🔬 condensed matter

Non-equilibrium dynamics of drift-diffusion process under threshold resetting

This paper investigates the emergence of non-equilibrium steady states in stochastic processes under threshold resetting, deriving a general condition linking the steady-state distribution to the ratio of mean local time and mean first-passage time, and demonstrating through a one-dimensional drift-diffusion model that this protocol induces rich intermediate-time dynamics such as anomalous relaxation and damped oscillations.

Original authors: Rahul Das, Satya N Majumdar, Arnab Pal

Published 2026-09-22

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CC BY 4.0

Original authors: Rahul Das, Satya N Majumdar, Arnab Pal

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 vast landscape of statistical physics, scientists study how random systems behave over time. Imagine a particle wandering through space, pushed by invisible forces and jostled by thermal noise. Usually, such a particle will drift aimlessly, spreading out until it covers a vast area, never settling into a predictable pattern. However, if you force this wandering particle to return to its starting point at random intervals, something remarkable happens: it stops spreading and settles into a stable, unchanging pattern known as a non-equilibrium steady state. This phenomenon, called stochastic resetting, has become a powerful tool for understanding everything from how animals search for food to how enzymes find their targets. For years, researchers have focused on a specific type of resetting where an external clock triggers the return, independent of the particle's actual journey. But in the real world, resets often happen because the system itself hits a limit, like a neuron firing when its electrical charge reaches a certain level or a stock trader selling when a price drops to a specific point. These are threshold-driven events, where the system's own motion triggers the reset. While these scenarios are common in nature and technology, the rules governing how they create stable patterns have remained largely a mystery.

A team of researchers has now cracked the code for these threshold-driven systems, revealing exactly when and how they settle into a steady state. By studying a particle moving with a constant push toward a boundary, they discovered a simple, universal rule: a stable pattern will form if and only if the particle can reach that boundary within a finite average amount of time. If the particle is likely to wander off forever without ever hitting the limit, no stable pattern will ever emerge. When a stable state does exist, the researchers found it can be described by a clear relationship between two fundamental quantities. The first is the average time the particle spends at any specific location during a single journey before it hits the boundary. The second is the average total time it takes to hit that boundary for the first time. The final, stable distribution of the particle is simply the ratio of these two times. In essence, the likelihood of finding the particle at a certain spot is determined by how long it lingers there compared to how long the entire trip takes.

To prove this theory and explore the journey toward stability, the team focused on a classic model of motion called drift-diffusion, where a particle is pushed by a steady wind while also jittering randomly. They tracked how the particle's position changed over time, from the moment it started until it finally settled into its steady pattern. They found that the path to stability is far from smooth. In situations where the pushing force is strong, the particle does not simply glide into its final resting place. Instead, it exhibits a rhythmic, oscillating behavior. The particle's average position and its spread across space rise and fall in damped waves before finally calming down. This happens because the strong push creates a synchronized rhythm: the particle travels to the boundary, resets, and travels back in a tight loop, creating periodic waves of probability. When the random jitter is stronger than the push, this rhythm breaks down, and the particle relaxes smoothly without any oscillations.

The study also uncovered a sharp transition in how the particle behaves at the very edges of its possible range. As time passes, the particle's movement creates a distinct "light cone," a boundary beyond which the particle simply cannot reach. Inside this moving boundary, the particle's behavior changes in a way that signals a phase transition, a sudden shift in the nature of its motion. The researchers showed that this transition is a second-order change, meaning the rate at which the particle spreads changes abruptly at the edge of this cone, even though the position itself remains continuous. This finding connects the behavior of threshold-resetting systems to other complex physical phenomena, suggesting a deep underlying order in how systems driven by internal limits evolve.

The implications of this work extend far beyond the theoretical. The rules derived here apply to any system where a process is interrupted by its own success or failure, from the firing of neurons in the brain to the management of computer networks that reset when they encounter errors. By showing that a finite travel time to a threshold is the key to stability, the researchers provide a clear criterion for designing systems that need to remain stable without external control. They also highlight the rich, complex dynamics that occur before stability is reached, revealing that the journey to equilibrium can be just as fascinating as the destination itself. This work transforms our understanding of event-driven processes, showing that even in a world of randomness, the simple act of hitting a limit can create a predictable, steady order.

Technical Summary: Non-equilibrium dynamics of drift–diffusion process under threshold resetting

Problem Statement
Stochastic resetting, where a system is periodically returned to an initial configuration, is a well-established mechanism for generating non-equilibrium steady states (NESS). However, most existing literature focuses on externally driven resetting, where reset events occur at random times determined by an independent clock or distribution. This paper investigates threshold resetting (TR), an event-driven protocol where the system resets only upon crossing a prescribed spatial boundary (threshold). Unlike external resetting, TR is intrinsically coupled to the system's own dynamics. While threshold-crossing events are ubiquitous in physical, biological, and engineered systems (e.g., integrate-and-fire neurons, financial stop-loss strategies), the non-equilibrium properties, specifically the emergence of steady states and relaxation dynamics under TR, remain largely unexplored. The central questions addressed are: Under what conditions does a NESS exist under TR? What is the physical structure of this steady state? How does the system relax to this state, particularly in the intermediate-time regime?

Methodology
The authors employ a renewal formalism to analyze a particle undergoing arbitrary stochastic dynamics in one dimension, starting at $x_0$ and resetting instantaneously to $x_0$ upon reaching the threshold at $x=0$ .

General Framework: They derive the propagator $P(x, t|x_0)$ by decomposing trajectories into those that have not yet hit the threshold and those that have undergone one or more reset events. This leads to a renewal equation relating the full propagator to the constrained propagator $G_+(x, t|x_0)$ (the process with an absorbing boundary at the threshold) and the first-passage time density $F(t|x_0)$ .
Laplace Space Analysis: By transforming the renewal equation into Laplace space, they express the propagator $\tilde{P}(x, s|x_0)$ entirely in terms of the underlying reset-free dynamics.
Steady-State Derivation: Using the final value theorem, they analyze the small- $s$ (long-time) behavior of the Laplace-transformed propagator to determine the conditions for the existence of a steady state $P_{ss}(x)$ .
Specific Model Application: To illustrate these general findings, the authors apply the framework to a drift-diffusion (DD) process with constant drift $v > 0$ toward the threshold and diffusion coefficient $D$ . They solve the associated Fokker-Planck equation to obtain exact analytical expressions for the propagator, moments, and mean-squared displacement (MSD). They utilize the dimensionless Péclet number ( $Pe = x_0 v / 2D$ ) to characterize the interplay between advection and diffusion.

Key Contributions and Results

General Criterion for NESS: The paper establishes a sufficient condition for the existence of a non-equilibrium steady state under threshold resetting: the mean first-passage time (MFPT) to the threshold must be finite. If the MFPT is finite, a NESS exists; if the particle can escape to infinity with non-zero probability (e.g., drift away from the threshold), no steady state forms.
Physical Representation of the Steady State: When a NESS exists, the authors derive a fundamental relation expressing the steady-state distribution as the ratio of two quantities determined solely by the underlying reset-free dynamics:
$P_{ss}(x) = \frac{\langle L(x|x_0) \rangle}{\langle T \rangle}$
where $\langle L(x|x_0) \rangle$ is the mean local time (MLT) spent at position $x$ during a single excursion between threshold crossings, and $\langle T \rangle$ is the mean first-passage time (MFPT). This provides a transparent physical interpretation: the steady-state probability density is the average time spent at $x$ per cycle divided by the average duration of a cycle.
Drift-Diffusion Dynamics:
- Steady-State Profile: For the DD process, the steady state exhibits a "kink" (discontinuity in the first spatial derivative) at the resetting position $x = x_0$ . The distribution vanishes at the threshold ( $x=0$ ), grows linearly near the threshold, and decays exponentially for $x > x_0$ due to the drift.
- Intermediate-Time Anomalies: The paper uncovers rich transient dynamics. In the drift-dominated regime ( $Pe \gg 1$ ), the relaxation to the steady state is characterized by damped oscillations in the mean position and MSD. These oscillations arise from the near-synchronization of renewal events; the particle drifts to the threshold and resets in coherent bursts, creating periodic probability waves.
- Diffusion-Dominated Regime: For low $Pe$ , diffusion broadens the first-passage time distribution, destroying the synchronization and leading to monotonic, exponential relaxation without oscillations.
Dynamical Phase Transition: The authors analyze the large-deviation behavior of the propagator at long times. They identify a second-order dynamical phase transition in the relaxation spectrum. The rate function $\phi(x/t)$ , which governs the tail of the propagator, exhibits a discontinuity in its second derivative at the "light cone" speed $x/t = v$ , separating the behavior inside and outside the drift front.

Significance and Claims
The paper claims to provide a general understanding of how event-driven (threshold) resetting shapes the spatial and temporal properties of stochastic processes. By linking the steady state to the ratio of MLT and MFPT, the work offers a universal framework applicable to a broad class of systems beyond the specific drift-diffusion example.

The authors emphasize that their findings reveal rich intermediate-time dynamics, including anomalous relaxation and oscillatory behavior, which are distinct from the typically monotonic relaxation seen in externally driven resetting. The identification of the dynamical phase transition in the relaxation spectrum further connects threshold resetting to broader concepts in non-equilibrium statistical physics.

The paper concludes modestly, suggesting that these results open avenues for investigating threshold resetting in heterogeneous and non-diffusive processes, many-particle systems (where threshold crossings induce effective correlations), and biological models like integrate-and-fire neurons or branching processes. It positions the work as a foundational step toward understanding the non-equilibrium properties of event-driven protocols.

Technical Summary: Non-equilibrium dynamics of drift–diffusion process under threshold resetting

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