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Monge-Ampère gravitating fluids. Least action principles and particle systems

Christian Léonard, Roya Mohayaee

TL;DR

This work revisits Monge-Ampère gravitation (MAG) from Brenier's original formulation, recapping its motivation and definition as a quadratic-optimal-transport-based modification of Newtonian gravity, with MAG forcing via the Monge–Ampère constraint and Brenier's polar factorization. It then extends MAG from discrete particle clouds to self-gravitating fluids using Otto–Wasserstein geometry, introducing an action functional for k-fluids that blends transport and Fisher-information terms through an $\varepsilon$-regularization. The paper develops an alternative, interpretable particle-system picture built on branching Brownian particles, derives the corresponding large-deviation rate functions, and connects to the Schrödinger problem and entropic interpolations, including a quantum-force correction term. A time-inhomogeneous extension with a coeffective $\kappa_s$ is formulated, preserving MAG's core least-action structure, and the final ε-MAG formulation for k-fluids yields a coherent framework linking branching-particle dynamics, optimal transport, and quantum-like forces. This provides a more transparent physical interpretation of MAG dynamics and a path toward numerical and analytical exploration in cosmology and transport.

Abstract

The Monge-Ampère gravitation theory (MAG) was introduced by Brenier 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, then Ambrosio, Baradat and Brenier 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 and propose another one which is easier to interpret and whose large deviation rate function is MAG's action functional for fluids. This model leads to a Gibbs conditioning principle that is an entropy minimization problem close to the Schrödinger problem. While the setting of the Schrödinger problem is a system of noninteracting particles, the particle system we work with is subject to some branching mechanism which regulates the thermal fluctuations and some quantum force which balances them.

Monge-Ampère gravitating fluids. Least action principles and particle systems

TL;DR

This work revisits Monge-Ampère gravitation (MAG) from Brenier's original formulation, recapping its motivation and definition as a quadratic-optimal-transport-based modification of Newtonian gravity, with MAG forcing via the Monge–Ampère constraint and Brenier's polar factorization. It then extends MAG from discrete particle clouds to self-gravitating fluids using Otto–Wasserstein geometry, introducing an action functional for k-fluids that blends transport and Fisher-information terms through an -regularization. The paper develops an alternative, interpretable particle-system picture built on branching Brownian particles, derives the corresponding large-deviation rate functions, and connects to the Schrödinger problem and entropic interpolations, including a quantum-force correction term. A time-inhomogeneous extension with a coeffective is formulated, preserving MAG's core least-action structure, and the final ε-MAG formulation for k-fluids yields a coherent framework linking branching-particle dynamics, optimal transport, and quantum-like forces. This provides a more transparent physical interpretation of MAG dynamics and a path toward numerical and analytical exploration in cosmology and transport.

Abstract

The Monge-Ampère gravitation theory (MAG) was introduced by Brenier 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, then Ambrosio, Baradat and Brenier 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 and propose another one which is easier to interpret and whose large deviation rate function is MAG's action functional for fluids. This model leads to a Gibbs conditioning principle that is an entropy minimization problem close to the Schrödinger problem. While the setting of the Schrödinger problem is a system of noninteracting particles, the particle system we work with is subject to some branching mechanism which regulates the thermal fluctuations and some quantum force which balances them.

Paper Structure

This paper contains 12 sections, 24 theorems, 325 equations, 4 figures.

Table of Contents

  1. MAG. Motivation and definition
  2. MAG. Action functional
  3. MAG. Particle system
  4. MAG for a fluid
  5. $\epsilon$- MAG for a $k$-fluid. Action functional
  6. $\epsilon$- MAG for a $k$-fluid. Newton equation
  7. Schrödinger problem
  8. Plugging $\kappa_s$ in
  9. $\epsilon$- MAG for a $k$-fluid. Particle system
  10. What remains to be done
  11. Least action principle
  12. Concentration of matter

Key Result

Proposition 4.11

Suppose that the (normalized) density $\mu_t=(Y_t)_{\#} P$ is the time marginal at time $t$ of some path measure $P\in\mathrm{P}(\Omega)$ which is supported by absolutely continuous sample paths, i.e. where $(Y_t) _{ t_0\le t\le t_1}$ stands for the canonical process on the path space and $\dot Y_t$ is some random vector, possibly depending (a priori) on the whole history of the path and satisfy

Figures (4)

  • Figure 1: Cosmic microwave background
  • Figure 2: Typical structures of the actual Universe Courtesy of Bruno Lévy
  • Figure 3: Optimal transport vs. N-body simulation, FMMS02.
  • Figure 4: The force field depends on $D$

Theorems & Definitions (60)

  • Definition 1.11: MAG's approximation of SNS, Bre11
  • Definition 1.12: MAG on $\mathbb{T}^d$, absolutely continuous fluid, Bre11
  • Definition 2.6
  • Definition 2.14: MAG dynamics allowing for matter concentration, Bre11
  • Definition 2.20: Action functional of MAG pushed by $D$, Bre11
  • Definition 2.24: $k$-mapping
  • Definition 2.26: Action functional of MAG pushed by $\{x_1,\dots, x_k\}$, Bre16
  • Definition 3.13: Action of $\epsilon$- MAG pushed by $\{x_1,\dots,x_k\}$
  • Definition 4.7: MAG's action for a fluid pushed by $\lambda$
  • Proposition 4.11
  • ...and 50 more