Nonequilibrium relaxation of a trapped particle in a near-critical Gaussian field

We study the nonequilibrium relaxational dynamics of a probe particle linearly coupled to a thermally fluctuating scalar field and subject to a harmonic potential, which provides a cartoon for an optically trapped colloid immersed in a fluid close to its bulk critical point. The average position of the particle initially displaced from the position of mechanical equilibrium is shown to feature long-time algebraic tails as the critical point of the field is approached, the universal exponents of which are determined in arbitrary spatial dimensions. As expected, this behavior cannot be captured by adiabatic approaches which assumes fast field relaxation. The predictions of the analytic, perturbative approach are qualitatively confirmed by numerical simulations.

Figure 1

Pictorial representation of the model: a colloidal particle is in contact with a fluctuating scalar field $\varphi \left(\mathbf{x}\right)$ and trapped by a harmonic potential.

Figure 2

Evolution of the average position $\langle X\left(t\right)\rangle$ of a harmonically trapped particle initially released out of equilibrium and coupled to a field evolving with model A dynamics, for various decreasing distances $r$ from the critical point. The plots show the analytical prediction up to $\mathcal{O}\left({\lambda }^2\right)$. The points are obtained from the numerical integration of Eq. (24) and they are joined by a linear interpolation to guide the eye. The exponents observed at criticality, $r=0$, agree with those predicted in Eq. (30). In these plots $\lambda =1$, $\nu =10$, $k=0.1$, $T=1$, $D=1$, $X_0=10$, and the interaction potential is Gaussian with $R=1$.

Figure 3

Evolution of the average position $\langle X\left(t\right)\rangle$ of a harmonically trapped particle initially released out of equilibrium and coupled to a field evolving with model B dynamics, for various decreasing distances $r$ from the critical point. The plots show the analytical prediction up to $\mathcal{O}\left({\lambda }^2\right)$. The points are obtained from the numerical integration of Eq. (24) and they are joined by a linear interpolation to guide the eye. The decay exponents agree with those predicted in Eq. (31). In these plots $\lambda$, $\nu$, $k$, $T$, $D$, and $X_0$ are set to unity and the interaction potential is Gaussian with $R=1$.

Figure 4

Average particle position $\langle X\left(t\right)\rangle$ in the presence of a coupling to the $n\mathrm{th}$ derivative of the field, as in Eq. (29). The plot shows the case of model A in $d=1$, and the observed decay exponents agree with those predicted in Eq. (33) for $z=2$. In this plot $\lambda$, $\nu$, $k$, $T$, $D$ and $X_0$ are set to unity and the interaction potential is Gaussian with $R=1$.

Figure 5

Independence of the long-time behavior of the average position of the colloid from the particular choice of the interaction potential $V\left(\mathbf{x}\right)$. The plot shows the correction $\langle X^{\left(2\right)}\left(t\right)\rangle$ for model A in $d=1$ (the leading order exponential term is irrelevant at long times). The exponent of the algebraic decay is not affected by the specific form of $V\left(\mathbf{x}\right)$, but only by the behavior of its Fourier transform $V_q$ for $q\to 0$, in agreement with the asymptotic expression in Eq. (32). On the contrary, the short-time behavior is sensitive to the particular choice of $V\left(\mathbf{x}\right)$. The forms of the interaction potential reported here are Gaussian $V_q=exp(-R^2{q}^2/2)$, box $V_q=\text{sinc}(Rq/2)$, cusp $V_q=1/(1+R^2{q}^2)$. Here $R$ indicates the linear size of the colloid. In the plot we set $r=0$ and all the other parameters to unity.

Figure 6

Relaxation of the average position $\langle X\left(t\right)\rangle$ toward equilibrium in $d=1$ critical model A when the initial position $X_0$ is chosen sufficiently large so as to emphasize the nonlinear response. Open blue circles represent the theoretical prediction in Eq. (24), while red filled circles are the results of numerical simulations performed at $T=0$ (we justify this choice and describe the simulation method in Sec. 5 and Appendix pp9). Parameters used in the simulation are $\nu =1$, $k=0.1$, $X_0=150$, $D=1$, $R=1$, $\lambda =0.25$, integration timestep $\mathrm{\Delta }t=0.01$ and lattice size $L=2048$.

Figure 7

Average position $\langle X\left(t\right)\rangle$ of the particle in numerical simulations of the noiseless equations of motion, corresponding to $T=0$, in $d=1$ (left) and $d=2$ (right). All the simulations are in excellent agreement with the long-time behavior predicted in Eqs. (30) and (31) (corresponding to the slopes indicated by the solid straight lines). We do not show here the full prediction in Eq. (24) for graphical clarity, as it is almost indistinguishable from the simulation points (but we do present such a comparison in Figs. 6 and 8). The parameters used in the simulation are $\nu =1$, $k=0.1$, $X_0=2$, $D=1$, $R=0.5$, $\lambda =0.25$, and $\mathrm{\Delta }t=0.01$. The system size is chosen to be $L=2048$ in the $d=1$ case, and $L=512$ in the $d=2$ case.

Figure 8

Average particle position $\langle X\left(t\right)\rangle$ during the relaxation to equilibrium in $d=1$ critical model B, in the presence of noise. Simulation results are plotted as a solid red line, while the blue dots represent the theoretical prediction in Eq. (24); they are shown to be in complete agreement. Parameters used in the simulation are $r=0$, $\nu =1$, $k=0.1$, $X_0=2$, $D=1$, $R=1$, $\lambda =0.25$, $\mathrm{\Delta }t=0.01$, $T=0.1$, $L=128$, and $N=7.7\times {10}^8$ realizations.

Figure 9

Average particle position $\langle X\left(t\right)\rangle$ during its relaxation toward equilibrium in $d=1$ critical model A, when the initial position $X_0$ is chosen sufficiently large so as to emphasize the nonlinear response. In the main plot, the various curves correspond to increasing values of $X_0$, and the associated crossover time $t_c$ is seen to shift toward larger times. In the inset, the same curves are collapsed according to the scaling form in Eq. (55) (see the main text). Parameters used in the simulation are $\nu =1$, $k=0.1$, $T=0$, $D=1$, $\lambda =0.25$, $R=1$, $\mathrm{\Delta }t=0.01$, and $L=8192$.

Figure 10

Perturbative correction to the average position $\langle X\left(t\right)\rangle$ during the relaxation to equilibrium in $d=1$ critical model B. Blue dots represent the theoretical prediction of $\langle X^{\left(2\right)}\left(t\right)\rangle$ in Eq. (24), while we plotted in different shades of red (and different dashing) the quantity $[\langle X\rangle -\langle X^{\left(0\right)}\rangle ]/{\lambda }^2$ estimated in numerical simulations for increasing values of $\lambda \in [0.25–2.00]$, from lightest to darkest (and from shortest to longest dashing). For each curve we subtracted from the data the purely exponential decay and divided by ${\lambda }^2$. For large values of $\lambda$, one observes qualitatively the same power-law decay at long times, whose onset is nonetheless delayed as $\lambda$ increases. Parameters used in the simulation are $T=0$, $\nu =1$, $k=0.1$, $X_0=2$, $D=1$, $R=1$, $\mathrm{\Delta }t=0.01$, and $L=128$.

Figure 11

Average particle position $\langle X\left(t\right)\rangle$ during its relaxation to equilibrium in $d=1$ critical model B. We chose a large value of the coupling constant $\lambda$, well beyond the perturbative regime where agreement is observed between simulation data and our analytical prediction. Here the theoretical prediction indeed fails to describe even the qualitative behavior of the average position at short times (see main text). Parameters used in the simulation are $\lambda =2$, $T=0$, $\nu =1$, $k=0.1$, $X_0=2$, $D=1$, $R=1$, $\mathrm{\Delta }t=0.01$, and $L=128$.