Resetting in a viscoelastic bath: the bath remembers

Technical Deep Dive: Resetting in a Viscoelastic Bath

Overview

This work studies the behavior of a Brownian particle (the "probe") that undergoes stochastic resetting while embedded in a viscoelastic medium. The key finding is that the memory effects of the viscoelastic environment lead to qualitatively different dynamics compared to the case of a Markovian (non-memory) environment.

Problem & Context

• Stochastic resetting, where a dynamical process is intermittently stopped and restarted from a fixed configuration, has emerged as a powerful framework for controlling stochastic dynamics.
• However, most prior work has assumed the underlying dynamics to be Markovian, where the medium responds instantly to the particle's motion.
• Many realistic environments, like polymeric and biological media, exhibit viscoelastic behavior with memory effects, where the medium retains a "memory" of the particle's past motion.
• Understanding stochastic resetting in viscoelastic media with memory is essential for describing realistic conditions, as the memory effects can qualitatively change the dynamics.

Methodology

• The authors model the viscoelastic medium using a single auxiliary "bath" particle coupled to the probe particle via a harmonic spring.
• At random reset times drawn from a Poisson process, only the probe particle is reset to the origin, while the bath particle continues to evolve.
• This ensures that the memory of the bath particle's past dynamics is retained across reset events, in contrast to prior works where the bath was also reset.
• The authors analyze this model analytically and numerically, focusing on the probe particle's position statistics.

Results

Instantaneous Resetting

• For strong bath memory (large γ), the stationary position distribution of the probe develops non-exponential, Gaussian tails, in contrast to the exponential tails in the Markovian case.
• The time-dependent position variance shows a two-step relaxation, reflecting the separation between fast probe-bath equilibration and slow bath relaxation.

Non-Instantaneous Resetting

• With a finite return speed for the probe, the stationary position distribution depends sensitively on the return speed, unlike the Markovian case where it is invariant.
• Slow returns allow the bath to relax more, leading to reduced position fluctuations compared to fast returns.
• Analytical expressions are derived for the limiting cases of very slow and very fast returns.

Interpretation

• The memory effects of the viscoelastic bath qualitatively change the dynamics compared to the Markovian case:
  • The stationary distribution deviates from the exponential form, developing Gaussian tails.
  • The time-dependent fluctuations show a two-step relaxation process.
  • The stationary distribution depends on the details of the resetting protocol, unlike the Markovian case.
• These effects arise due to the persistence of bath correlations across reset events, which influence the subsequent probe dynamics.

Limitations & Uncertainties

• The model uses a single auxiliary bath particle, which is a minimal representation of a viscoelastic medium. More complex bath models may reveal additional features.
• The analysis is limited to the case of constant-velocity return protocols. Other return dynamics could lead to further insights.
• Experimental verification in colloidal systems with controlled viscoelastic properties would be valuable to confirm the theoretical predictions.

What Comes Next

• Exploring resetting in viscoelastic baths with power-law memory kernels, which arise in spatially extended systems and near critical points.
• Investigating resetting in nonequilibrium baths where active fluctuations coupled with memory may produce even richer behavior.
• Studying the interplay between resetting and other non-Markovian effects, such as inertial dynamics or fluctuating parameters.

Sources:

• Resetting in a viscoelastic bath: the bath remembers