Recoil of a driven tracer in a correlated medium

TL;DR

The paper addresses recoil of a driven tracer moving through a slowly relaxing correlated medium modeled by a linear coupling to a Gaussian field. It develops a nonlinear, non-Markovian effective tracer dynamics via integrating out the field and analyzes it perturbatively in the coupling $\lambda$, revealing a recoil opposite to the dragging velocity after release. The recoil amplitude is captured by a universal scaling function $\mathcal{F}^{A/B}$ of $R/\xi$, $t/\tau_R$, and $v\tau_R/R$, with model A and B showing distinct relaxation (exponential vs algebraic) and critical-point behaviors; near criticality, the recoil can diverge in low dimensions and decay algebraically with exponent $(2-d)/z$ in the long-time limit. The results offer qualitative guidance for driven colloids in near-critical media and suggest experimental tests, while highlighting avenues for refinement including hydrodynamics, nonlinear field interactions, and active-bath extensions.

Abstract

We study the stochastic dynamics of a Brownian particle after it is suddenly released from a harmonic trap moving with constant velocity through a fluctuating correlated medium, described by a scalar Gaussian field with relaxational dynamics and in contact with a thermal bath. We show that, after the release, the particle exhibits recoil, i.e., it moves in the direction opposite to the drag. As expected, this effect vanishes if the field equilibrates instantaneously. The final value of the average position of the particle is reached algebraically in time in the case of conserved dynamics of the field or for non-conserved dynamics at the critical point. Our predictions are expected to be relevant, at least qualitatively, to driven colloidal particles in liquid media close to critical points.

Figures (8)

• Figure 1: Schematic representation of a Brownian particle (black sphere), driven by a moving optical trap (providing the potential represented by the dark yellow line) and interacting with a fluctuating correlated field $\phi$ (dark blue line), which is preferably enhanced in the vicinity of the particle (see \ref{['eq:Hint']} with $\lambda>0$). (a) While the particle is driven, i.e., for $t<0$, the field attains an average configuration (blue line) which is stationary and spatially modulated in the reference frame co-moving with the trap. At time $t = 0^-$, the particle is subject to the harmonic force $\mathbf{F}_{\kappa}$ exerted by the moving trap, to the field-induced force $\mathbf{f}$ defined in Eq. \ref{['eq:force_def']} which pulls the particle along the gradient of the field, and to the friction force $\mathbf{F}_{\nu}$. (b) At time $t = 0^+$, the particle is released from its confinement and thus it starts moving backwards (recoil), due to the fact that $\mathbf{f}$ remains nonvanishing due to the slow relaxation of the field, and that the friction changes both its sign and amplitude $\mathbf{F}_{\nu} \mapsto \mathbf{\tilde{F}_{\nu}} = - \mathbf{f}$.
• Figure 2: Dependence of the recoil amplitude $|\Delta X|$ on time, for the non-critical model A in spatial dimension $d=1$, in the absence of noise ($T=0$) and for various values of the correlation length $\xi$ of the field. The interaction of the particle with the field is assumed to be Gaussian as in Eq. \ref{['eq:Gauss_kernel']} with $R=1$, while the other parameters of the model are $\kappa = 0.5$, $D=1$, $\nu =1$, $\lambda=0.1$, and $v=1$. The solid lines correspond to the predictions of the perturbative calculation reported in \ref{['zeroT_rec']}, while the dashed lines are the result of a numerical simulation in which the particle was dragged through the medium for $t \in [0, 250]$, before being released from its confinement at time $t = 250$.
• Figure 3: Scaling function $\mathcal{F}^A$ of the recoil for model A in spatial dimension $d=3$, see \ref{['zeroT_rec_scale']}. Panel (a) shows $\mathcal{F}^A$ as a function of the dimensionless time $\tilde{t}$ with $\tilde{v}=1$, and various values of the dimensionless correlation length $\xi/R$. The dashed line corresponds to the long-time behavior at the critical point reported in \ref{['eq:crit_recoil_asympt']}, where $z = 2$. Panel (b) shows the dependence of the long-time limit $\mathcal{R}^A$ of the scaling function $\mathcal{F}^A$ of the recoil (see \ref{['eq:range_def']}) as a function of $\xi/R$, for various values of the scaled driving velocity $\tilde{v}$. The behaviors of these curves for large values of $\xi/R$, given by \ref{['eq:range_asympt']}, are reported as dashed lines.
• Figure 4: Scaling function $\mathcal{F}^B$ of the recoil for model B in spatial dimension $d=3$, see \ref{['zeroT_rec_scale']}. Panel (a) shows $\mathcal{F}^B$ as a function of the dimensionless time $\tilde{t}$ with $\tilde{v}=1$, and various values of the dimensionless correlation length $\xi/R$. The dashed brown line corresponds to the long-time behavior at the critical point reported in \ref{['eq:crit_recoil_asympt']}, where $z = 4$, and the dashed purple line corresponds to the long-time behavior in the non-critical case reported in \ref{['eq:subcrit_B_recoil_asympt']}. Panel (b) shows the dependence of the long-time limit $\mathcal{R}^B$ of the scaling function $\mathcal{F}^B$ of the recoil (see \ref{['eq:range_def']}) as a function of $\xi/R$, for various values of the scaled driving velocity $\tilde{v}$. The behaviors of these curves for large values of $\xi/R$, given by \ref{['eq:range_asympt']}, are reported as dashed lines.
• Figure 5: Dimensionless recoil range $\tilde{v} \times \mathcal{R}^{A/B}$ (see Eqs. \ref{['zeroT_rec_scale_def']} and \ref{['eq:range_def']}) as a function of the dimensionless driving velocity $\tilde{v}$ in spatial dimension $d=3$, for (a) model A or (b) model B, and various values of the correlation length $\xi$. In order to ease the comparison between these two dynamics, the dashed lines in panel (b) correspond to the behavior of the recoil range for model A at small or large velocities. As indicated in the inset, for large velocities, the curves corresponding to models A and B coincide.