Heterodyne Scattering Model

The heterodyne package implements the two-component heterodyne scattering model from He et al. PNAS 2024 (Equations S-95 to S-98), generalized to nonequilibrium conditions with time-dependent transport coefficients.

Theoretical Foundation

The package implements Equation S-95 (general time-dependent form) from He et al. PNAS 2024, which uses transport coefficients J(t) for nonequilibrium dynamics.

Model Equation (Equation S-95):

\[\begin{split}c_2(\vec{q}, t_1, t_2, \phi) = 1 + \frac{\beta}{f^2} \Bigg[ [x_r(t_1)x_r(t_2)]^2 e^{-q^2 \int_{t_1}^{t_2} J_r(t) dt} + \\ [x_s(t_1)x_s(t_2)]^2 e^{-q^2 \int_{t_1}^{t_2} J_s(t) dt} + \\ 2x_r(t_1)x_r(t_2)x_s(t_1)x_s(t_2) e^{-\frac{1}{2}q^2 \int_{t_1}^{t_2} [J_s(t)+J_r(t)] dt} \cos\left[q \cos(\phi) \int_{t_1}^{t_2} v(t) dt\right] \Bigg]\end{split}\]

where \(f^2 = [x_s(t_1)^2 + x_r(t_1)^2][x_s(t_2)^2 + x_r(t_2)^2]\).

Key Parameters:

• J_r(t), J_s(t): Separate time-dependent transport coefficients for reference and sample [Ų/s]

• x_s(t_1), x_s(t_2): Sample fraction evaluated at times t₁ and t₂ (each in [0,1])

• x_r(t) = 1 - x_s(t): Reference fraction

• v(t): Time-dependent mean velocity [nm/s]

• φ: Relative angle = φ₀ - φ_scattering (flow angle minus scattering angle)

• β: Contrast factor (implicit in experimental measurements)

• f²: Two-time normalization factor

Relationship to Equilibrium Form: For equilibrium Wiener processes, J = 6D where D is the traditional diffusion coefficient. The “commonly used heterodyne equation” (Equation S-98) is this equilibrium simplification. This package implements the more general S-95.

Nonequilibrium Implementation: This package parameterizes J(t) = J₀·t^α + J_offset to capture nonequilibrium dynamics including aging, yielding, and shear banding phenomena. Parameters labeled “D” (D₀, α, D_offset) are actually transport coefficient parameters (J₀, α, J_offset) for historical compatibility.

Time-Dependent Fraction:

The fraction evolves according to:

\[f(t) = f_0 \times \exp(f_1 \times (t - f_2)) + f_3\]

with physical constraint: \(0 \leq f(t) \leq 1\) for all times.

14-Parameter System

The model uses 14 parameters organized into five groups:

Reference Transport Parameters (3):

Parameter	Range	Units	Physical Meaning
D₀_ref	[1.0, 1×10⁶]	Ų/s	Reference field transport coefficient at t=1s
α_ref	[-2.0, 2.0]	dimensionless	Reference transport time-scaling exponent
D_offset_ref	[-100, 100]	Ų/s	Reference baseline transport coefficient

Sample Transport Parameters (3):

Parameter	Range	Units	Physical Meaning
D₀_sample	[1.0, 1×10⁶]	Ų/s	Sample field transport coefficient at t=1s
α_sample	[-2.0, 2.0]	dimensionless	Sample transport time-scaling exponent
D_offset_sample	[-100, 100]	Ų/s	Sample baseline transport coefficient

Velocity Parameters (3):

Parameter	Range	Units	Physical Meaning
v₀	[1×10⁻⁵, 10.0]	nm/s	Reference velocity at t=1s
β	[-2.0, 2.0]	dimensionless	Velocity time-scaling exponent
v_offset	[-0.1, 0.1]	nm/s	Baseline velocity contribution

Fraction Parameters (4):

Parameter	Range	Units	Physical Meaning
f₀	[0.0, 1.0]	dimensionless	Fraction amplitude
f₁	[-1.0, 1.0]	s⁻¹	Exponential rate of fraction change
f₂	[0.0, 200.0]	s	Time offset for fraction dynamics
f₃	[0.0, 1.0]	dimensionless	Baseline fraction value

Flow Angle (1):

Parameter	Range	Units	Physical Meaning
φ₀	[-10, 10]	degrees	Flow direction angle

Physical Interpretation

Transport Coefficient Contribution:

The time-dependent transport coefficients for reference and sample fields are:

\[\begin{split}D_{\text{ref}}(t) &= D_{0,\text{ref}} \times t^{\alpha_{\text{ref}}} + D_{\text{offset,ref}} \\ D_{\text{sample}}(t) &= D_{0,\text{sample}} \times t^{\alpha_{\text{sample}}} + D_{\text{offset,sample}}\end{split}\]

This captures:

• Anomalous transport: α ≠ 0 indicates sub-diffusive (α < 0) or super-diffusive (α > 0) dynamics

• Separate field dynamics: Independent reference and sample transport properties

• Aging effects: Time-dependent transport in nonequilibrium systems

• Baseline transport: Constant background transport contributions

Velocity Contribution:

The time-dependent velocity is:

\[v(t) = v_0 \times t^\beta + v_{\text{offset}}\]

This describes:

• Flow acceleration/deceleration: β controls temporal evolution of flow

• Steady flow: β = 0 with v₀ > 0 gives constant flow velocity

• Transient flow: Non-zero β captures time-dependent flow dynamics

Fraction Dynamics:

The time-dependent fraction mixing describes:

• Component evolution: How reference and sample contributions change over time

• Relaxation dynamics: f₁ controls the rate of exponential relaxation

• Equilibrium state: f₃ represents the long-time steady-state fraction

• Initial conditions: f₀ and f₂ control the amplitude and temporal offset

Configuration Examples

Full 14-Parameter Heterodyne Configuration:

{
  "metadata": {
    "config_version": "1.0",
    "analysis_mode": "heterodyne"
  },
  "initial_parameters": {
    "parameter_names": [
      "D0_ref", "alpha_ref", "D_offset_ref",
      "D0_sample", "alpha_sample", "D_offset_sample",
      "v0", "beta", "v_offset",
      "f0", "f1", "f2", "f3",
      "phi0"
    ],
    "values": [1000.0, -0.5, 100.0, 1000.0, -0.5, 100.0, 0.01, 0.5, 0.001, 0.5, 0.0, 50.0, 0.3, 0.0],
    "active_parameters": ["D0_ref", "alpha_ref", "D0_sample", "alpha_sample", "v0", "beta", "f0", "f1"]
  },
  "parameter_space": {
    "bounds": [
      {"name": "D0_ref", "min": 1.0, "max": 1000000, "type": "Normal"},
      {"name": "alpha_ref", "min": -2.0, "max": 2.0, "type": "Normal"},
      {"name": "D_offset_ref", "min": -100, "max": 100, "type": "Normal"},
      {"name": "D0_sample", "min": 1.0, "max": 1000000, "type": "Normal"},
      {"name": "alpha_sample", "min": -2.0, "max": 2.0, "type": "Normal"},
      {"name": "D_offset_sample", "min": -100, "max": 100, "type": "Normal"},
      {"name": "v0", "min": 1e-5, "max": 10.0, "type": "Normal"},
      {"name": "beta", "min": -2.0, "max": 2.0, "type": "Normal"},
      {"name": "v_offset", "min": -0.1, "max": 0.1, "type": "Normal"},
      {"name": "f0", "min": 0.0, "max": 1.0, "type": "Normal"},
      {"name": "f1", "min": -1.0, "max": 1.0, "type": "Normal"},
      {"name": "f2", "min": 0.0, "max": 200.0, "type": "Normal"},
      {"name": "f3", "min": 0.0, "max": 1.0, "type": "Normal"},
      {"name": "phi0", "min": -10, "max": 10, "type": "Normal"}
    ]
  }
}

Simplified Configuration (Fewer Active Parameters):

For initial exploration, you can fix some parameters:

{
  "initial_parameters": {
    "parameter_names": [
      "D0_ref", "alpha_ref", "D_offset_ref",
      "D0_sample", "alpha_sample", "D_offset_sample",
      "v0", "beta", "v_offset",
      "f0", "f1", "f2", "f3",
      "phi0"
    ],
    "values": [1000.0, -0.5, 0.0, 1000.0, -0.5, 0.0, 0.01, 0.0, 0.0, 0.5, 0.0, 50.0, 0.3, 0.0],
    "active_parameters": ["D0_ref", "D0_sample", "v0", "f0"]  // Optimize only 4 parameters
  }
}

Analysis Workflow

1. Initial Exploration:

Start with a subset of active parameters:

# Optimize only diffusion parameters
heterodyne --config config.json --method classical

2. Incremental Complexity:

Gradually add more parameters:

# Add velocity parameters
# Edit config to include v0, beta in active_parameters
heterodyne --config config.json --method classical

3. Full Optimization:

Optimize all relevant parameters:

# Full parameter optimization with robust methods
heterodyne --config config.json --method all

4. Robust Optimization for Noisy Data:

Use robust methods for experimental data with uncertainty:

# Wasserstein DRO for outlier resistance
heterodyne --config config.json --method robust

Parameter Selection Guidelines

Start with Essential Parameters:

• D₀_ref, D₀_sample: Core transport dynamics for both fields

• v₀: Flow velocity (if flow present)

• f₀: Reference/sample mixing amplitude

Add Complexity as Needed:

• α_ref, α_sample: For time-dependent transport in each field

• β: If flow shows time-dependent behavior

• f₁, f₂: If fraction mixing shows temporal dynamics

• D_offset_ref, D_offset_sample, v_offset: For baseline corrections

• f₃: For steady-state fraction adjustment

• φ₀: For flow direction refinement

Physical Constraints:

The package automatically enforces:

• D(t) ≥ 1×10⁻¹⁰: Positive diffusion coefficient

• v(t) ≥ 1×10⁻¹⁰: Positive velocity

• 0 ≤ f(t) ≤ 1: Valid fraction range

Best Practices

1. Validate Experimental Data:

heterodyne --config config.json --plot-experimental-data

2. Start Simple:

Begin with fewer active parameters and add complexity incrementally.

3. Check Convergence:

Monitor chi-squared values and parameter uncertainties in results.

4. Use Robust Methods for Noisy Data:

Wasserstein DRO, scenario-based, or ellipsoidal methods handle uncertainty better than classical optimization.

5. Physical Interpretation:

Ensure fitted parameters have physically meaningful values and interpretations.

Troubleshooting

Poor Convergence:

• Reduce number of active parameters

• Adjust initial parameter values

• Try different optimization methods

Unphysical Parameters:

• Check parameter bounds in configuration

• Verify experimental data quality

• Review fraction constraint: 0 ≤ f(t) ≤ 1

High Chi-Squared:

• Increase number of active parameters

• Use robust optimization methods

• Check for systematic errors in data

Fraction Constraint Violations:

• Adjust f₀, f₁, f₂, f₃ bounds

• Ensure f(t) stays within [0, 1] for all times

• Review fraction dynamics physical interpretation

See Also

• Configuration Guide - Detailed configuration guide

• heterodyne.analysis.core - Core Analysis Engine - Core analysis API

• Performance Optimization Guide - Optimization strategies

• Quick Start Guide - Quick start tutorial