAG) was introduced by Brenier in 2011 to obtain an approximate solution of the early Universe reconstruction problem. It is a modification of Newtonian gravitation which is based on quadratic optimal transport. Later, Brenier in 2016, then Ambrosio, Baradat and Brenier in 2020 discovered a double large deviation principle for Brownian particles whose rate function is precisely MAG's action functional. In the present article, following Brenier we first recap MAG's theory. Then, we slightly extend it from particles to fluid. This allows us to revisit the Ambrosio-Baradat-Brenier particle system. We propose another particle system which is easier to interpret in physics and whose large deviation rate function is half the way to MAG's action functional for fluids. While the setting of the Schr\"odinger problem is a system of noninteracting particles, our particle system is subject to some splitting mechanism which regulates the thermal fluctuations. This gives rise to some conditional Gibbs principle that leaves us with an action functional on the fluid space which is MAG's action functional plus an extra term associated with thermal fluctuations. In order to recover MAG's action functional, we have to remove this extra term. To do so, we propose to add some quantum force field on the Otto-Wasserstein manifold of fluids to balance the thermal fluctuations. A microscopic description of a system of particles leading to a conditional Gibbs principle whose action functional generates such a quantum force remains a challenging open problem.