Transport Coefficient J(t)¶

The transport coefficient \(J(t)\) is the central quantity in the homodyne framework. It encodes how the variance of particle displacements grows in time and provides a direct bridge between microscopic stochastic dynamics and macroscopic rheological observables through a generalized Green-Kubo relation.

Definition¶

The transport coefficient admits three equivalent representations.

Variance form (definition):

(1)¶\[J(t) \;=\; \frac{d}{dt}\,\mathrm{Var}\!\left[x(t)\right]\]

This is the instantaneous rate of growth of the mean-squared displacement (MSD) of a particle at time \(t\) (He et al. PNAS 2025, Eq. S-38).

Covariance form:

(2)¶\[J(t) \;=\; 2\,\mathrm{Cov}\!\left[x(t),\, v(t)\right]\]

where \(v(t) = \dot{x}(t)\) is the instantaneous particle velocity. The factor of 2 arises from the symmetry \(\mathrm{Cov}[x, v] = \frac{1}{2} \frac{d}{dt}\mathrm{Var}[x]\).

Green-Kubo form (microscopic origin):

(3)¶\[J(t) \;=\; 2 \int_0^{t} \mathrm{Cov}\!\left[v(t),\, v(t')\right] dt'\]

Equation (3) is the generalized Green-Kubo relation: the transport coefficient equals twice the integral of the velocity autocorrelation function (VACF) from \(0\) to \(t\). For a stationary process this reduces to the classical \(J = 2\int_0^\infty C_v(\tau)\,d\tau = 2D\).

Physical Interpretation¶

The transport coefficient \(J(t)\) has units of \(\text{length}^2 / \text{time}\), identical to a diffusion coefficient. It measures how rapidly positional uncertainty accumulates at time \(t\).

Note

In equilibrium, \(J\) is constant: \(J = 2D\), where \(D\) is the Stokes-Einstein diffusion coefficient. The homodyne framework generalizes this to non-stationary processes where \(J(t)\) can vary with time (e.g., aging, yielding).

The diffusion integral that enters the correlation function is:

(4)¶\[\mathcal{D}(t_1, t_2) \;=\; \int_{t_1}^{t_2} J(t') \, dt' \;=\; \mathrm{Var}\!\left[x(t_2) - x(t_1)\right]\]

This equals the variance of the net displacement, confirming that \(J\) directly controls how the Debye-Waller factor decays in the correlation function.

Homodyne Implementation¶

The homodyne package parameterizes a time-dependent diffusion rate as a power law with offset:

(5)¶\[D(t) \;=\; D_0 \cdot t^\alpha + D_\mathrm{offset}\]

where \(D_0 > 0\) is the diffusion prefactor, \(\alpha\) is the anomalous exponent (\(\alpha = 0\) for normal diffusion, \(\alpha < 0\) for sub-diffusion, \(\alpha > 0\) for super-diffusion), and \(D_\mathrm{offset} \geq 0\) is a constant background diffusion.

The diffusion integral entering the correlation function (5) is computed by cumulative trapezoidal integration of \(D(t)\) on the experimental time grid:

(6)¶\[\mathcal{D}(t_1, t_2) \;=\; \int_{t_1}^{t_2} D(t') \, dt'\]

The numerical integration is implemented in homodyne.core.physics_utils (shared trapezoidal kernels) and used by both homodyne.core.jax_backend (NLSQ meshgrid mode) and homodyne.core.physics_cmc (CMC element-wise mode).

Warning

Convention: The \(D_0\) parameter in homodyne absorbs a factor of 2 from the formal transport coefficient. For standard Brownian motion, the physical Stokes-Einstein diffusion coefficient is \(D_\mathrm{SE} = k_B T / (6\pi\eta R_h)\), while homodyne’s \(D_0 = 2 D_\mathrm{SE}\). This is because the Siegert relation (8) squares \(c_1\), and the correlation function depends on \(|c_1|^2 = \exp(-q^2 \mathcal{D})\) where \(\mathcal{D} = \int D(t')\,dt'\). For the standard result \(|c_1|^2 = e^{-2 q^2 D_\mathrm{SE}\,\tau}\) to hold, we need \(D_0 = 2 D_\mathrm{SE}\).

Connection to Physical Diffusion Coefficient¶

In homodyne’s parameterization (where \(D_0\) absorbs the factor of 2; see warning above), the mean-squared displacement is:

(7)¶\[\mathrm{MSD}(t) \;\equiv\; \mathrm{Var}[x(t) - x(0)] \;=\; \int_0^t D(t')\,dt'\]

The effective physical (Stokes-Einstein) diffusion coefficient at time \(t\) is half the homodyne value:

\[D_\mathrm{SE}(t) \;=\; \frac{\mathrm{MSD}(t)}{2t}\]

For \(\alpha = 0\) (standard Brownian motion): \(D_\mathrm{SE}\) is a constant. For \(\alpha < 0\) (sub-diffusion with aging): particles slow down over time.

Table of J(t) for Classical Processes¶

The following table summarizes \(J(t)\) for Langevin processes that arise naturally in soft matter. See Classical Langevin Processes for full derivations.

Transport coefficients for classical stochastic processes¶
Process	\(J(t)\)	Regime
Wiener (free diffusion)	\(2D\)	\(D\) = constant
Anomalous diffusion	\(2D_0 t^{\alpha}\)	\(\alpha \in (-1, 1]\)
Ornstein-Uhlenbeck	\(2D\!\left(1 - e^{-\gamma t}\right)^2\)	Confinement radius \(\sqrt{D/\gamma}\)
Brownian oscillator (underdamped)	\(\displaystyle 2D\frac{\gamma^2}{\omega_s^2} e^{-\gamma t}\sin^2(\omega_s t)\)	\(\omega_s^2 = \omega_0^2 - \gamma^2/4 > 0\)
Brownian oscillator (overdamped)	\(\displaystyle 8D\frac{\gamma^2}{\gamma_s^2} e^{-\gamma t}\sinh^2\!\left(\tfrac{1}{2}\gamma_s t\right)\)	\(\gamma_s^2 = \gamma^2 - 4\omega_0^2 > 0\)

where \(\gamma\) is the friction coefficient, \(\omega_0\) is the trap frequency, \(\omega_s = \sqrt{\omega_0^2 - \gamma^2/4}\) and \(\gamma_s = \sqrt{\gamma^2 - 4\omega_0^2}\). Here \(D\) is the physical Stokes-Einstein diffusion coefficient; in homodyne’s parameterization \(D_0 = 2D\) (see (5)).

Relationship to Rheology¶

The Green-Kubo form (3) links \(J(t)\) to the complex shear modulus \(G^*(\omega)\) of the suspending medium through the generalized Stokes-Einstein relation (GSER):

\[D(\omega) \;=\; \frac{k_B T}{6\pi R \eta(\omega)} \;=\; \frac{k_B T}{6\pi R}\,\frac{1}{G^*(\omega)}\]

where \(R\) is the particle radius, \(k_B T\) is the thermal energy, and \(\eta(\omega)\) is the frequency-dependent viscosity. Measuring \(J(t)\) from XPCS data therefore yields a non-invasive probe of local viscoelastic properties.

For the yielding transition studied in He et al. PNAS 2025, the time-evolution of \(J(t)\) during the rheological loading protocol distinguishes:

Repulsive suspensions: Andrade creep (\(\gamma \sim t^{1/3}\)) maps to \(J(t) \propto t^{-2/3}\) — a sub-diffusive, aging transport coefficient.
Attractive suspensions: heterogeneous shear banding produces non-Gaussian displacement distributions not captured by a single \(J(t)\).

See Yielding Dynamics of Colloidal Suspensions for details.