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Grand-Canonical Optimal Transport

Simone Di Marino, Mathieu Lewin, Luca Nenna

TL;DR

This work extends optimal transport by introducing Grand-Canonical OT (GC-OT), where the number of marginals is not fixed and the optimization occurs over a grand-canonical ensemble of symmetric plans with a fixed average density $\rho$. It establishes existence, duality, and structural properties (notably $\textbf{c}$-monotonicity) for GC-OT, and links it to the classical multi-marginal problem via a convex-hull formalism, showing how GC-OT decomposes into combinations of integer-marginal problems. A substantial part analyzes pairwise repulsive costs, including the Coulomb interaction, deriving tight bounds on the support of minimizers, truncation schemes, and detailed Coulomb-specific results in 1D and higher dimensions. Entropic regularization at temperature $T>0$ is developed, leading to a Gibbs-state framework with a corresponding dual problem; the zero-temperature limit recovers the GC-OT energy and the finite-temperature behavior is connected to Poisson and Gibbs states, with regularity results for dual potentials under natural growth assumptions. The paper thus provides a comprehensive mathematical foundation for GC-OT, covering existence, duality, support, asymptotics, and regularization, with implications for statistical mechanics, density functional theory, and large-scale multi-agent transport problems.

Abstract

We study a generalization of the multi-marginal optimal transport problem, which has no fixed number of marginals $N$ and is inspired of statistical mechanics. It consists in optimizing a linear combination of the costs for all the possible $N$'s, while fixing a certain linear combination of the corresponding marginals.

Grand-Canonical Optimal Transport

TL;DR

This work extends optimal transport by introducing Grand-Canonical OT (GC-OT), where the number of marginals is not fixed and the optimization occurs over a grand-canonical ensemble of symmetric plans with a fixed average density . It establishes existence, duality, and structural properties (notably -monotonicity) for GC-OT, and links it to the classical multi-marginal problem via a convex-hull formalism, showing how GC-OT decomposes into combinations of integer-marginal problems. A substantial part analyzes pairwise repulsive costs, including the Coulomb interaction, deriving tight bounds on the support of minimizers, truncation schemes, and detailed Coulomb-specific results in 1D and higher dimensions. Entropic regularization at temperature is developed, leading to a Gibbs-state framework with a corresponding dual problem; the zero-temperature limit recovers the GC-OT energy and the finite-temperature behavior is connected to Poisson and Gibbs states, with regularity results for dual potentials under natural growth assumptions. The paper thus provides a comprehensive mathematical foundation for GC-OT, covering existence, duality, support, asymptotics, and regularization, with implications for statistical mechanics, density functional theory, and large-scale multi-agent transport problems.

Abstract

We study a generalization of the multi-marginal optimal transport problem, which has no fixed number of marginals and is inspired of statistical mechanics. It consists in optimizing a linear combination of the costs for all the possible 's, while fixing a certain linear combination of the corresponding marginals.
Paper Structure (25 sections, 27 theorems, 247 equations, 3 figures)

This paper contains 25 sections, 27 theorems, 247 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Main properties of Grand-Canonical Optimal Transport
  3. Subsystems are often grand-canonical
  4. The Grand-Canonical Optimal Transport problem
  5. Grand-canonical probabilities
  6. Well-posedness and existence of optimizers
  7. Relation with the multi-marginal problem
  8. Support for pairwise repulsive costs
  9. Support and truncation
  10. $\mathbf{c}$--monotonicity
  11. Repulsive pairwise costs
  12. Bounded repulsive cost
  13. Unbounded repulsive cost with a control on $1/c_2$
  14. Unbounded repulsive asymptotically doubling cost
  15. The 3D Coulomb cost
  16. ...and 10 more sections

Key Result

Theorem 2.13

Let $\Omega\subset\mathbb{R}^d$ be any Borel set. Let $\mathbf{c}=(c_n)_{n \geqslant 0}$ be a superstable family of lower semi-continuous costs. Then any finite $\mathcal{C}(\rho)$ admits a minimizer $\mathbb{P}^*$. Moreover $\rho\mapsto \mathcal{C}(\rho)$ is convex and lower semi-continuous for the

Figures (3)

  • Figure 1: For $t=0.7$, the measure $\rho = \frac{1}{2} \sum_{i=1}^6 \delta_{x_i}$ has the unique optimal grand canonical probability $\mathbb{P}^*_2=\delta_{(x_1,x_2)}/2$, $\mathbb{P}^*_4= \delta_{(x_3,x_4,x_5,x_6)})/2$. In particular, $\mathcal{C}(\rho)<\mathcal{C}_3(\rho)$.
  • Figure 2: Coulomb cost of all the extreme points \ref{['eq:extremal_pt']} for the $x_j$ as in Figure \ref{['fig:losange']}, as functions of the length $t\in(0,1)$ (left), with a zoom around the value $t=0.7$ (right). The grand-canonical extreme points $(|I|=2$) are in red whereas the canonical ones $(|I|=3$) are in blue. For $t\simeq0.7$, $p^{\{1,2\}}$ has the lowest possible Coulomb cost. Its curve has been made thicker in the drawing.
  • Figure 4: Examples of the points $\{y_1^k , \ldots y_{6^k}^k\}$ for $k=2,3$ and the values $\ell_2=5$ and $\ell_3=25$, when we start from the configuration in Figure \ref{['fig:losange']} with $t=0.7$.

Theorems & Definitions (78)

  • Example 2.1: Usual multi-marginal probabilities
  • Example 2.2: Poisson states
  • Definition 2.3: Pairwise costs
  • Example 2.4: Interacting classical particles
  • Remark 2.5: Sub-additivity
  • Example 2.6: Triviality of the pairwise harmonic cost
  • Example 2.7: Mapping two densities with different masses
  • Definition 2.8: Stability
  • Example 2.9: Unstability of the repulsive pairwise harmonic cost
  • Example 2.10: No-agent cost
  • ...and 68 more