Theory & Physics¶
The heterodyne XPCS analysis framework implements the two-component scattering theory developed by He et al. for studying nonequilibrium dynamics under flow conditions. The theoretical foundation spans two publications:
[He2024] (doi:10.1073/pnas.2401162121): Introduces the generalized two-time correlation framework, the transport coefficient \(J(t)\), and the integral formulation connecting microscopic velocity statistics to measurable scattering correlations.
[He2025] (doi:10.1073/pnas.2514216122): Extends the theory to multi-component heterodyne detection, derives the two-component interference model, and demonstrates extraction of flow velocity from cross-correlation oscillations.
The key insight is that heterodyne scattering — where a static reference field interferes with scattered light from a moving sample — produces an oscillatory cross-correlation term whose frequency encodes the sample velocity. Combined with the transport coefficient formalism, this enables simultaneous measurement of diffusion, flow velocity, and composition dynamics from a single two-time correlation measurement.
All equations reference the Supporting Information (SI) numbering from the PNAS publications. Physical quantities use angstrom-based units throughout: \(q\) in \(\text{\AA}^{-1}\), \(D_0\) in \(\text{\AA}^2/\text{s}^\alpha\), and velocities in \(\text{\AA/s}\).
Important
Numerical integration only. The implementation always evaluates the
transport and velocity integrals
(\(\int_{t_i}^{t_j} J(t')\, dt'\) and
\(\int_{t_i}^{t_j} v(t')\, dt'\)) numerically via
trapezoid_cumsum. Analytical antiderivatives are never
substituted, even for special cases (e.g., constant \(J\)). The
power-law parameterization \(J(t) = D_0 t^\alpha + D_\mathrm{offset}\)
has no useful closed-form antiderivative for general \(\alpha\),
so numerical integration is the only correct approach.