Step 22: Kinetic Fluctuation-Dissipation Validation

[anjor]

Validate Hermite implementation against analytical theory (Thesis §2.6.1, Chapter 3):

1. Create tests/test_kinetic_fdt.py with:

   • Drive single k-mode with Gaussian white noise forcing
   • Run to saturation (energy reaches steady state)
   • Measure time-averaged |gₘ|² vs. m for m=0,1,...,M
   • Compare with analytical predictions from linear Vlasov theory

2. Expected results (Thesis Figs 3.1, 3.3, B.1):

   • Phase mixing spectrum: |gₘ|² ∝ m^(-α) for some α > 0
   • Phase unmixing spectrum: Different scaling
   • Total spectrum: Weighted sum of mixing/unmixing
   • Should match analytical expressions (Eqs 3.37, 3.58 in thesis)

3. Acceptance criteria:

   • Numerical spectra agree with theory within 10%
   • Agreement holds for multiple k-values
   • Scaling exponents match analytically predicted values
   • "The dotted lines in Fig. 3.3 are not fits to the numerical spectra, but are the exact expressions"

4. This is a critical benchmark:

   • Validates linear kinetic physics
   • Tests Hermite coupling, collisions, and closures
   • If this fails, kinetic implementation is incorrect

Dependencies: Step 21 (Phase mixing diagnostics)
Next Step: Step 23

[anjor]

Status Check ✓ Still needed - No implementation yet

Current State

• No test_kinetic_fdt.py test file exists
• No validation against analytical Vlasov theory
• No measurements of |gₘ|² spectrum vs analytical predictions

Context

This is a critical benchmark for validating the kinetic implementation. Per thesis §2.6.1 and Chapter 3, the numerical Hermite moment spectrum should match analytical predictions from linear Vlasov theory.

What this tests:

• Hermite moment coupling (g0_rhs, g1_rhs, gm_rhs correctness)
• Collision operator implementation
• Closure scheme validity
• Phase mixing vs phase unmixing balance

Why It's Critical

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

If this test fails, the entire kinetic implementation is incorrect - this is the gold standard validation.

Current Kinetic Implementation Status

✅ Hermite hierarchy equations implemented (PR #46, #50)
✅ Collision operators implemented (Issue #23)
✅ Closures working (Issue #24 - partial)
✅ Time integration working (PR #47, #50)
❌ No validation against analytical theory yet

Dependencies

Should be implemented before serious kinetic turbulence studies:

• Depends on Issue Step 21: Phase Mixing Diagnostics #26 (Phase mixing diagnostics) for detailed analysis
• Depends on Issue Step 24: Forcing Mechanisms #29 (Forcing) for driving single k-mode
• Issue Step 20: Field Line Following Diagnostics #25 (Field line following) optional but helpful

Suggested Implementation Order

1. Implement Gaussian white noise forcing (Issue Step 24: Forcing Mechanisms #29 subset)
2. Implement phase mixing diagnostics (Issue Step 21: Phase Mixing Diagnostics #26 subset)
3. Create test_kinetic_fdt.py with analytical comparison
4. Validate against thesis Figs 3.1, 3.3, B.1

Priority

HIGH - This is the definitive test of kinetic physics correctness. Should be done before publishing any kinetic turbulence results.

Keeping open - critical validation needed. 🎯