🔬 condensed matter

Heterogeneous planar diffusion with axisymmetric power-law decaying diffusion coefficient under stochastic resetting

This paper analytically investigates two-dimensional heterogeneous diffusion with a radially decaying coefficient under stochastic resetting, deriving exact probability densities, stationary states, and first-passage properties for various noise interpretations and memory effects, while demonstrating how these factors influence mean squared displacement saturation and the existence of an optimal resetting rate.

Original authors: Trifce Sandev, Sebastien Fumeron, Malte Henkel, Ervin K. Lenzif

Published 2026-09-22

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CC BY 4.0

Original authors: Trifce Sandev, Sebastien Fumeron, Malte Henkel, Ervin K. Lenzif

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 microscopic world of cells and complex materials, particles rarely move in a straight, predictable line. Instead, they drift through crowded, uneven landscapes where the ease of movement changes depending on exactly where the particle is. This phenomenon, known as heterogeneous diffusion, means that a molecule might zip through an open space only to get stuck or slow down significantly when it encounters a dense cluster of other molecules. To make sense of this erratic behavior, scientists often rely on a concept called stochastic resetting. Imagine a search party looking for a lost hiker in a vast forest; if the searchers wander too far without finding anything, they might decide to return to the starting point and begin again. In physics, this "return to start" mechanism is a powerful tool that prevents a particle from wandering off forever, eventually forcing it into a stable, predictable pattern of movement even in a chaotic environment.

A team of researchers has now taken this idea and applied it to a specific, challenging scenario: a particle moving in a flat, two-dimensional plane where the ability to diffuse weakens as the particle moves further from the center. In this model, the further a particle travels from its origin, the harder it becomes to move, with the ease of movement dropping off sharply with distance. The researchers wanted to understand how the combination of this difficult terrain and the frequent "returns to start" would affect the particle's journey. They examined three different mathematical ways to describe how the particle interacts with its environment, a distinction that arises because the rules of motion change depending on how one interprets the random jolts the particle receives. By treating all three interpretations together, they were able to map out exactly where the particle is likely to be found after a long time, how far it travels on average, and how long it takes to reach a specific target.

The study reveals that when a particle is subjected to this constant resetting in a difficult, uneven environment, it does not simply settle into a uniform cloud around the center. Instead, it forms a specific, stable shape that is distinct from what is seen in simpler, uniform environments. This final state is governed by complex mathematical curves that describe how the particle concentrates near the center but still has a chance to be found further out. Crucially, the researchers found that the particle's average distance from the center stops growing after a while. Instead of spreading out infinitely, the particle's wandering is capped, and it settles into a steady average distance that depends on how frequently it is reset. If the reset happens too rarely, the particle wanders too far; if it happens too often, the particle never gets a chance to explore. There is a specific, optimal frequency of resetting that keeps the particle in the most efficient state of exploration.

The team also investigated how long it takes for the particle to find a target located at the center, a scenario relevant to how proteins might search for specific sites within a cell. They discovered that there is a "sweet spot" for the resetting rate that minimizes the time it takes to find the target. If the particle is reset too slowly, it wastes time wandering aimlessly; if it is reset too quickly, it never gets close enough to the target to find it. This optimal rate changes depending on how the particle's movement is mathematically interpreted, but the principle remains the same: a balanced return to the start makes the search most efficient. Furthermore, the researchers looked at what happens if the particle gets "stuck" for periods of time, a phenomenon known as memory, where the particle's past movements influence its future steps. They found that when these memory effects are strong, the optimal resetting rate must increase. In other words, if the particle tends to linger or get trapped, it needs to be sent back to the start more frequently to ensure it eventually finds its target.

One of the most striking findings concerns a threshold time, which measures how long it takes for half of a group of particles to spread beyond a certain distance. In a normal search, resetting helps the particles find a target faster. However, when looking at how far they spread, resetting acts as a trap. The researchers found that if the resetting rate is too high, the particles never manage to spread beyond a certain radius, no matter how long they wait. The system undergoes a transition where the particles become permanently confined near the center, and the time it takes for them to escape effectively becomes infinite. This creates a sharp boundary: below a certain critical rate, the particles can eventually escape; above it, they are forever held back. This behavior is unique to this type of difficult, uneven environment and does not occur in simpler, uniform settings.

Finally, the researchers extended their work to include these memory effects, modeling a situation where the particle's movement is interrupted by long pauses, similar to a traveler who stops to rest for unpredictable amounts of time. They derived a new set of rules describing how the particle behaves under these conditions. The results showed that the presence of memory changes the landscape of the search. As the memory effects become stronger—meaning the particle waits longer between moves—the optimal strategy for finding a target shifts. The system requires a higher frequency of resetting to overcome the delays caused by the memory. The study provides a complete mathematical description of these behaviors, offering a clear picture of how randomness, difficult terrain, and memory interact to shape the movement of particles in complex systems. This work not only deepens the theoretical understanding of diffusion but also offers a framework for analyzing real-world processes where movement is hindered by the environment and interrupted by external forces.

Technical Summary: Heterogeneous Planar Diffusion with Axisymmetric Power-Law Decaying Diffusion Coefficient under Stochastic Resetting

Problem Statement
The paper investigates two-dimensional isotropic heterogeneous diffusion characterized by a radially decaying diffusion coefficient, specifically $D(\rho) = D_0/\rho^2$ . This model is relevant for systems such as intracellular protein target searches where macromolecular crowding induces radially dependent diffusivity. A critical feature of the study is the treatment of multiplicative noise, which renders the stochastic process interpretation-dependent. The authors analyze the system under three distinct interpretations of the Langevin equation: Itô ( $\alpha=0$ ), Stratonovich ( $\alpha=1/2$ ), and Hänggi–Klimontovich ( $\alpha=1$ ). Furthermore, the study incorporates Poissonian stochastic resetting, where the particle is returned to the origin at a constant rate $\bar{r}$ , a mechanism used to model homing or refuge-seeking behaviors. The work extends to include memory effects via a subordination (continuous-time random walk) approach with a power-law kernel.

Methodology
The authors employ a unified analytical framework to derive exact solutions for the probability density function (PDF), mean squared displacement (MSD), and first-passage properties across the three noise interpretations.

Renewal Theory: The core methodology utilizes the renewal equation to relate the PDF with resetting, $P_{\bar{r}}$ , to the PDF without resetting, $P$ . This is solved in Laplace space, where the relationship is given by $\tilde{P}_{\bar{r}}(\rho, s) = \frac{s+\bar{r}}{s} \tilde{P}(\rho, s+\bar{r})$ .
Fokker-Planck Formalism: The authors derive the corresponding Fokker-Planck equations for the radial coordinate under each interpretation, incorporating the resetting term as a loss of probability at $\rho$ and a gain at the origin.
Subordination Approach: To model memory effects, the system is generalized using coupled Langevin equations involving an operational time $u$ and an inverse $\alpha$ -stable subordinator. This leads to a fractional Fokker-Planck equation involving a memory kernel $\eta(t)$ . The solution is obtained via a subordination integral, linking the memory-dependent process to the standard heterogeneous diffusion solution.
First-Passage Analysis: The mean first-passage time (MFPT) to an absorbing boundary at the origin is calculated using the survival probability. Additionally, a "threshold time" $t_{1/2}$ is defined as the time required for half the population to spread beyond a radius $R$ , analyzing the confinement transition induced by resetting.

Key Contributions and Results

Non-Equilibrium Stationary State (NESS):
The study derives exact expressions for the NESS induced by resetting. Unlike the Laplace distribution found in homogeneous diffusion under resetting, the NESS for this heterogeneous system is governed by modified Bessel functions of the second kind, $K_\nu(z)$ . The specific form depends on the noise interpretation ( $\alpha$ ). For instance, in the Itô case ( $\alpha=0$ ), the stationary state involves $K_0$ , while the Hänggi–Klimontovich case ( $\alpha=1$ ) yields an exponential form.
Mean Squared Displacement (MSD):
The MSD, $\langle \rho^2(t) \rangle$ , is shown to saturate at long times due to the resetting mechanism. In the long-time limit, the MSD scales as $\langle \rho^2 \rangle \sim \bar{r}^{-1/2}$ . In the presence of memory (power-law kernel with exponent $\mu$ ), the long-time saturation scales as $\langle \rho^2 \rangle \sim \bar{r}^{-\mu/2}$ . The time-dependent MSD in the memory case is expressed using the three-parameter Mittag-Leffler function.
First-Passage Properties and Optimal Resetting:
The MFPT to an absorbing boundary exhibits a non-monotonic dependence on the resetting rate $\bar{r}$ . There exists an optimal resetting rate that minimizes the MFPT. The authors provide explicit analytical formulas for the MFPT for all three interpretations, noting that the optimal resetting rate can be determined by solving the condition $\partial/\partial\bar{r}\langle T_{\bar{r}} \rangle = 0$ . Graphical results confirm the existence of this minimum for each interpretation.
- Memory Effects on Optimization: When memory is introduced, the optimal resetting rate increases as the memory exponent $\mu$ decreases. This is attributed to the increased waiting times of the particle, necessitating more frequent resets to prevent trapping.
Threshold Time and Confinement Transition:
A distinct phenomenon is observed regarding the threshold time $t_{1/2}$ (time for 50% of the population to exceed radius $R$ ). Unlike the MFPT, which is minimized at a finite rate, $t_{1/2}$ increases monotonically with $\bar{r}$ and diverges at a critical rate $\bar{r}^*$ . Above $\bar{r}^*$ , the stationary state retains more than half the population within $R$ , meaning the threshold time ceases to exist. This "confinement transition" is unique to heterogeneous diffusion under resetting and has no counterpart in homogeneous diffusion. The critical rate $\bar{r}^*$ varies by interpretation, with the Hänggi–Klimontovich prescription requiring the weakest resetting to induce confinement.

Significance and Claims
The paper claims to provide the first detailed analytical investigation of two-dimensional heterogeneous diffusion with a $1/\rho^2$ decay under stochastic resetting, explicitly addressing the ambiguity of multiplicative noise interpretations. The authors highlight that the interplay between spatial heterogeneity, noise interpretation, and resetting leads to stationary states and first-passage behaviors that fundamentally differ from standard homogeneous diffusion. Specifically, the identification of a confinement transition for the threshold time, distinct from the optimization of the MFPT, is presented as a novel finding. The extension to memory effects via subordination further elucidates how environmental history (waiting times) modulates the efficiency of resetting strategies. The work is positioned as a theoretical foundation for understanding diffusion in complex, disordered, and crowded two-dimensional media.

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