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On Minimizing Phase Space Energies

Michael Updike, Nicholas Bohlsen, Hong Qin, Nathaniel Fisch

TL;DR

This work analyzes the maximal energy that can be extracted from a particle distribution under phase-space evolution constrained to Hamiltonian symplectomorphisms. By focusing on a linear, quadratic regime, the authors reduce the problem to tractable trace-minimization over affine maps, obtaining explicit formulas for the linear Gardner energy $E_{SL(2n)}$ and the linear Gromov energy $E_{Sp(2n)}$ in terms of the moments $N$, $V$, $H$ and the symplectic eigenvalues. They show $E_{Sp(2n)}$ generally exceeds $E_{SL(2n)}$, with equality only in special cases (e.g., $n=1$), and provide concrete examples including a Gromov non-squeezing-style bound and an ellipsoid equivalence criterion. The results connect rigorous mathematical constraints (Gardner vs Gromov, Williamson's theorem, symplectic eigenvalues) to phase-space energy extraction, offering a framework for benchmarking nonlinear limits and guiding future coarse-grained or data-driven approaches. The work lays a foundation for phase-space engineering by linking energy minimization to explicit, computable linear-algebraic quantities, while highlighting the gap to fully nonlinear dynamics and practical implementation.

Abstract

A primary technical challenge for harnessing fusion energy is to control and extract energy from a non-thermal distribution of charged particles. The fact that phase space evolves by symplectomorphisms fundamentally limits how a distribution may be manipulated. While the constraint of phase-space volume preservation is well understood, other constraints remain to be fully appreciated. To better understand these constraints, we study the problem of extracting energy from a distribution of particles using area-preserving and symplectic linear maps. When a quadratic potential is imposed, we find that the maximal extractable energy can be computed as trace minimization problems. We solve these problems and show that the extractable energy under linear symplectomorphisms may be much smaller than the extractable energy under special linear maps. The method introduced in the present study enables an energy-based proof of the linear Gromov non-squeezing theorem.

On Minimizing Phase Space Energies

TL;DR

This work analyzes the maximal energy that can be extracted from a particle distribution under phase-space evolution constrained to Hamiltonian symplectomorphisms. By focusing on a linear, quadratic regime, the authors reduce the problem to tractable trace-minimization over affine maps, obtaining explicit formulas for the linear Gardner energy and the linear Gromov energy in terms of the moments , , and the symplectic eigenvalues. They show generally exceeds , with equality only in special cases (e.g., ), and provide concrete examples including a Gromov non-squeezing-style bound and an ellipsoid equivalence criterion. The results connect rigorous mathematical constraints (Gardner vs Gromov, Williamson's theorem, symplectic eigenvalues) to phase-space energy extraction, offering a framework for benchmarking nonlinear limits and guiding future coarse-grained or data-driven approaches. The work lays a foundation for phase-space engineering by linking energy minimization to explicit, computable linear-algebraic quantities, while highlighting the gap to fully nonlinear dynamics and practical implementation.

Abstract

A primary technical challenge for harnessing fusion energy is to control and extract energy from a non-thermal distribution of charged particles. The fact that phase space evolves by symplectomorphisms fundamentally limits how a distribution may be manipulated. While the constraint of phase-space volume preservation is well understood, other constraints remain to be fully appreciated. To better understand these constraints, we study the problem of extracting energy from a distribution of particles using area-preserving and symplectic linear maps. When a quadratic potential is imposed, we find that the maximal extractable energy can be computed as trace minimization problems. We solve these problems and show that the extractable energy under linear symplectomorphisms may be much smaller than the extractable energy under special linear maps. The method introduced in the present study enables an energy-based proof of the linear Gromov non-squeezing theorem.

Paper Structure

This paper contains 14 sections, 2 theorems, 35 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Analysis
  3. Linear Gardner Energy
  4. Linear Gromov Energy
  5. Example 1
  6. Example 2
  7. Example 3: Symplectic Equivalence of Ellipsoids
  8. Example 4: Gromov's non-Squeezing Theorem
  9. Discussion
  10. Appendix
  11. Degenerate Potentials
  12. Gardner's Restacking and Hamiltonian Approximations
  13. Linear vs. Nonlinear Operations
  14. Quadratic Hamiltonian and Linear Maps

Key Result

Theorem 1

For any $2n \times 2n$, symmetric, positive-definite matrices $V,H$ with respective symplectic eigenvalues $\lambda_{1}^{V} \geq \hdots \geq \lambda_{2n}^{V} > 0$ and $\lambda_{1}^{H} \geq \hdots \geq \lambda_{2n}^{H} > 0$,

Figures (2)

  • Figure 1: Minimal energies for Example \ref{['ex1']}.
  • Figure 2: Minimal energies for Example \ref{['ex2']}.

Theorems & Definitions (4)

  • Theorem 1
  • proof
  • Theorem 2
  • proof