Classical Stochastic Processes

The generalized transport coefficient framework subsumes several classical stochastic models as special cases. This section catalogs these limits, following PNAS 2024 SI Section 4 (Eqs. S-108 through S-127), and shows how each maps onto the power-law parameterization used by the heterodyne package.

Wiener Process (Standard Diffusion)

The Wiener process describes free Brownian motion with a constant diffusion coefficient \(D\). The transport coefficient is:

\[J(t) \;=\; 2D\]

and the internal field correlation is:

\[c_{1,\mathrm{in}}(q, t_1, t_2) \;=\; \exp\!\left(-\frac{q^2}{2} \int_{t_1}^{t_2} J(t')\, dt'\right)\]

This is the simplest case: the position variance grows linearly in time, \(\mathrm{Var}\!\left[x(t)\right] = \int_0^t J(t')\, dt'\), and the correlation decays monotonically with the integrated transport.

Power-law mapping: \(\alpha = 0\), \(D_0 = 2D\), \(D_\mathrm{offset} = 0\).

Ornstein-Uhlenbeck Process

The Ornstein-Uhlenbeck (OU) process models diffusion in a harmonic potential with restoring rate \(\gamma\):

\[J(t) \;=\; 2D\left(1 - e^{-\gamma t}\right)^2\]

At short times (\(t \ll 1/\gamma\)), the particle is ballistic and \(J(t) \approx 2D\gamma^2 t^2\). At long times (\(t \gg 1/\gamma\)), the velocity decorrelates and \(J(t) \to 2D\), recovering Fickian diffusion.

The OU process is not exactly representable by the power-law parameterization, but for time windows much longer than \(1/\gamma\) it is well approximated by \(\alpha = 0\).

Brownian Oscillator

The Brownian oscillator describes a particle in a harmonic potential with natural frequency \(\omega_0\) and damping rate \(\gamma\). It exhibits two regimes:

Overdamped (\(\gamma > 2\omega_0\)):

\[J(t) = 2D \left(1 - e^{-\gamma t/2} \left[\cosh(\Omega t) + \frac{\gamma}{2\Omega}\sinh(\Omega t)\right]\right)^2\]

where \(\Omega = \sqrt{(\gamma/2)^2 - \omega_0^2}\).

Underdamped (\(\gamma < 2\omega_0\)):

\[J(t) = 2D \left(1 - e^{-\gamma t/2} \left[\cos(\omega t) + \frac{\gamma}{2\omega}\sin(\omega t)\right]\right)^2\]

where \(\omega = \sqrt{\omega_0^2 - (\gamma/2)^2}\).

In both cases, \(J(t) \to 2D\) at long times. The underdamped case exhibits oscillatory transients in \(J(t)\) that can produce non-monotonic correlation decay at short lag times.

Advection-Diffusion

The advection-diffusion model combines Brownian diffusion with a constant drift velocity \(v_0\). Using the internal/external factorization:

\[c_1(q, t_1, t_2) \;=\; \exp\!\left(-\frac{q^2}{2} \int_{t_1}^{t_2} J(t')\, dt'\right)\; \exp\!\left(i\, q \int_{t_1}^{t_2} v(t')\, dt'\right)\]

The first factor is the diffusive decay (transport integral); the second is the advective phase shift (velocity integral). In a homodyne measurement, the phase vanishes under the modulus squared:

\[g_2(q, \tau) \;=\; 1 + \beta\, \exp\!\left(-q^2 \int_0^{\tau} J(t')\, dt'\right)\]

so the velocity is invisible. In a heterodyne measurement, the velocity produces observable oscillations in the cross-correlation (see Heterodyne Scattering Model).

Power-law mapping: \(\alpha = 0\), \(D_0 = 2D\), \(D_\mathrm{offset} = 0\), with velocity \(v(t) = v_0\) (i.e., \(\beta_v = 0\), \(v_\mathrm{offset} = v_0\)).

Power-Law Transport

The general power-law parameterization used in the heterodyne package (PNAS 2024 SI Eq. S-105):

\[J(t) \;=\; D_0\, t^\alpha + D_\mathrm{offset}\]

This phenomenological form captures the leading-order behavior of a wide class of transport processes:

Parameters	Model	Physical scenario
\(\alpha = 0,\; D_\mathrm{offset} = 0\)	Wiener	Free diffusion
\(\alpha < 0\)	Subdiffusive	Viscoelastic confinement, crowded environments
\(0 < \alpha < 1\)	Weakly superdiffusive	Persistent random walks
\(\alpha = 1\)	Ballistic	Uniform acceleration
\(D_\mathrm{offset} \neq 0\)	Mixed	Baseline diffusion with time-dependent correction

The offset \(D_\mathrm{offset}\) allows the model to accommodate processes that have a finite transport rate at \(t = 0\) (e.g., a Wiener component superimposed on a power-law anomalous process).

Summary of Limits

Process	\(J(t)\)	Short-time	Long-time
Wiener	\(2D\)	\(2D\)	\(2D\)
OU	\(2D(1-e^{-\gamma t})^2\)	\(\sim t^2\)	\(2D\)
Oscillator	(see above)	\(\sim t^2\)	\(2D\)
Advection-diffusion	\(2D\)	\(2D\)	\(2D\)
Power-law	\(D_0 t^\alpha + D_\mathrm{offset}\)	\(D_\mathrm{offset}\)	\(\sim t^\alpha\)