Dynamics of a tracer trapped in a correlated medium in the presence of a wall

TL;DR

The paper develops an exact nonlinear, non-Markovian description of a tracer trapped near a wall inside a fluctuating Gaussian medium with correlation length $ξ$, where Dirichlet boundary conditions generate a repulsive Casimir-like force. By integrating out the Gaussian field, the authors derive an effective tracer dynamics featuring a wall-dependent memory kernel and a field-induced noise, reducing to a Markovian form in the adiabatic limit. A perturbative analysis in the coupling $λ$ in 1D (point-like particle) reveals two contributions to the two-time correlator: a Casimir term that renormalizes the trap and shifts fluctuations, and a memory term that decays algebraically as $t^{-1/2}$ at criticality and exhibits bulk-wall crossover behavior. The study also analyzes the power spectral density, showing non-monotonic memory contributions that enhance correlations at zero frequency and depend on the wall distance and correlation length, with implications for near-critical soft matter and microrheology.

Abstract

We describe the random motion of a particle immersed in a thermally fluctuating medium and harmonically trapped at a certain distance from a wall. The medium, modeled by a Gaussian field with a tunable correlation length $ξ$, is linearly coupled to the particle and evolves according to dissipative relaxational dynamics. Dirichlet boundary conditions imposed on the field at the wall give rise to a repulsive fluctuation-induced force acting on the particle, causing a shift in its average position and a renormalization of the strength of the harmonic trap. We describe the effective overdamped dynamics of the particle, which features a nonlinear memory term depending on the wall-particle separation. We show that the two-time correlation function of the particle position features a memory-induced term that depends on the distance of the particle from the wall. At the critical point, this term decays algebraically upon increasing time and it displays a crossover from the behavior observed in the bulk to that corresponding to having the particle at the wall.

Figures (5)

• Figure 1: (a) Cartoon of the model under investigation: it comprises the tracer particle (black sphere), spatially confined by an optical trap (providing the potential represented by the dark yellow line) and interacting with a fluctuating correlated field $\phi$ (with the dark blue line respesenting its average) which vanishes at the wall according to the boundary condition \ref{['BCs']}. Due to the coupling \ref{['eq:Hint']} between the particle and the field, assuming $\lambda$ positive (negative), it is energetically favorable for the particle to occupy the position corresponding to the maximum (minimum) of the field. Due to the Dirichlet boundary conditions \ref{['BCs']} at the wall and the particle acting as a source of the field in \ref{['eq:Hint']}, this position is farther away from the wall than the minimum of the trap at $X_0$. This field gives rise to a repulsive fluctuation-induced force $\mathbf{F_C} = \mathbf{\hat{e}_1}F_{\rm C}$ causing a shift of the particle position, counterbalanced by the force $\mathbf{F}_{\kappa}$ due to the trap. (b) Plot of the scaling function $\Phi$ introduced in \ref{['eq:Cas_scaling']}, in the case of $d=1$, for the Gaussian \ref{['eq:Gauss_ker']}, the exponential \ref{['eq:exp_ker']} and the point-like \ref{['eq:kernel_point_like_lim']} kernels modeling the shape of the particle, see \ref{['eq:Hint']}. Note that, assuming $\xi \gg R$, the different curves coincide for $X_1/\xi \gtrsim 0.5$.
• Figure 2: (a) Memory-induced contribution $\lambda^2 C_{c, \mathrm{memo}}^{(2)}(t)$ to the correlation function in Eq. \ref{['C_c_mem']} as a function of time $t$ for $X_0=10$. Note that this contribution vanishes at $t=0$ since it takes a finite amount of time for the memory to "build up". After an initial growth $\sim t^{3/2}$ stemming from the $\sim t^{-1/2}$ behavior of the memory \ref{['mem_small_t']} integrated in \ref{['C_c_mem']}, $\lambda^2 C_{c, \mathrm{memo}}^{(2)}(t)$ attains its maximal value at $t \approx 2$ and then, upon further increasing $t$, it decreases either exponentially for finite $\xi$, or algebraically $\sim t^{-1/2}$ for $\xi \to \infty$. (b) Bulk ($\mathrm{F^+}$) and wall ($\mathrm{F^-}$) critical contributions to the memory given by Eqs. \ref{['memo_crit_pl']} and \ref{['memo_crit_min']}, respectively, as functions of time $t$. We note that $\mathrm{F^+}$ and $\mathrm{F^-}$ coincide at long times. In the inset, we plot $\mathrm{F^\pm}$ on a log-log scale in order emphasize the different order of magnitudes of the bulk and the wall contributions at short times. Quantities in this figure are plotted for $\lambda^2/(2 \kappa) = \ell_T = \sqrt{D/\omega_0} =1$.
• Figure 3: Memory-induced contribution $\lambda^2 C_{c, \mathrm{memo}}^{(2)}(t)$ to the correlation function in Eq. \ref{['C_c_mem']} at the critical point $r=0$ and as a function of time $t$. (a) Plot on a log-log scale of $\lambda^2 C_{c, \mathrm{memo}}^{(2)}(t)$, which highlights the algebraic decay $\propto t^{-1/2}$ at long times. Additionally to the plots obtained for finite values of $X_0$, we present the asymptotic curves corresponding to $X_0 = 0$ (upper curve) and $X_0 \to \infty$ (lower curve). For finite values of $X_0$, one observes a cross over from the bulk asymptotic to the wall asymptotic. (b) Logarithmic derivative (see \ref{['eq:log_der_def']}) of $\lambda^2C_{c, \mathrm{memo}}^{(2)}(t)$ as a function of time $t$. At short times, the values of the logarithmic derivative approaches $3/2$, corresponding to a $\propto t^{3/2}$ growth of the memory-induced contribution, whereas it takes the value $-1/2$ at long times, corresponding to the algebraic behavior discussed above. The bumps in the plots indicate the times at which the crossovers from the bulk to the wall asymptotic occur. Note that the larger $X_0$, the longer the crossover time. Both plots correspond to $\lambda^2/(2 \kappa) = \ell_T = \sqrt{D/\omega_0} =1$.
• Figure 4: Bulk term of the memory-induced contribution $\lambda^2 S_{\mathrm{memo}, +}^{(2)}(\omega)$ to the power spectral density as a function of $\omega$, for various values of the correlation length $\xi$. (a) For large values of $\xi$, one observes $\lambda^2S^{(2)}_{\mathrm{memo}, +}(\omega) \propto \omega^{-1/2}$ (dotted line) at small but finite values of $\omega$. (b) At larger values of $\omega$, instead, $\lambda^2S^{(2)}_{\mathrm{memo}, +}(\omega)$ shows a non-monotonic behavior and, in fact, after becoming negative, it approaches zero from below as $\omega \to\infty$. The various curves are compared with the one obtained at the critical point for strong confinement, see \ref{['Spl_T0']} (dashed line). The solid lines in both panels were obtained with $\lambda^2/(2 \kappa) = \ell_T = \sqrt{D/\omega_0} =1$.
• Figure 5: Wall term of the memory-induced contribution $\lambda^2 S_{\mathrm{memo}, -}^{(2)}(\omega)$ to the power spectral density for various values of the correlation length $\xi$, as a function of (a) $\omega$ with fixed $X_0=10$ or (b) $X_0$ with $\omega=0$. Note that $\lambda^2 S_{\mathrm{memo}, -}^{(2)}(\omega)$ is generically a non-monotonic function of its variables and, for a fixed value of $X_0$, it attains a finite value for $\omega \to 0$, the sign of which depends on $\xi$, see panel (a). The dashed line in panel (b) corresponds to $(\lambda^2/\xi)S_{\mathrm{memo}, -}^{(2)}(\omega= 0)$ in the limit of strong confinement, given by Eq. \ref{['S0_scaling']}, which changes sign at $X_0/\xi =0.5$ and attains its minimal value at $X_0/\xi =1$. The solid lines in both panels are plotted for $\lambda^2/(2 \kappa) = \ell_T = \sqrt{D/\omega_0} =1$.