Implement exact KRMHD dispersion relation for quantitative FDT validation

[anjor]

Overview

Replace phenomenological response function in validation.py with exact KRMHD dispersion relation to enable quantitative FDT validation at 10% accuracy level.

Current Status

PR #66 implemented exact plasma physics special functions (plasma dispersion function Z(ζ), modified Bessel functions I_m(b)) and correct power laws (m^(-3/2), m^(-1/2)). However, the kinetic_response_function() uses a phenomenological approximation:

response = kinetic_factor / (1.0 + abs(zeta)**2)  # Phenomenological

This provides qualitatively correct behavior but lacks:

• Exact normalization from full dispersion relation
• Proper k⊥ρ_s dependence in resonance width
• FLR corrections to susceptibility

Required Implementation

Implement exact KRMHD dispersion relation from:

• Howes et al. (2006) ApJ 651:590, Equations 14-15
• Thesis Equation 3.37 (exact phase mixing spectrum)

The full dispersion relation has the form:

D(k,ω) = 1 - χ(k,ω) = 0

where χ(k,ω) includes:

• Plasma dispersion function Z(ζ) for Landau resonance ✅ (already implemented)
• Modified Bessel functions for FLR effects ✅ (already implemented)
• Proper k⊥ρ_s dependence in susceptibility
• Exact normalization factors

Algorithm

1. Solve dispersion relation D(k,ω) = 0 for given k
2. Compute linear susceptibility χ(k,ω) from solution
3. Use |χ|² in spectrum formula instead of phenomenological factor

Success Criteria

1. Quantitative agreement: Numerical FDT spectrum matches analytical within 10% for all m
2. Physics validation:
   • Correct Landau damping rate vs analytical theory
   • Proper FLR corrections at k⊥ρ_s ~ 1
   • Accurate critical moment m_crit vs k∥v_th/ν
3. Benchmarking: Match thesis Figures 3.1, 3.3, B.1

References

Primary sources:

• Howes, G. G. et al. (2006) ApJ 651:590 - KRMHD dispersion relation (Eq. 14-15)
• Thesis Chapter 3 - Analytical FDT theory (Eq. 3.37, 3.58)
• Thesis Figs 3.1, 3.3, B.1 - Benchmark comparisons

Background:

• Schekochihin, A. A. et al. (2009) ApJS 182:310 - Kinetic cascades
• Adkins & Schekochihin (2017) arXiv:1709.03203 - Phase mixing power laws

Implementation Notes

Current infrastructure (ready to use):

• ✅ Plasma dispersion function: plasma_dispersion_function(zeta)
• ✅ Plasma dispersion derivative: plasma_dispersion_derivative(zeta)
• ✅ Modified Bessel ratios: modified_bessel_ratio(m, x)
• ✅ FLR correction factors: flr_correction_factor(m, k_perp, rho_s)
• ✅ 15 unit tests validating all special functions

New work required:

1. Implement full KRMHD dispersion relation D(k,ω)
2. Numerical solver for D(k,ω) = 0 (root finding)
3. Extract linear susceptibility χ(k,ω) from solution
4. Replace phenomenological response with exact χ
5. Validate against thesis benchmarks

Complexity estimate:

• Medium-high (requires root finding of complex dispersion relation)
• Essential for production validation studies
• Well-defined physics problem with clear benchmarks

Testing Strategy

1. Unit tests: Verify dispersion relation solver

   • Test k⊥→0 (fluid limit): should recover MHD dispersion
   • Test k⊥ρ_s~1 (kinetic regime): check FLR corrections
   • Test collisionless vs collisional limits

2. Integration tests: Compare with thesis figures

   • Fig 3.1: Hermite spectrum vs m
   • Fig 3.3: Phase mixing/unmixing regimes
   • Fig B.1: Detailed benchmarks

3. Quantitative validation: Enforce 10% criterion

   • For each k-mode, verify |numerical - analytical| / analytical < 0.1
   • Test multiple k∥/k⊥ ratios
   • Test multiple collision frequencies ν

Priority

Medium-High - Required for production validation studies

Current status: Qualitative FDT validation works (PR #66)
Target status: Quantitative FDT validation at 10% level

Related

• PR Implement exact FDT analytical expressions from thesis #66: Exact FDT expressions (plasma dispersion + Bessel functions) ✅ merged
• Issue Step 22: Kinetic Fluctuation-Dissipation Validation #27: Kinetic FDT validation (infrastructure) ✅ complete
• Thesis validation requirement: "10% agreement on normalized spectrum"

Notes

Per thesis: "The dotted lines in Fig. 3.3 are not fits to the numerical spectra, but are the exact expressions"

This is the final piece needed for production-quality FDT validation!

[anjor]

Status Update

All infrastructure is now complete! ✅

PR #66 has been successfully merged (Oct 17), providing all the exact analytical functions needed:

• ✅ Plasma dispersion function Z(ζ) and its derivative
• ✅ Modified Bessel functions I_m(b) with ratios
• ✅ FLR correction factors
• ✅ 15 unit tests validating all special functions

Ready for implementation! The exact KRMHD dispersion relation solver can now be implemented using these building blocks. No blockers remain.

Next steps:

1. Implement full dispersion relation D(k,ω) from Howes et al. (2006) Eq. 14-15
2. Add numerical root finder for D(k,ω) = 0
3. Extract linear susceptibility χ(k,ω) and use in FDT validation
4. Target: 10% quantitative agreement with thesis Figures 3.1, 3.3, B.1

[anjor]

Revisiting this issue in context of multi-species KRMHD planning.

Physics clarification:
KRMHD assumes k⊥ρᵢ << 1 in its asymptotic ordering (thesis Ch. 1). The dynamics (physics.py, hermite.py) correctly implement this truncated model.

Issue #75 is about validation methodology: using the exact linear dispersion relation (which includes full FLR effects via Bessel functions) to validate the truncated KRMHD nonlinear evolution. This tests whether the k⊥ρᵢ << 1 approximation is valid in our parameter regime.

Deferral rationale:
For upcoming multi-species work (energy partitioning between H+/He++/e⁻ in k⊥ρᵢ << 1 regime):

• Science requires 20-30% accuracy on heating rate ratios, not 10% precision on absolute rates
• Current phenomenological validation (validation.py:278-279) provides adequate qualitative checks
• Multi-species extension is a higher priority than single-species quantitative refinement

Recommendation: Defer as low-priority enhancement. Valid for future quantitative validation work, but not blocking multi-species implementation.

[anjor]

Related Work Completed:

PR #66 implemented the foundational special functions needed for exact KRMHD dispersion:

• ✅ Plasma dispersion function Z(ζ) and derivative Z'(ζ)
• ✅ Modified Bessel function ratios I_m(b)/I_0(b)
• ✅ FLR correction factors
• ✅ 15 unit tests validating all functions

Current Status:

The kinetic_response_function() in validation.py still uses a phenomenological approximation:

response = kinetic_factor / (1.0 + abs(zeta)**2)  # Phenomenological

This provides qualitatively correct behavior but lacks exact normalization from the full KRMHD dispersion relation.

Next Steps:

Implementing the exact dispersion relation requires:

1. Numerical root finding for D(k,ω) = 0 (complex dispersion relation)
2. Extracting linear susceptibility χ(k,ω) from the solution
3. Replacing phenomenological response with |χ|²

This would enable quantitative FDT validation at 10% accuracy level as required for production validation studies.