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Energy dissipation and global convergence of a discrete normalized gradient flow for computing ground states of two-component Bose-Einstein condensates

Zixu Feng, Lunxu Liu, Qinglin Tang

TL;DR

This work addresses computing ground states of two-component Bose-Einstein condensates via a discrete normalized gradient flow (GFSI) and provides rigorous proofs of energy dissipation and global convergence. By reformulating GFSI with a Lagrange multiplier, the authors derive an equivalent scheme for MBECs with Josephson coupling and rotation and establish an energy-dissipation law, along with a provable global convergence to stationary states. They prove that, for a symmetric positive-definite interaction matrix with nonnegative entries, the GFSI sequence is energy-dissipative and globally convergent, with a time-step bound that depends on the interaction strength $k_m$. Numerical experiments on bounded domains corroborate the theory, illustrate the energy decay across different discretizations, and reveal how larger inter-component coupling restricts the allowable time step, thereby validating the theoretical results and providing practical guidance for simulations.

Abstract

The gradient flow with semi-implicit discretization (GFSI) is the most widely used algorithm for computing the ground state of Gross-Pitaevskii energy functional. Numerous numerical experiments have shown that the energy dissipation holds when calculating the ground states of multicomponent Bose-Einstein condensates (MBECs) with GFSI, while rigorous proof remains an open challenge. By introducing a Lagrange multiplier, we reformulate the GFSI into an equivalent form and thereby prove the energy dissipation for GFSI in two-component scenario with Josephson junction and rotating term, which is one of the most important and topical model in MBECs. Based on this, we further establish the global convergence to stationary states. Also, the numerical results of energy dissipation in practical experiments corroborate our rigorous mathematical proof, and we numerically verified the upper bound of time step that guarantees energy dissipation is indeed related to the strength of particle interactions.

Energy dissipation and global convergence of a discrete normalized gradient flow for computing ground states of two-component Bose-Einstein condensates

TL;DR

This work addresses computing ground states of two-component Bose-Einstein condensates via a discrete normalized gradient flow (GFSI) and provides rigorous proofs of energy dissipation and global convergence. By reformulating GFSI with a Lagrange multiplier, the authors derive an equivalent scheme for MBECs with Josephson coupling and rotation and establish an energy-dissipation law, along with a provable global convergence to stationary states. They prove that, for a symmetric positive-definite interaction matrix with nonnegative entries, the GFSI sequence is energy-dissipative and globally convergent, with a time-step bound that depends on the interaction strength . Numerical experiments on bounded domains corroborate the theory, illustrate the energy decay across different discretizations, and reveal how larger inter-component coupling restricts the allowable time step, thereby validating the theoretical results and providing practical guidance for simulations.

Abstract

The gradient flow with semi-implicit discretization (GFSI) is the most widely used algorithm for computing the ground state of Gross-Pitaevskii energy functional. Numerous numerical experiments have shown that the energy dissipation holds when calculating the ground states of multicomponent Bose-Einstein condensates (MBECs) with GFSI, while rigorous proof remains an open challenge. By introducing a Lagrange multiplier, we reformulate the GFSI into an equivalent form and thereby prove the energy dissipation for GFSI in two-component scenario with Josephson junction and rotating term, which is one of the most important and topical model in MBECs. Based on this, we further establish the global convergence to stationary states. Also, the numerical results of energy dissipation in practical experiments corroborate our rigorous mathematical proof, and we numerically verified the upper bound of time step that guarantees energy dissipation is indeed related to the strength of particle interactions.
Paper Structure (11 sections, 11 theorems, 83 equations, 3 figures, 2 tables)

This paper contains 11 sections, 11 theorems, 83 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The constrained energy minimization and basic properties
  3. Notations
  4. Properties of $\mathcal{H}_{\Psi}$ and $\mathcal{A}_{\Psi}^a(\cdot,\cdot)$
  5. The reformulation of GFSI algorithm and energy dissipation
  6. The GFSI algorithm
  7. The energy dissipation
  8. Global convergence of the GFSI
  9. Numerical experiments
  10. Conclusion
  11. Proof of Lemma \ref{['existence of ground state']}

Key Result

Lemma 2.1

There exists $\Psi_g \in \mathcal{M}$, such that $\Psi_g$ is a global minimizer of the constrained minimization problem (mini problem).

Figures (3)

  • Figure 5.1: Energy evolution that decrease at each step for various time steps $\tau= 0.1$, $0.2$, $0.5$, $1.0$ and different mesh sizes h= $1/8$, $1/16$, $1/32$ in Case 1 (upper) and Case 2 (lower) of Example \ref{['example: energy decrease']}.
  • Figure 5.2: Contour plots of the converged functions $|\psi_1|^2$ and $|\psi_2|^2$ in Case 3 (upper) and Case 4 (lower) of Example \ref{['example: energy decrease']}.
  • Figure 5.3: The energy evolution does not satisfy energy-diminishing property in Case 5, Case 6 and Case 7 (from left to right) for mesh size $h = 1/8$ and various time steps $\tau$.

Theorems & Definitions (30)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Remark 2.1
  • Remark 3.1
  • Theorem 3.1
  • Lemma 3.1
  • ...and 20 more