Theory & Physics

Homodyne implements the transport coefficient framework developed by He et al. (PNAS 2024, 2025) for X-ray Photon Correlation Spectroscopy (XPCS) analysis of soft matter under nonequilibrium conditions. This section provides the complete theoretical foundation underlying every computation in the package.

The central quantity is the transport coefficient \(J(t)\), which connects microscopic particle dynamics to macroscopic rheological observables via a generalized Green-Kubo relation. From \(J(t)\), the package constructs the full two-time intensity correlation function \(c_2(\vec{q}, t_1, t_2)\) — avoiding the equilibrium assumption embedded in the standard \(g_2(q, \tau)\) representation.

Overview of Sections

Theory

• Transport Coefficient J(t)
  • Definition
  • Physical Interpretation
  • Homodyne Implementation
  • Connection to Physical Diffusion Coefficient
  • Table of J(t) for Classical Processes
  • Relationship to Rheology
• Correlation Functions in XPCS
  • Position Density and Scattered Field
  • First-Order Correlation Function
  • Second-Order Correlation Function
  • Siegert Relation
  • Equilibrium Approximation: g₂(q, τ)
  • Two-Time Correlation Matrix
  • Model Fitting
• Homodyne Scattering: Laminar Flow Model
  • Taylor-Couette Geometry
  • Gap Integration and the sinc² Modulation
  • Full Laminar Flow Equation
  • Angular Dependence
  • Homodyne Implementation Parameters
  • Code Example
  • Connection to Rheology
• Heterodyne Scattering: Multi-Component Models
  • Physical Motivation
  • N-Component General Formula
  • Two-Component Case: Static + Flowing
  • Three-Component Case: Two Flowing + One Static
  • Normalization Factor f²
  • Oscillatory Patterns as Diagnostic
  • Comparison to Homodyne
• Classical Langevin Processes
  • The Langevin Equation
  • Wiener Process (Standard Diffusion)
  • Ornstein-Uhlenbeck Process (Inertial Brownian Motion)
  • Brownian Oscillator (Underdamped)
  • Brownian Oscillator (Overdamped)
  • Comparison Table
  • Non-Gaussian Corrections
• Yielding Dynamics of Colloidal Suspensions
  • Overview: Repulsive vs Attractive Suspensions
  • Repulsive Suspensions: Andrade Creep
  • Attractive Suspensions: Shear Banding and Delayed Yielding
  • Bond Dynamics and Interfacial Layers
  • Non-Affine Displacements
  • Connection to XPCS Measurements
  • Summary of Physical Signatures
• Theoretical Framework
  • Mathematical Foundation
  • Parameter Sets
  • Experimental Parameters
  • Integral Functions
  • Scattering Geometry
  • Physical Interpretation
  • Chi-Squared Objective Function
  • Uncertainty Quantification
  • Numerical Implementation
  • References
• Analysis Modes
  • Mode Overview
  • Static Mode
  • Laminar Flow Mode
  • Parameter Bounds and Priors (NLSQ & CMC)
  • Per-Angle Scaling
  • Mode Selection Guidelines
  • Physical Constraints
  • Configuration Example
• Per-Angle Scaling and Anti-Degeneracy
  • The Parameter Absorption Degeneracy
  • Per-Angle Modes
  • Mathematical Formulation
  • Parameter Count Summary
  • When to Use Each Mode
  • Implementation Reference
• Anti-Degeneracy Defense System
  • Introduction
  • Theoretical Background
  • Layer 1: Fourier Reparameterization
  • Layer 2: Hierarchical Optimization
  • Layer 3: Adaptive CV Regularization
  • Layer 4: Gradient Collapse Detection
  • Layer 5: Shear-Sensitivity Weighting
  • Cross-Path Uniformity
  • Configuration Reference
  • Usage Tutorial
  • API Reference
  • See Also
• Computational Methods
  • Non-Linear Least Squares (NLSQ)
  • Consensus Monte Carlo (CMC)
• References and Citations
  • Primary References
  • XPCS Methodology
  • Stochastic Processes
  • Bayesian Methods
  • Optimization
  • JAX and Scientific Stack
  • Rheology
  • Citing Homodyne
  • Acknowledgments
  • Contact

Quick Physics Reference

Static mode (\(n_\mathrm{params} = 3\)):

\[c_2(q, t_1, t_2) = 1 + \beta(t_1, t_2) \exp\!\left(-q^2 \int_{t_1}^{t_2} J(t)\,dt\right)\]

Laminar flow mode (\(n_\mathrm{params} = 7\)):

\[c_2(\vec{q}, t_1, t_2) = 1 + \beta(t_1, t_2) \exp\!\left(-q^2 \int_{t_1}^{t_2} J(t)\,dt\right) \operatorname{sinc}^2\!\left(\tfrac{1}{2} q h \int_{t_1}^{t_2} \dot{\gamma}(t) \cos\phi\,dt\right)\]

Parameter table:

Symbol	Parameter name	Physical meaning
\(D_0\)	D0	Diffusion prefactor (\(\text{Å}^2/\text{s}\))
\(\alpha\)	alpha	Diffusion anomalous exponent (0 < α ≤ 1)
\(D_\mathrm{offset}\)	D_offset	Constant diffusion background
\(\dot{\gamma}_0\)	gamma_dot_0	Shear rate prefactor (\(\text{s}^{-1}\))
\(\beta\)	beta	Shear rate power-law exponent
\(\dot{\gamma}_\mathrm{offset}\)	gamma_dot_t_offset	Constant shear rate background
\(\phi_0\)	phi_0	Azimuthal reference angle (rad)

See Transport Coefficient J(t) for derivation of \(J(t)\), and Homodyne Scattering: Laminar Flow Model for the full laminar-flow equation.