Nonequilibrium forces following quenches in active and thermal matter

Nonequilibrium systems with conserved quantities like density or momentum are known to exhibit long-ranged correlations. This, in turn, leads to long-ranged fluctuation-induced (Casimir) forces, predicted to arise in a variety of nonequilibrium settings. Here, we study such forces, which arise transiently between parallel plates or compact inclusions in a gas of particles, following a change (“quench”) in temperature or activity of the medium. Analytical calculations, as well as numerical simulations of passive or active Brownian particles, indicate two distinct forces: (i) The immediate effect of the quench is adsorption or desorption of particles of the medium to the immersed objects, which in turn initiates a front of relaxing (mean) density. This leads to time-dependent density-induced forces. (ii) A long-term effect of the quench is that density fluctuations are modified, manifested as transient (long-ranged) (pair-)correlations that relax diffusively to their (short-ranged) steady-state limit. As a result, transient fluctuation-induced forces emerge. We discuss the properties of fluctuation-induced and density-induced forces as regards universality, relaxation as a function of time, and scaling with distance between objects. Their distinct signatures allow us to distinguish the two types of forces in simulation data. Our simulations also show that a quench of the effective temperature of an active medium gives rise to qualitatively similar effects to a temperature quench in a passive medium. Based on this insight, we propose several scenarios for the experimental observation of the forces described here.

Figure 1

Sketch of two parallel plates, separated by a distance $L$ along the $z$ axis, immersed in a medium. The lateral extensions of the plates are assumed to be much larger than $L$. In Sec. 3a, the medium is an ideal gas of active or passive Brownian particles, subject to a quench in temperature or activity. In Sec. 4 the same system with interactions is studied.

Figure 2

Force on the plate at $z=0$ from Eq. (13) as a function of dimensionless time $t^{*}=D_{0}t/L^2$ for ${\alpha }_1={\alpha }_2={\alpha }_1^{\left(o\right)}\equiv \alpha$. The curve approaches 2 for large times. Inset: Asymptotic scaling $\sim {\left(t^{*}\right)}^{-1/2}$ (black line) of the difference between the force and its steady-state value, as given in Eq. (14).

Figure 3

Pressure on the plates separated by a distance $L$, after a quench from $k_B{T}_I=0$ to $k_B{T}_F=0.5$ (simulation units, see Sec. 2), measured in 2D simulations for PBPs (a) and ABPs (b); $t^{*}=D_{0}t/L^2$ with $D_0={\mu }_0{k}_B{T}_F$. Due to the adsorbed density, the steady-state pressure deviates from $P=k_B{T}_F{\rho }_0$ (black dashed line) by a term proportional to $1/L$; see Eq. (16). Insets show the $L$-dependence of the final pressure, with the straight lines corresponding to slope $\alpha =-0.89$ found from Eq. (17).

Figure 4

Comparison of the data of Fig. 3 to Eq. (12) (solid black line), for passive (a) and active (b) particles, with $\alpha =-0.89$ obtained from Eq. (17). The dashed and dotted lines are the leading behaviors at short and long times, respectively. For the active case, $D_0$ was reduced to $\simeq 0.75{\mu }_0{k}_B{T}_{\mathrm{eff}}$ to obtain agreement.

Figure 5

Sketch of two inclusions with characteristic sizes ${\sigma }_i$, separated by a center-to-center distance $L$. The surrounding medium is a suspension of noninteracting active or passive Brownian particles, undergoing a temperature quench. Throughout, the limit ${\sigma }_i\ll L$ is assumed.

Figure 6

Simulation results of the force between two identical inclusions modeled by the potential in Eq. (23) in $d=2$, after a quench from infinite temperature to a finite $T_F$ for ideal PBPs. Solid black curve gives Eq. (24) using $a=1$. The maxima in the simulation curves have been normalized to unity, which yields the temperature-dependent amplitude $a$ shown in the inset. $L=6,\sigma =1,V_0=2\pi$, and system size $72\times 12$ with periodic boundary conditions. The force is repulsive.

Figure 7

Simulation results of the force between two identical inclusions modeled by the potential in Eq. (23) in $d=1$, after a quench from infinite temperature to a finite $T_F$ for ideal PBPs. The solid black curve shows Eq. (22). At low temperatures, the particles are trapped between the two inclusions which become more and more like impenetrable “plates,” so that the curve approaches the shape of the curve in Fig. 2. $\sigma =1,V_0=\sqrt{2\pi }$, and system size is 72 with periodic boundary conditions.

Figure 8

The time-dependent bulk correlation function, relating two points separated by a distance $X$, as given in Eq. (31). We show different dimensions $d$.

Figure 9

Pressure exerted on the inside surfaces of two parallel plates after a temperature quench to $T_F=0.5$ from simulations (data points). The only difference to Fig. 3 is the presence of an interaction potential $U$. (a) PBPs with $\lambda =1$ and ${\rho }_0=0.936$, giving $P^{\prime \prime }\left({\rho }_0\right)=0.674$ and $c^{\left(2\right)}=-1.35$. $t^{*}=\mu m/L^2$ is found with Eq. (26). (b) ABPs with $\lambda =2$ and ${\rho }_0=0.909$, giving $P^{\prime \prime }\left({\rho }_0\right)=4.06$ and $c^{\left(2\right)}=-3.61$. For both: System size $10\times 2000$. The data is averaged over $6\times {10}^4$ runs and the final pressure is measured in longer simulations $t^{*}>200$. The theory curves are the sum of Eq. (16) (with $\alpha$ chosen for best fit) and Eqs. (35) or (39), respectively.

Figure 10

The function ${\mathrm{\Theta }}^{\left(1\right)}(t^{*},\gamma )$ from Eq. (B7) (i.e., $d=1)$ for various values of $\gamma =\sigma /L$. For $\gamma \ne 0$ the force begins at some nonzero value, since the potentials overlap appreciably. As $\gamma \to 0$, the curves collapse onto the red master curve, which begins at 0.

Figure 11

Force between two inclusions modeled by Eq. (23) in $d=1$ after a quench from infinite temperature to a finite $T_F$ for a passive ideal gas. The theory curve is given by Eq. (B7). $\sigma =1,L=6$ (i.e., $\gamma =1/6)$, $k_{B}T=5$.