Resetting in a viscoelastic bath: the bath remembers

Abstract

We study stochastic resetting of a probe particle in a viscoelastic environment where only the probe is reset while the medium retains memory of its past dynamics. Using a minimal model with finite correlation time, we analyze the competition between the resetting timescale and the viscoelastic relaxation timescale. This interplay leads to nonequilibrium steady states that differ qualitatively from those of Markovian Brownian motion with resetting. In particular, strong memory effects produce stationary position distributions with non-exponential tails. For instantaneous resets, we derive the limiting steady-state distributions analytically and compute exactly the time dependent leading non-vanishing moments. We also investigate non-instantaneous resetting via constant-velocity return protocols. In contrast to overdamped Brownian motion, where steady-state fluctuations are independent of the return dynamics, we find that in a viscoelastic medium the fluctuations depend on the reset velocity. This protocol dependence arises from the finite memory of the environment and highlights the role of environmental correlations in resetting-induced steady states.

Figures (12)

• Figure 1: (a) Schematic of a colloidal probe (in red circle) in a viscoelastic medium, represented by the black mesh. (b) Our model models the medium by means of another fluctuating particle (in dark grey sphere) coupled to it by a spring (see Eqs. \ref{['eq:model']}). At each resetting event, the position of the probe is reset without affecting the medium directly. (c) Instantaneous Reset: Representative trajectories of the probe for different viscoelastic time-scales in the presence of resetting.
• Figure 2: A plot of the extracted time-scales $\tau_i$ and the corresponding weights $A_i$ in Eq. \ref{['eq:msd']} in the small [panels (a)] and large [panels (b)] $\gamma$ limit.
• Figure 3: Mean squared displacement $\langle x^2(t)\rangle$ of the probe in presence of stochastic resetting. Panel (a) shows the two-step relaxation in the large $\gamma$ limit ($\gamma = 10$ and different values of $r$) as predicted from Eq. \ref{['eq:msd']}. Panel (b) shows $\langle x^2(t)\rangle$ in the small $\gamma$ limit ($\gamma = 0.1$ and different values of $r$) which has a single step relaxation as predicted. In both the figures the symbols denote numerical simulations while the dashed black lines are obtained from inverting Eq. \ref{['eq:m2s']} for the specific parameter values.
• Figure 4: Stationary position distribution $P(x)$ of the probe under instantaneous resetting for different viscoelastic time-scales $\gamma$.Panel (a) shows $P(x)$ for small resetting rates and different values of $\gamma$.The dotted black lines denote the Markovian Brownian motion limit Eq. \ref{['eq:BM']}, while the dashed line denotes the analytical prediction Eq. \ref{['int:psx']} for large $\gamma$. Panel (b) shows the independence of $P(x)$ with $\gamma$ for large resetting rates, as predicted in Eq. \ref{['eq:markovreset']}.
• Figure 5: Assymptotic behavior of the stationary position distribution at the tails and near the origin respectively. Panel (a) shows the scaled position distribution, the scaling prediction in Eq. \ref{['eq:scaling']} is illustrated by a collapse of the distributions obtained numerically for different values of $r$, and $\gamma=100$. The symbols denote the numerical simulations, while Gaussian scaling function is shown with dashed black line. Panel (b) shows the behavior of the distribution near the origin which shows an exponential decay as predicted in Eq. \ref{['eq:scaling']} (shown as a black dashed line).