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Fractional kinetics equation from a Markovian system of interacting Bouchaud trap models

This paper demonstrates that a Markovian partial exclusion process on a random trapping environment converges to a fractional kinetics equation in dimensions $d \ge 2$ , while in one dimension ( $d=1$ ), it converges to a singular quasi-diffusion known as FIN diffusion.

Original authors: Alberto Chiarini, Simone Floreani, Federico Sau

Published 2026-09-15

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CC BY 4.0

Original authors: Alberto Chiarini, Simone Floreani, Federico Sau

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of physics, scientists often try to understand how the chaotic, jittery motion of individual atoms or molecules gives rise to the smooth, predictable flow of fluids or heat that we see in the everyday world. This process of connecting the tiny, random steps of a single particle to the large-scale behavior of a crowd is known as a hydrodynamic limit. Usually, when particles move randomly through a uniform space, their collective behavior follows a standard rule: they spread out at a steady, predictable rate, much like a drop of ink dispersing in a glass of still water. This is called diffusion, and it is the foundation for understanding how heat travels or how gases mix. However, nature is rarely perfectly uniform. In many real-world materials, such as porous rocks or complex biological tissues, the environment is full of obstacles and irregularities that can trap particles, causing them to get stuck for long periods before moving again. When this happens, the spreading slows down dramatically, a phenomenon known as sub-diffusion. For decades, physicists have struggled to find a simple, microscopic model that could explain why these systems sometimes behave in a way that defies standard diffusion, leading to equations that involve fractional time steps rather than ordinary ones.

A team of researchers has now constructed a specific model of interacting particles that successfully bridges this gap, showing how a system of simple, memoryless particles can collectively produce a complex, memory-like behavior on a large scale. The scientists studied a system of particles moving on a grid, where each location on the grid acts as a "trap" with a specific capacity. These traps are not uniform; some are small and hold only a few particles, while others are massive and can hold many. The size of these traps is determined by a random distribution where extremely large traps are rare but possible. The particles themselves follow a simple rule: they try to hop to neighboring sites, but they are blocked if the destination is already full. This creates a crowded, interactive environment where particles must wait for space to open up. The researchers then watched how this system evolved over time, scaling up the view to see the big picture.

What they discovered depends entirely on the dimension of the space the particles occupy. In two or three dimensions, the system behaves in a surprising way. Even though every single particle moves according to simple, standard rules with no memory of its past, the collective density of the particles spreads out according to a strange, new equation. This equation, known as a fractional kinetics equation, describes a process where the rate of change depends on the entire history of the system, not just the current moment. It is as if the crowd of particles, acting together, develops a kind of collective memory that slows their spread in a specific, mathematically precise way. The randomness of the individual traps and the interactions between particles effectively wash out, leaving behind a smooth, deterministic pattern that follows this fractional rule. This finding is significant because it proves that a system made entirely of simple, Markovian components—where the future depends only on the present—can generate a macroscopic behavior that looks like a complex, non-Markovian process.

The story changes completely when the researchers looked at the same system in just one dimension, like particles moving along a single line. Here, the behavior is even more erratic. Instead of smoothing out into a predictable pattern, the large-scale density of the particles remains frozen in a random, jagged landscape determined by the specific arrangement of the traps. The particles do not spread out evenly; instead, they cluster in huge piles at the locations of the largest traps, and these piles barely move over time. The resulting equation describing this motion is a random diffusion process, where the path of the particles is dictated by a random, irregular measure rather than a smooth, uniform space. In this one-dimensional case, the randomness of the environment does not disappear; it becomes the very fabric of the macroscopic world, dictating how the particles can move.

The researchers achieved these results by using a clever mathematical trick called duality, which allowed them to study the complex crowd by looking at the behavior of just a few particles. This method bypassed the need for the heavy, traditional tools usually required to prove such limits, offering a direct line of sight from the microscopic rules to the macroscopic outcome. By carefully adjusting the time scale at which they observed the system, they were able to isolate the specific moments where the trapping effect becomes dominant. Their work provides the first rigorous example of an interacting particle system that naturally rescales to a time-fractional equation in higher dimensions, and a random quasi-diffusion in one dimension. This confirms that the strange, slow spreading seen in many physical systems is not just a mathematical curiosity, but a natural consequence of particles interacting in a rugged, random landscape. The study does not suggest that these findings apply to every material, nor does it claim to solve all problems of sub-diffusion, but it firmly establishes a concrete, microscopic mechanism that generates these complex behaviors from simple, local rules.

Technical Summary: Fractional Kinetics Equation from a Markovian System of Interacting Bouchaud Trap Models

Problem Statement
The paper addresses the derivation of macroscopic hydrodynamic limits (HL) for interacting particle systems evolving in random environments. While the HL for the Simple Symmetric Exclusion Process (SSEP) in random conductances is well-established (yielding the heat equation or quasi-diffusions), there is a lack of rigorous microscopic models that rescale to time-fractional sub-diffusive equations (specifically, the Fractional Kinetics Equation, FKE). The authors aim to construct a Markovian interacting particle system in a heavy-tailed random trapping environment that, upon appropriate space-time rescaling, converges to a deterministic measure satisfying a fractional PDE in dimensions $d \ge 2$ , and a random quasi-diffusion in $d=1$ .

Model Definition
The authors introduce a system of Interacting Bouchaud Trap Models (BTM) on the lattice $\mathbb{Z}^d$ .

Environment: The environment is defined by a collection of i.i.d. random variables $\alpha = (\alpha_x)_{x \in \mathbb{Z}^d}$ with heavy-tailed distributions satisfying $Q(\alpha_0 > u) \sim u^{-\beta}$ for $\beta \in (0, 1)$ .
Particle Dynamics: The system consists of particles evolving on $\mathbb{Z}^d$ $Z^{d}$ subject to a partial exclusion rule. At each site $x$ $x$ , the maximum occupancy is $\alpha_x$ $α_{x}$ . The jump rate from $x$ $x$ to a neighbor $y$ $y$ is proportional to $\alpha_x^{a-1} \alpha_y^a (1 - \eta(y)/\alpha_y)$ $α_{x}^{a - 1} α_{y}^{a} (1 - η (y) / α_{y})$ , where $\eta(y)$ $η (y)$ is the number of particles at $y$ $y$ and $a \in [0, 1]$ $a \in [0, 1]$ is a parameter controlling asymmetry.
- The term $(1 - \eta(y)/\alpha_y)$ enforces the exclusion rule: jumps to occupied sites are suppressed, and the "capacity" of a site is determined by the trap size $\alpha_x$ .
- When $a=0$ , the model is symmetric; when $a \in (0, 1]$ , it is asymmetric.
Key Feature: Unlike independent BTMs, these particles interact via the exclusion rule, yet the system retains stochastic self-duality, a property crucial for the analysis.

Methodology
The proof of the hydrodynamic limit relies on the stochastic self-duality of the interacting system, avoiding the traditional martingale problem approach combined with one- and two-block estimates, which the authors note are difficult to extend to time-fractional limits.

Duality Relations: The authors utilize a duality relation between the interacting particle system and a system of independent BTMs (and pairs of interacting BTMs). This allows the expectation of the empirical field to be reduced to the study of single-particle scaling limits.
Empirical Fields: Two distinct empirical fields are analyzed:
- The Empirical Density Field ( $X^n_t$ ): Scales with $n^{-d/\beta}$ .
- The Empirical Frequency Field ( $Z^n_t$ ): Scales with $n^{-d}$ and weights particles by the local capacity $\alpha_x$ (i.e., $\eta(x)/\alpha_x$ ).
Scaling Limits of Single Particles: The analysis leverages known scaling limits of the single-particle BTM:
- For $d \ge 2$ , the rescaled BTM converges to the Fractional Kinetics process (a time-changed Brownian motion).
- For $d = 1$ , it converges to the FIN diffusion (a time-changed Brownian motion with a singular speed measure).
Decomposition: The proof decomposes the empirical field into four terms: dynamic noise, initial condition variance, initial condition mean, and the deterministic limit term. The authors show that the noise and variance terms vanish, while the mean term converges to the solution of the target PDE.

Key Results

The paper establishes the hydrodynamic limit for the empirical frequency field $Z^n_t$ and the density field $X^n_t$ under specific time rescalings $\theta_n$ :

Case $d \ge 2$ :
- Time Rescaling: $\theta_n \sim n^{2/\beta}$ (with a logarithmic correction for $d=2$ ).
- Frequency Field ( $Z^n_t$ ): Converges to a deterministic measure with density $\rho_t$ solving the Fractional Kinetics Equation:
  $\frac{\partial^\beta \rho_t}{\partial t^\beta} = D_{\text{eff}} \Delta \rho_t$
  where $\partial^\beta/\partial t^\beta$ is the Caputo fractional derivative. The randomness of the environment homogenizes, resulting in a deterministic macroscopic limit.
- Density Field ( $X^n_t$ ): Converges to a random measure $\rho_0 W$ , where $W$ is a random measure supported on a Poisson point process. The macroscopic dynamics are "frozen" in the sense that the randomness of the environment persists in the limit.
Case $d = 1$ :
- Time Rescaling: $\theta_n \sim n^{1+1/\beta}$ .
- Frequency Field ( $Z^n_t$ ): Converges to a random measure with density $\rho_t$ solving a random quasi-diffusion equation:
  $\frac{\partial \rho_t}{\partial t} = \frac{\partial}{\partial W} \frac{\partial \rho_t}{\partial x}$
  Here, the derivative is interpreted in the sense of the random measure $W$ (the FIN diffusion speed measure).
- Density Field ( $X^n_t$ ): Also converges to a random measure $\rho_t W$ .

Significance and Claims
The authors claim this work provides the first example of an interacting particle system (specifically, interacting BTMs) that rescales to a time-fractional sub-diffusive equation.

Novelty in Methodology: The paper highlights that previous works on trap models relied on martingale problems, which struggle with non-Markovian (fractional) limits. By exploiting stochastic duality, the authors establish a direct connection between the pre-limit interacting system and the hydrodynamic limit, successfully handling time-fractional setups.
Homogenization vs. Randomness: A significant finding is the distinction between the two empirical fields. For $d \ge 2$ , the frequency field exhibits homogenization (randomness vanishes, leading to a deterministic fractional PDE), whereas the density field retains the environmental randomness. This contrasts with the $d=1$ case, where both fields remain random.
Microscopic Origin of Fractional Kinetics: The work demonstrates that a purely Markovian microscopic system of interacting particles can generate a macroscopic non-Markovian process (the Fractional Kinetics process) through the mechanism of heavy-tailed trapping times and partial exclusion.

The paper concludes that the derived fractional kinetics equation is a rigorous hydrodynamic limit of a specific microscopic interacting dynamics, bridging the gap between microscopic trap models and macroscopic fractional PDEs.

Technical Summary: Fractional Kinetics Equation from a Markovian System of Interacting Bouchaud Trap Models

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