Stochastically-Forced Diffusion Model

• The stochastically-forced diffusion model is a non-equilibrium framework characterized by reciprocal, random pair interactions in a lattice gas setting.
• Its microscopic construction involves single-particle hops and pairwise attractive moves that break detailed balance while conserving center-of-mass.
• The continuum theory predicts a q-dependent effective temperature that can reverse typical diffusion slowdowns, leading to enhanced tracer mobility.

Searching arXiv for the primary paper and closely related work on stochastic diffusion and stochastic forcing. Searching for (Alston et al., 26 Aug 2025). In contemporary arXiv usage, a stochastically-forced diffusion model can denote several classes of non-equilibrium diffusion theories; in the specific sense developed for non-motile active matter, it is a lattice-gas and continuum field theory in which reciprocal but randomly fluctuating pairwise interactions generate active suspensions and enhance the diffusion of a weakly coupled tracer, even in the absence of self-propulsion or non-reciprocity (Alston et al., 26 Aug 2025). The central result is that stochastic pair forces minimally break detailed balance, create a $q$-dependent effective temperature that rises at short wavelength, and thereby open a distinct route to $D_{\rm eff}>D_y$ in dense suspensions.

1. Microscopic construction

The model is formulated on a one-dimensional periodic lattice $\mathcal L$ of spacing $a$. At each site $i\in\mathcal L$, the occupation number is $n_i\in\{0,1,\dots,\kappa\}$, where $\kappa$ is the site carrying capacity. Two classes of moves are allowed. The first is a single-particle hop, i.e. a partial-exclusion process, in which a particle at $i$ hops to $i\pm1$ with rate

$$\alpha_{i,\pm1}(n)=D_r\,n_i\Bigl(1-\frac{n_{i\pm1}}{\kappa}\Bigr),$$

with $D_{\rm eff}>D_y$0 setting the microscopic diffusive timescale. The second is a pairwise “attractive” hop in which two particles on $D_{\rm eff}>D_y$1 simultaneously hop to $D_{\rm eff}>D_y$2 with rate

$D_{\rm eff}>D_y$3

The second rule is the source of activity. It breaks detailed balance, drives an effective attraction, and ultimately phase separation, yet preserves reciprocity and center-of-mass. This combination is the defining feature of the construction: the system is non-equilibrium not because of self-propelled particles or non-reciprocal forces, but because the pairwise dynamics inject stochasticity while retaining reciprocal interactions (Alston et al., 26 Aug 2025).

2. Coarse-grained dynamical theory

The coarse-grained description starts from the master equation, introduces the Martin–Siggia–Rose action, expands $D_{\rm eff}>D_y$4, and keeps only leading gradient terms. In the continuum limit, fixing $D_{\rm eff}>D_y$5 but restricting to long wavelengths, the density field $D_{\rm eff}>D_y$6 obeys

$D_{\rm eff}>D_y$7

Linearization about a uniform average $D_{\rm eff}>D_y$8 proceeds by setting $D_{\rm eff}>D_y$9. The resulting Langevin equation is

$\mathcal L$0

with coefficients

$\mathcal L$1

$\mathcal L$2

$\mathcal L$3

$\mathcal L$4

The noises $\mathcal L$5 and $\mathcal L$6 are independent, zero-mean, unit-variance white noises. The two-noise structure is essential: the $\mathcal L$7 term is equilibrium-like, whereas the $\mathcal L$8 term encodes the non-equilibrium stochastic pair forcing (Alston et al., 26 Aug 2025).

3. Effective temperature and short-wavelength amplification

In Fourier space, with $\mathcal L$9, the two noise sources combine into an effective-temperature spectrum

$a$0

with $a$1. Equivalently,

$a$2

Because of the $a$3 term, the effective temperature grows at short wavelength. In the model’s interpretation, this growth reflects the nonequilibrium pairwise noise $a$4. The effective temperature is therefore not a flat equilibrium control parameter; it is a mode-dependent spectrum. This distinction is central to the ensuing tracer dynamics, because the enhanced short-scale agitation can outweigh the usual fluctuation-induced slowdown and reverse the sign of the correction to diffusion (Alston et al., 26 Aug 2025).

4. Tracer coupling and renormalized diffusion

A tracer at position $a$5, with bare mobility $a$6, is weakly coupled to the fluctuating density field through

$a$7

Integrating out $a$8 perturbatively by a Dean–Demery path-integral gives, at zero external force, the self-diffusion coefficient

$a$9

where $i\in\mathcal L$0 is the tracer’s bare diffusion, $i\in\mathcal L$1 is the field’s deterministic relaxation rate, and $i\in\mathcal L$2 is the coupling form-factor, taken here as $i\in\mathcal L$3. In one dimension, with $i\in\mathcal L$4 and $i\in\mathcal L$5, this reduces to Eq. (30) in the supplementary material.

The equilibrium comparison is explicit. In equilibrium Model B, where $i\in\mathcal L$6 and $i\in\mathcal L$7, the fluctuation-dissipation theorem holds and the correction is negative, so $i\in\mathcal L$8. The same occurs when the stochastic-force amplitude vanishes, $i\in\mathcal L$9: the $n_i\in\{0,1,\dots,\kappa\}$0 growth in $n_i\in\{0,1,\dots,\kappa\}$1 disappears, equilibrium-like noise is recovered, and fluctuations slow the tracer. In the dense-suspension limit $n_i\in\{0,1,\dots,\kappa\}$2 (or $n_i\in\{0,1,\dots,\kappa\}$3), both $n_i\in\{0,1,\dots,\kappa\}$4 and $n_i\in\{0,1,\dots,\kappa\}$5; the field becomes quasi-frozen, $n_i\in\{0,1,\dots,\kappa\}$6, and $n_i\in\{0,1,\dots,\kappa\}$7. By contrast, intermediate densities with $n_i\in\{0,1,\dots,\kappa\}$8 generate high-$n_i\in\{0,1,\dots,\kappa\}$9 noise and can reverse the sign of the integral, yielding $\kappa$0. This is the article’s precise sense of enhanced diffusion (Alston et al., 26 Aug 2025).

5. Relation to other stochastic diffusion frameworks

The phrase “stochastically-forced diffusion model” is not unique to non-motile active matter. In the literature represented here, it spans several distinct constructions. Donev et al. derive a stochastic advection-diffusion equation for tracer concentration from fluctuating Stokes dynamics in the large-Schmidt-number limit, with a divergence-free white-in-time random velocity $\kappa$1, and show that the ensemble mean obeys Fick’s law with $\kappa$2 while individual realizations exhibit giant fluctuations (Donev et al., 2013). The “Multinomial Diffusion Equation” is instead a microscopic mass-conserving discrete-particle model on a lattice whose continuum limit reproduces the classical stochastic diffusion PDE, but which remains faithful at low particle density where the classical SDE fails (Balter et al., 2010).

A different family places the stochasticity in the diffusivity itself rather than in pairwise forces or fluctuating fluxes. The generalized grey Brownian motion and diffusing-diffusivity models both yield strictly linear growth of the MSD while allowing non-Gaussian displacement statistics; in the DD model, the propagator is non-Gaussian at short times and crosses over to Gaussian at long times (Sposini et al., 2018). A bounded diffusing-diffusivity model driven by symmetric dichotomous noise likewise produces a short-time PDF with a logarithmic divergence at the origin, Gaussian tails modulated by a power law, and ordinary Gaussian diffusion at long times (Lee et al., 13 Apr 2026). These comparisons clarify that the non-motile active-matter construction is not a random-diffusivity model in disguise; its stochasticity is carried by reciprocal pair interactions and their induced density-field noise.

6. Significance, misconceptions, and scope

A common misconception is that enhanced diffusion in active suspensions requires self-propulsion, non-reciprocity, or explicit motility. The non-motile active-matter model excludes that identification: reciprocal but randomly fluctuating interactions already suffice to generate active suspensions and enhance tracer diffusion. Another frequent misconception is to treat $\kappa$3 as a thermodynamic temperature. Here it is explicitly $\kappa$4-dependent, and the short-wavelength increase is the diagnostic of the nonequilibrium pairwise forcing rather than an equilibrium fluctuation-dissipation parameter (Alston et al., 26 Aug 2025).

The scope of the construction is equally specific. It is derived from a one-dimensional periodic lattice gas, coarse-grained in a long-wavelength limit, linearized about a uniform average density, and evaluated for a weakly coupled tracer. Within that regime, the model yields a two-noise continuum field theory, a short-scale elevated effective temperature, and a renormalized tracer diffusivity whose correction changes sign when $\kappa$5 is large enough. The authors summarize the outcome as a generic route to enhanced diffusion via purely stochastic, center-of-mass-conserving forces. A plausible implication is that stochastic pair forcing should be regarded as an independent mechanism of transport renormalization in dense non-equilibrium suspensions, rather than as a secondary correction to more familiar motility-based activity.