Glossary

This glossary defines terms used in homodyne documentation and XPCS analysis.

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anomalous diffusion

Diffusion in which the mean-squared displacement scales as \(\langle r^2(t) \rangle \propto t^\alpha\) with \(\alpha \neq 1\). Sub-diffusion (\(\alpha < 1\)) is seen in gels, glasses, and crowded environments. Super-diffusion (\(\alpha > 1\)) occurs in active systems.

APS

Advanced Photon Source — a synchrotron X-ray facility at Argonne National Laboratory (Illinois, USA). The primary source of XPCS data for which homodyne was developed.

APS-U

Advanced Photon Source Upgrade — the upgraded APS facility with a new multi-bend achromat storage ring providing dramatically higher brightness and coherent flux.

ArviZ

A Python package for exploratory analysis of Bayesian models. Provides diagnostics (R-hat, ESS, BFMI), posterior visualization, and model comparison tools. Used in homodyne for CMC result analysis.

Brownian motion

Random thermal motion of particles suspended in a fluid, described by the Langevin equation. Characterized by normal diffusion (\(\alpha = 1\)).

chi-squared

A statistic measuring goodness of fit: \(\chi^2 = \sum_i (y_i - f_i)^2 / \sigma_i^2\). The reduced chi-squared \(\chi^2_\nu = \chi^2 / (n - p)\) normalizes by degrees of freedom. Values near 1 indicate a good fit.

CMC

Consensus Monte Carlo — a divide-and-conquer MCMC method that partitions data into shards, runs NUTS sampling on each shard independently, then combines posteriors using precision-weighted averaging. Enables Bayesian inference on large datasets that would be infeasible with full-data MCMC.

coherence length

The spatial extent over which X-rays maintain a definite phase relationship. The transverse coherence length determines the speckle size; the longitudinal coherence length determines the energy resolution of the correlation.

contrast

The amplitude of the correlation function at zero lag time, denoted \(\beta\) (also called speckle contrast or coherence factor). Values range from 0 (fully incoherent) to 1 (fully coherent). Typical experimental values: 0.01–0.5.

Couette geometry

A concentric-cylinder shear cell geometry used in rheology-XPCS experiments. The inner cylinder (rotor) rotates while the outer cylinder (stator) is fixed, creating simple shear flow in the gap.

D0

The reference diffusion coefficient \(D_0\) in homodyne’s model. Controls the amplitude of the diffusion kernel and is the primary parameter of interest for characterizing particle dynamics. Units: Ų/s.

D_offset

The baseline (time-independent) diffusion coefficient in homodyne’s model. Adds a linear contribution to the diffusion kernel: \(J \supset D_\text{offset} (t_2 - t_1)\). Physically represents a fast background diffusion component.

decorrelation

The process by which the normalized correlation function \(g_2(t)\) decays from 1+:math:beta to the baseline (offset) as lag time increases. Fast decorrelation means fast dynamics.

divergent transitions

In NUTS/HMC sampling, transitions where the numerical integrator produces energy errors exceeding a threshold. High divergence rates (> 10%) indicate problems with the posterior geometry or sampler configuration.

DLS

Dynamic Light Scattering — the optical (visible light) analogue of XPCS. Uses the same physical principles but with visible laser light instead of X-rays. Limited to dilute, transparent samples and larger length scales.

effective sample size (ESS)

The number of truly independent samples a Markov chain is equivalent to, accounting for chain autocorrelation. A rule of thumb: ESS > 400 for reliable posterior summaries.

ergodic

A system is ergodic if time averages equal ensemble averages. Standard XPCS analysis assumes ergodicity; XPCS with two-time correlations can detect and accommodate non-ergodic behavior (aging, non-stationary dynamics).

g1

The normalized first-order (field) correlation function: \(g_1(q, t) = \langle E^*(q, 0) E(q, t) \rangle / \langle |E(q)|^2 \rangle\). For Brownian particles: \(g_1(q, t) = \exp(-q^2 D_0 t^\alpha)\).

g2

The normalized second-order (intensity) correlation function: \(g_2(q, t) = \langle I(q, 0) I(q, t) \rangle / \langle I(q) \rangle^2\). Related to \(g_1\) by the Siegert relation: \(g_2 = 1 + \beta |g_1|^2\).

gamma_dot_0

The reference shear rate \(\dot\gamma_0\) (units: s⁻¹). Amplitude of the time-dependent shear rate in homodyne’s laminar flow model.

GAP

See: gap distance

gap distance

The stator-rotor separation in a Couette shear cell, denoted \(h\). Configured as gap_distance in YAML (in µm). Internally stored in Å (1 µm = 10⁴ Å).

HDF5

Hierarchical Data Format 5 — a file format for large numerical datasets, widely used at synchrotron facilities for storing raw and reduced XPCS data.

HMC

Hamiltonian Monte Carlo — a family of MCMC algorithms that use Hamiltonian dynamics to propose distant moves, reducing random-walk behavior. NUTS is an adaptive variant of HMC.

homodyne detection

Detection of scattered X-rays without mixing with a reference beam. Produces \(C_2 \propto |g_1|^2\) via the Siegert relation. Contrast with heterodyne detection, where scattered and reference beams are mixed.

intermediate scattering function (ISF)

See: g1

JAX

A Python library for high-performance numerical computing with automatic differentiation and JIT compilation via XLA. Homodyne uses JAX for all numerical computations.

JIT compilation

Just-in-Time compilation — compiling code at runtime rather than ahead-of-time. JAX’s JIT compilation traces Python functions and compiles them to optimized XLA code on first invocation; subsequent calls are fast.

laminar_flow

Analysis mode for samples under laminar shear (e.g., Couette geometry). Uses 7 physical parameters (D₀, α, D_offset, γ̇₀, β, γ̇_offset, φ₀) and includes the sinc² factor for shear-induced decorrelation.

Latin Hypercube Sampling (LHS)

A stratified random sampling method that ensures even coverage of parameter space. Used in homodyne’s multi-start NLSQ to generate diverse starting points.

Levenberg-Marquardt

A damped least-squares algorithm that interpolates between gradient descent (far from minimum) and Gauss-Newton (near minimum). The basis of homodyne’s trust-region NLSQ solver.

NLSQ

Non-Linear Least Squares — a class of optimization methods for minimizing a sum of squared residuals with respect to nonlinear parameters. Homodyne uses the NLSQ package with a trust-region LM solver.

NumPyro

A probabilistic programming library built on JAX and NumPy, providing NUTS/HMC samplers with automatic differentiation. Homodyne’s CMC backend uses NumPyro.

NUTS

No-U-Turn Sampler — an adaptive extension of HMC that automatically chooses the trajectory length, avoiding the need to tune it manually. Used in homodyne’s CMC backend (via NumPyro).

offset

The baseline level of the correlation function at large lag times, where \(C_2 \to 1\) in the ideal case. Deviations from 1 indicate incoherent background contributions.

per-angle mode

Controls how homodyne handles angle-to-angle variations in speckle contrast and background offset. Options: auto (recommended), constant, individual, fourier.

phi_0

The angular offset \(\phi_0\) (degrees) between the shear flow direction and the detector coordinate system. Used in laminar_flow mode to orient the sinc² angular dependence.

R-hat

The Gelman-Rubin convergence statistic. Values near 1.0 indicate that multiple Markov chains have converged to the same distribution. Rule of thumb: R-hat < 1.05.

relaxation time

The characteristic timescale at which the correlation function decays to 1/e of its initial value. For normal diffusion: \(\tau_q \sim (q^2 D_0)^{-1}\).

shard

In CMC, a subset of the data assigned to one worker process. The shard size (max_points_per_shard) controls the trade-off between NUTS accuracy (larger shards) and computation time (smaller shards).

shear rate

The rate of deformation in a shear flow, \(\dot\gamma = dv_x/dy\) (s⁻¹). In Couette geometry, the average shear rate equals the rotor angular velocity times the ratio of rotor radius to gap width.

Siegert relation

The relation \(g_2(q,t) = 1 + \beta |g_1(q,t)|^2\), connecting the measurable intensity correlation function \(g_2\) to the intermediate scattering function \(g_1\).

sinc

\(\mathrm{sinc}(x) = \sin(\pi x) / (\pi x)\). The shear-induced term in homodyne’s laminar flow model contains \(\mathrm{sinc}^2(\cdot)\), which creates zeros (nulls) at specific lag times when the shear displacement equals multiples of \(1/q\).

speckle

A random, granular intensity pattern produced when coherent radiation (light or X-rays) is scattered by a disordered sample. Speckle patterns change as scatterers move, encoding the dynamics in their temporal fluctuations.

speckle contrast

See: contrast

static mode

Analysis mode for equilibrium samples with pure diffusive dynamics. Uses 3 physical parameters (D₀, α, D_offset). No angular dependence expected.

Stokes-Einstein equation

Relates the diffusion coefficient to particle size: \(D_0 = k_BT / (6\pi\eta R_h)\), where \(k_BT\) is thermal energy, \(\eta\) is viscosity, and \(R_h\) is the hydrodynamic radius.

sub-diffusion

Diffusion with \(\alpha < 1\) in the mean-squared displacement \(\langle r^2 \rangle \propto t^\alpha\). Characteristic of caged motion in dense suspensions, gels, and glasses.

super-diffusion

Diffusion with \(\alpha > 1\). Seen in active particle systems, anomalous transport in heterogeneous media, and driven systems.

Taylor-Couette geometry

See: Couette geometry

two-time correlation function (C2)

The main observable in XPCS: \(C_2(t_1, t_2)\) correlates scattering intensities at two absolute times rather than a single lag. Captures non-stationary dynamics. Shape: (n_phi, n_t1, n_t2).

uv

A fast Python package manager and project tool used to manage homodyne’s virtual environment and dependencies. All project commands use uv run.

wavevector

The scattering vector \(q = (4\pi/\lambda) \sin(\theta/2)\), where \(\lambda\) is the X-ray wavelength and \(\theta\) is the scattering angle. In XPCS, \(q\) selects the length scale probed: large \(q\) → short distances.

XLA

Accelerated Linear Algebra — a domain-specific compiler for linear algebra operations, used by JAX as its execution backend. XLA JIT-compiles and optimizes computational graphs for CPU (and GPU/TPU).

XPCS

X-ray Photon Correlation Spectroscopy. A technique using coherent X-rays to measure the dynamics of materials via temporal correlations of scattered intensities.

YML / YAML

YAML Ain’t Markup Language — a human-readable data serialization format. All homodyne configuration is written in YAML files loaded by ConfigManager.