conservative-adaptive-rank-quantum-kinetics

name	conservative-adaptive-rank-quantum-kinetics
description	Conservative adaptive rank methodology for quantum kinetic simulations — ACA SVD with Fermi-Dirac reconstruction preserving discrete macroscopic invariants near machine precision.

Conservative Adaptive Rank Quantum Kinetics

Description

Methodology for conservative adaptive rank simulation of quantum kinetic systems (Wigner-Poisson). Combines ACA SVD compression with Fermi-Dirac-type macroscopic reconstruction to preserve discrete total energy and other invariants near machine precision.

Activation Keywords

• adaptive rank quantum simulation
• Wigner-Poisson system
• conservative kinetic simulation
• Fermi-Dirac reconstruction
• quantum kinetic method
• ACA SVD quantum
• 自适应秩量子模拟
• 维格纳-泊松系统
• 守恒量子动力学

Core Concepts

Key Finding (arXiv:2606.20234)

A conservative adaptive rank method for the 1D1V Wigner-Poisson system that reduces phase-space evolution cost while preserving macroscopic invariants needed for physical fidelity.

Key Components

1. Sampling-based adaptive rank Wigner-Poisson update — adapts numerical rank to phase-space complexity
2. Conservative density-momentum solve — provides local macroscopic updates
3. Fermi-Dirac-type reconstruction — transfers macroscopic updates to kinetic solution (motivated by quantum-statistical structure)
4. Global quadratic moment correction — enforces discrete total energy constraint at kinetic level
5. ACA SVD representation — allows rank to adapt to nonlocal Wigner operator and self-consistent Poisson field

Novelty vs Classical Methods

Unlike Maxwell-Boltzmann-type corrections used in classical kinetic settings, the reconstruction uses Fermi-Dirac-type form motivated by quantum-statistical structure.

Methodology

Step 1: Setup Adaptive Rank Framework

1. Initialize phase-space representation with ACA SVD
2. Set target accuracy for macroscopic invariants
3. Configure adaptive rank bounds

Step 2: Wigner-Poisson Update

1. Apply sampling-based adaptive rank update
2. Nonlocal Wigner operator generates phase-space complexity
3. Self-consistent Poisson field coupling
4. Numerical rank adapts automatically to complexity

Step 3: Conservative Macroscopic Correction

1. Local density-momentum solve: Compute macroscopic updates locally
2. Fermi-Dirac reconstruction: Transfer updates to kinetic solution
   • Uses Fermi-Dirac form (NOT Maxwell-Boltzmann)
   • Motivated by quantum-statistical structure of the model
3. Global quadratic moment correction: Enforce discrete total energy at kinetic level

Step 4: Validation

• Conservation errors should be near machine precision
• Adaptive ranks should remain bounded
• Phase-space dynamics should capture benchmark behavior

Step 5: Formulation Comparison

Two compatible correction strategies:

• Local density-momentum + global energy: Uses local correction plus global total energy
• Globally conservative: Mass, momentum, and energy all globally conserved
• Both produce nearly identical results for periodic benchmarks

Usage Patterns

Pattern 1: Quantum Kinetic Simulation

For simulating Wigner-Poisson dynamics:

1. Use adaptive rank compression for efficiency
2. Apply Fermi-Dirac reconstruction for quantum statistics
3. Verify conservation near machine precision

Pattern 2: Benchmark Validation

Standard test cases:

• Two-stream instability
• Strong Landau damping
• Bump-on-tail instability
• Various quantum parameter H values

Pattern 3: Classical-to-Quantum Extension

When extending classical kinetic methods to quantum:

• Replace Maxwell-Boltzmann correction with Fermi-Dirac
• Add global quadratic moment correction for energy
• Both correction strategies are compatible with adaptive rank compression

Error Handling

Unbounded Adaptive Rank

If rank grows without bound:

• Check quantum parameter H range
• Verify ACA SVD tolerance settings
• Ensure Fermi-Dirac reconstruction is properly applied

Conservation Drift

If conservation errors exceed machine precision:

• Verify global quadratic moment correction is applied
• Check Fermi-Dirac reconstruction consistency
• Ensure density-momentum solve is converged

Resources

• arXiv:2606.20234 "A conservative adaptive rank method for the Wigner-Poisson system"
• Related skills: quantum-statistical-estimation-framework

Notes

• Validated for 1D1V periodic setting
• Both correction strategies (local+global vs fully global) produce nearly identical results
• Conservation errors near machine precision achieved
• Applicable to quantum semiconductor and plasma simulations