Dynamics of a homogeneous active dumbbell system

We analyze the dynamics of a two-dimensional system of interacting active dumbbells. We characterize the mean-square displacement, linear response function, and deviation from the equilibrium fluctuation-dissipation theorem as a function of activity strength, packing fraction, and temperature for parameters such that the system is in its homogeneous phase. While the diffusion constant in the last diffusive regime naturally increases with activity and decreases with packing fraction, we exhibit an intriguing nonmonotonic dependence on the activity of the ratio between the finite-density and the single-particle diffusion constants. At fixed packing fraction, the time-integrated linear response function depends nonmonotonically on activity strength. The effective temperature extracted from the ratio between the integrated linear response and the mean-square displacement in the last diffusive regime is always higher than the ambient temperature, increases with increasing activity, and, for small active force, monotonically increases with density while for sufficiently high activity it first increases and next decreases with the packing fraction. We ascribe this peculiar effect to the existence of finite-size clusters for sufficiently high activity and density at the fixed (low) temperatures at which we worked. The crossover occurs at lower activity or density the lower the external temperature. The finite-density effective temperature is higher (lower) than the single dumbbell one below (above) a crossover value of the Péclet number.

Figure 1

A sketch of an active dumbbell molecule.

Figure 2

Mean-square displacement as a function of the delay time hereafter called $t$ measured after $t^{\prime }=100$ in a system of interacting passive dumbbells with the surface fractions given in the key at $T=0.001$ (double logarithmic scale).

Figure 3

Rescaled mean-square displacement in systems of interacting passive dumbbells with the surface fractions given in the key. The data in Fig. 2 are divided by $2dt$ and presented as a function of $1/t$ to identify the diffusive regime as a plateau with height given by the diffusion constant. The single-passive-dumbbell diffusive constant takes the value $D_{\mathrm{c}.\mathrm{m}.}^{\mathrm{pd}}=5\times {10}^{-5}$ for these parameters, consistently with the trend of the numerical results for the many-particle system.

Figure 4

Upper panel: In linear-log scale, the diffusion constant of an interacting system, as a function of surface fraction $\varphi$, in four cases: the passive dumbbell system displayed with (green) crosses $\times$; the passive colloidal system shown with (red) pluses $+$; and the active dumbbell system with $F_{\mathrm{act}}=0.001$ and $F_{\mathrm{act}}=0.1$ drawn with (pink) triangles $▵$ and (blue) squares $\square$, respectively. In all cases the systems are in contact with the same thermal bath at temperature $T=0.001$ and interact with the same WCA potential given in Eq. (2). Lower panel: In double linear scale, the same data relative to its single-particle limit, as a function of surface fraction $\varphi$. The solid line represents the analytic prediction by Tokuyama-Oppenheim (TO) for a colloidal system with the same parameters [75].

Figure 5

Mean-square displacement of active dumbbells for various values of $F_{\mathrm{act}}$ given in the key at temperature $T=0.001$ and linear size of the system $l=100$. The behavior of a single dumbbell is shown with thinner lines while the one of a system of interacting dumbbells with $\varphi =0.1$ is shown with thicker lines. The average has been taken over 200 thermal histories. The inertial and angular time scales are signaled with vertical black lines at $t_I=0.1$ and $t_a=5000$, respectively. The floating time scale $t^{*}$ is proportional to the inverse square active force, $F_{\mathrm{act}}^{-2}$. See the text for a discussion of its effect.

Figure 6

Schematic phase diagram of the dumbbell system. Inside the curve, at high Péclet numbers, the system undergoes phase separation into two phases characterized by two different densities. Here, typically, large and stable clusters are observed, as shown in the snapshot on the right. The dumbbells are frozen and point preferentially towards the center of the cluster. The enlargement shows the border of such a cluster, with the green part, corresponding to the head of the dumbbells, pointing inside, and the tails (red part) pointing outside. For small Péclet numbers the system does not show the formation of such large clusters and a typical configuration is presented. The location of the transition line is based on the results of simulations with $F_{\mathrm{act}}=1$ (see Refs. [24, 25] for further details); the right snapshot is taken at $T=0.01$ and the one on the left at $T=0.05$.

Figure 7

Clustering effects in the active many-body system. Density is $\varphi =0.4$, temperature is $T=0.05$, and from left to right and top to bottom the active forces are $F_{\mathrm{act}}=0.01,0.1,0.3,0.5,0.7,1$. The side of the box is $l=100$.

Figure 8

Clustering effects in the active many-body system. In all snapshots the active force strength is $F_{\mathrm{act}}=0.5$ and the temperature is $T=0.05$, while the densities are $\varphi =0.1,0.2,0.3,0.4$ (in reading order). The side of the box is $l=100$.

Figure 9

The structure factor of the active sample with $\varphi =0.1$ (upper panel) and $\varphi =0.4$ (lower panel) for different active force strengths given in the key. See the text for a discussion.

Figure 10

Pdf of the horizontal component of the center-of-mass dumbbell velocity, ${v_{\mathrm{c}.\mathrm{m}.}}_x$, in a system with $\varphi =0.1$ at $T=0.05$, for three values of the active force, $F_{\mathrm{act}}=0.01,0.1,1$. Data are shown in linear-log scale. The values of the kinetic temperature, $T_{\mathrm{kin}}=0.0500,0.0501,0.0596$, respectively, used in the Gaussian fit shown with continuous and dashed lines are very close to the result for a single molecule; see Eq. (35). At $\varphi =0.3$, for the same set of active forces, we have checked that the pdf is also well represented by a Gaussian distribution with $T_{\mathrm{kin}}$ given by Eq. (35).

Figure 11

The diffusion constant in the active system at $T=0.05,D_A(F_{\mathrm{act}},\varphi )$, as a function of $F_{\mathrm{act}}$ for five values of the system's surface fraction given in the key. Normal scale is used. The continuous (orange) line is the analytic expression for $\varphi =0$, see Eq. (41). The dashed (red) line is a fit of the data to the form $D_A(0,0.2)+a{F}_{\mathrm{act}}^2$ with $D_A(0,0.2)=0.00153$ obtained from the passive data at this temperature and $a=0.273$ (the $F_{\mathrm{act}}^2$ dependence found for the single molecule) for $\varphi =0.2$ (green crosses $\times$). The dashed (blue) line is a fit to the form $D_A(0,0.2)+a{F}_{\mathrm{act}}^{\alpha }$ with $D_A(0,0.2)=0.00153,a=0.259$, and $\alpha \simeq 1.56$ (the fit used in Refs. [56, 57], though $\alpha \simeq 2.3$ was found in that case for the low density used). The corrections to this fit will be studied in the next figures and discussed in the text.

Figure 12

The diffusion constant, as extracted from the mean-square displacement of the dumbbells' center of mass, as a function of $\varphi$, in a system of active molecules at $T=0.05$. Upper panel: A linear-ln scale is used due to the large variation in the absolute values of $D_A(F_{\mathrm{act}},\varphi )$. The values of $F_{\mathrm{act}}$ are given in the key. The values at $\varphi =0$, shown with larger symbols, correspond to the theoretical value of $D_A$ for a single dumbbell given by Eq. (41). The data for relatively small $\varphi$ are rather well fitted by a straight line, suggesting $ln{D}_A(F_{\mathrm{act}},\varphi )=ln{D}_A(F_{\mathrm{act}},0)-b\left(F_{\mathrm{act}}\right)\varphi$ with $D_A(F_{\mathrm{act}},0)$ the single-active-molecule diffusion constant. In the lower panel the data are normalized by the diffusion constant of the single molecule and they are presented in linear scale. Nonmonotonic behavior with respect to $F_{\mathrm{act}}$ is observed, suggesting that the fitting function $b\left(F_{\mathrm{act}}\right)$ should be nonmonotonic.

Figure 13

Upper panel: The exponential $e^{-b\left(F_{\mathrm{act}}\right)\varphi }$ for three densities given in the key. Original data are represented by open symbols while the filled symbols represent the values obtained from the fit in Eq. (49) of $\frac{D_A(F_{\mathrm{act}},\varphi )}{D_A(F_{\mathrm{act}},0)}$ performed in Fig. 12. The lines are guides to the eye obtained with a spline of the filled points. Lower panel: The correct TO description of data, $\frac{D_A(0,\varphi )}{D_A(0,0)}={[1+H\left(\varphi \right)]}^{-1}$ for $F_{\mathrm{act}}=0$ (with red solid line), confronted to the exponential approximation in Eq. (49), which in this case yields $b(F_{\mathrm{act}}=0)=2.4$ (with green dashed line). The two curves are very close to each other up to $\varphi \lesssim 0.3$.

Figure 14

The induced displacement or linear susceptibility for a system of active dumbbells at $T=0.05$ and surface fractions $\varphi =0.1$ (above) and $\varphi =0.3$ (below). The different values of the active force used are given in the key. The magnitude of the applied field is $f=0.01$. One sees a nonmonotonic dependence of $\mu (F_{\mathrm{act}},\varphi )$ on $F_{\mathrm{act}}$, similarly to what we observed in $D_A(F_{\mathrm{act}},\varphi )/D_A(F_{\mathrm{act}},0)$.

Figure 15

Upper panel: The asymptotic slope of the integrated linear response $\mu (F_{\mathrm{act}},\varphi )$ as a function of $\varphi$ for various active forces given in the key. The two curves are exponential fits to the data for $F_{\mathrm{act}}=0.01,0.1$; the prefactor is $1/\left(2\gamma \right)=0.05$. Lower panel: The ratio $\mu (F_{\mathrm{act}},\varphi )/\mu (F_{\mathrm{act}},0)$ is plotted as a function of $F_{\mathrm{act}}$ for the density values given in the key. The scale on the right side gives the values of $\mu (F_{\mathrm{act}},\varphi )$. With open symbols, the ratios between numerical data; with filled symbols the result of the exponential fit of data. The curves are nonmonotonic and the relative motility is enhanced for active forces of the order of $0.1\lesssim F_{\mathrm{act}}\lesssim 0.4$.

Figure 16

Parametric plot ${\Delta }^2\left(\chi \right)$ for three values of the activity, $F_{\mathrm{act}}=0.01,0.1,1$; two values of the surface concentration $\varphi =0.1,0.3$; and $T=0.05$. Double logarithmic representation. The effective temperature can be read from the offset in the $y$ direction of the projection of the straight line in the long-time regime after the shoulder, $ln{T}_{\mathrm{eff}}=ln{\Delta }^2-ln2\chi$.

Figure 17

Fitted values of $T_{\mathrm{eff}}$ from the parametric plot in Fig. 16 for five values of $\varphi$ given in the key, all at the temperature $T=0.05$. In this representation, all data seem to be in good agreement with the $T_{\mathrm{eff}}$ formula for a single dumbbell, Eq. (47), which is represented with the solid line. However, the actual values of $T_{\mathrm{eff}}$ are given in Table 1, where one sees that there is, though, a weak difference in the data for different $\varphi$: for $F_{\mathrm{act}}=0.1,T_{\mathrm{eff}}$ tends to increase with increasing $\varphi$, while for $F_{\mathrm{act}}=1,T_{\mathrm{eff}}$ tends to decrease with increasing $\varphi$.

Figure 18

The ratio between the finite-density effective temperature and the single-molecule-limit one as a function of the strength of the active force, $T_{\mathrm{eff}}(F_{\mathrm{act}},\varphi )/T_{\mathrm{eff}}(F_{\mathrm{act}},0)$, where $T_{\mathrm{eff}}(F_{\mathrm{act}},0)=T[1+{\text{Pe}}^2/8]$. The temperature is $T=0.05$. See the text for a discussion.