Recovery of mechanical pressure in a gas of underdamped active dumbbells with Brownian noise

In contrast with a gas at thermodynamic equilibrium, the mean force exerted on a wall by a gas of active particles usually depends on the confining potential, thereby preventing a proper definition of mechanical pressure. In this paper, we investigate numerically the properties of a gas of underdamped self-propelled dumbbells subject to Brownian noise of increasing intensity, in order to understand how the notion of pressure is recovered as noise progressively masks the effects of self-propulsion and the system approaches thermodynamic equilibrium. The simulations performed for a mobile asymmetric wall separating two chambers containing an equal number of active dumbbells highlight some subtle and unexpected properties of the system. First, Brownian noise of moderate intensity is sufficient to let mean forces equilibrate for small values of the damping coefficient, while much stronger noise is required for larger values of the damping coefficient. Moreover, the displacement of the mean position of the wall upon increase of the intensity of the noise is not necessarily monotonous and may instead display changes of direction. Both facts actually reflect the existence of several mechanisms leading to the rupture of force balance, which tend to displace the mean position of the wall towards different directions and display different robustness against an increase of the intensity of Brownian noise. This work therefore provides a clear illustration of the fact that driving an autonomous system towards (or away from) thermodynamic equilibrium may not be a straightforward process, but may instead proceed through the variations of the relative weights of several conflicting mechanisms.

Figure 1

(a) Schematic diagram of a dumbbell, showing the two particles located at positions ${\mathbf{R}}_{2j-1}$ (tail) and ${\mathbf{R}}_{2j}$ (head), the string connecting them, and the self-propulsion force applied to each particle and directed from the tail to the head of the dumbbell. (b) Schematic diagram of the confinement chambers. Fixed walls are shown as black solid lines and the mobile wall as red dotted lines. $x_w$ denotes the abscissa of the median line of the mobile wall. Also shown is the position ${\mathbf{R}}_k$ of a particle that has penetrated inside a fixed wall and its projection $\mathbf{p}\left({\mathbf{R}}_k\right)$ on the surface of the wall. The repelling force exerted by the wall on this particle is proportional to $\parallel {\mathbf{R}}_k-\mathbf{p}\left({\mathbf{R}}_k\right)\parallel$. The force constants associated with the repulsion potential on the left side of the mobile wall ($h_L$) and the right side of the mobile wall ($h_R$) are different.

Figure 2

Evolution, as a function of the damping coefficient γ, of the root mean square translational velocity $\sqrt{\langle v_{\mathrm{t}}^2\rangle }$ of a single dumbbell enclosed in a single confinement chamber for $v_0=2$ and 14 values of $v_{\mathrm{B}}$ ranging from 0.0 to 2.0, as well as for the system without self-propulsion ($v_0=0$ and $v_{\mathrm{B}}=2$); see the legend. Note that the four plots for values of $v_{\mathrm{B}}$ ranging from 0.0 to 0.05 nearly superpose. The plots shown here were obtained for $h_w=4.0$, but the plots obtained for $h_w=0.4$ are almost identical. Each plot was obtained by integrating the equations of motion for $2.0\times {10}^{12}$ time steps, with γ increasing regularly between 0 and 1, and averaging $\sqrt{\langle v_{\mathrm{t}}^2\rangle }$ over $2.0\times {10}^9$ successive steps.

Figure 3

Representative trajectories of the active dumbbell without Brownian noise ($v_{\mathrm{B}}=0$, left column) and with Brownian noise ($v_{\mathrm{B}}>0$, right column) for the modified model with a single confinement chamber and (a) $h_w=4.0$ and $\gamma =0.21$, (b) $h_w=0.4$ and $\gamma =0.52$, and (c) $h_w=0.4$ and $\gamma =1.00$. The confinement chamber moves towards the right each time it is hit by the dumbbell. Represented in this figure are only its initial (in blue) and final (in red) positions. Each trajectory is integrated for 1000 time units.

Figure 4

Evolution, as a function of the damping coefficient γ, of the mean relative position of the mobile wall $\langle x_w\rangle /L_x$ for $N=500$ active dumbbells obeying Eq. (3) obtained from simulations with (a) $v_0=2$ and values of $v_{\mathrm{B}}$ ranging from 0 to 2 (see the legend), or (b) $v_0=0$ and $v_{\mathrm{B}}=2$. The plot in (b) corresponds to a system at thermodynamic equilibrium and those in (a) to systems out of thermodynamic equilibrium. Each plot was computed from a single simulation integrated for $5.0\times {10}^9$ steps, with γ increasing regularly from 0 to 1 and $\langle x_w\rangle$ being averaged over $5.0\times {10}^7$ successive steps.

Figure 5

Evolution, as a function of the damping coefficient γ, of the mean relative position of the mobile wall $\langle x_w\rangle /L_x$ for $N=50$ active dumbbells obeying Eq. (3) with $v_0=2$ and ten different values of $v_{\mathrm{B}}$ ranging from 0 to 2 (see the legend). Each plot was computed from the average of eight different simulations integrated for ${10}^{10}$ steps, with γ increasing regularly from 0 to 1 and $\langle x_w\rangle$ being averaged over ${10}^8$ successive steps. See Fig. 14 for a graph showing the same results plotted as a function of $\gamma /v_B^2$ instead of γ and Sec. 4 for a discussion of the choice of the abscissa axis.

Figure 6

Evolution, as a function of the mean random velocity $v_{\mathrm{B}}$, of the mean relative position of the mobile wall $\langle x_w\rangle /L_x$ for $N=50$ active dumbbells obeying Eq. (3) with $v_0=2$ and five different values of the damping coefficient γ ranging from 0.13 to 1.0 (see the legend). Each plot was computed from the average of eight different simulations integrated for ${10}^{10}$ steps, with $v_{\mathrm{B}}$ increasing regularly from 0 to 2 and $\langle x_w\rangle$ being averaged over ${10}^8$ successive steps.

Figure 7

Evolution of $\mathrm{\Delta }{v}_w$ as a function of the damping coefficient γ for $v_0=2$ and values of $v_{\mathrm{B}}$ ranging (a) from 0.0 to 0.3, and (b) from 0.5 to 2.0; see the legend. Also shown as a red solid line in (b) is the plot obtained for the system without self-propulsion, that is, for $v_0=0$ and $v_{\mathrm{B}}=2$. These plots were obtained for the modified model with a single confinement chamber and a single active dumbbell enclosed therein. $\mathrm{\Delta }{v}_w$ is the difference between the values of the mean displacement of the chamber towards the right per unit time obtained for $h_w=4.0$ and $h_w=0.4$. See the caption of Fig. 2 for computational details.

Figure 8

Evolution of $f/\sqrt{\langle v_{\mathrm{t}}^2\rangle }$ as a function of the damping coefficient γ for $h_w=0.4, v_0=2$ and values of $v_{\mathrm{B}}$ ranging (a) from 0.0 to 0.3, and (b) from 0.5 to 2.0; see the legend. Also shown as a red solid line in (b) is the plot obtained for the system without self-propulsion, that is, for $v_0=0$ and $v_{\mathrm{B}}=2$. These plots were obtained for the modified model with a single confinement chamber and a single active dumbbell enclosed therein. $f$ is the number of times the dumbbell hits the right wall per unit time and $\sqrt{\langle v_{\mathrm{t}}^2\rangle }$ the root mean square translational velocity of the dumbbell (see Fig. 2). The horizontal gray dot-dashed line is just a guideline for the eyes. See the caption of Fig. 2 for computational details.

Figure 9

Same as Fig. 8, but for $h_w=4.0$ instead of $h_w=0.4$.

Figure 10

Probability density $p\left(\theta \right)$ for the velocity vector of the center of mass of the dumbbell to be oriented with an angle θ with respect to the $x$ axis for $\gamma =0.70$, values of $v_{\mathrm{B}}$ in the range $0\le v_{\mathrm{B}}\le 0.3$, and (a) $h_w=0.4$ or (b) $h_w=4.0$. The plots are symmetric with respect to the $\theta =0$ axis. They were obtained for the modified model with a single confinement chamber and a single active dumbbell enclosed therein. Each plot was computed by integrating the equations of motion for $3.0\times {10}^{11}$ time steps, with $v_{\mathrm{B}}$ increasing regularly between 0 and 0.3, and averaging $p\left(\theta \right)$ over ${10}^9$ successive steps for each bin.

Figure 11

Evolution, as a function of the damping coefficient γ, of the mean relative position of the mobile wall $\langle x_w\rangle /L_x$ for $N=500$ active dumbbells obeying Eq. (3) with $v_0=2$ and $v_{\mathrm{B}}=0$. The solid red line was obtained for a wall mass $m_w=2$, as all other simulations discussed in this paper, while the dashed blue line was obtained for $m_w=20$. Each plot was computed from the average of five simulations integrated for $5.0\times {10}^9$ steps, with γ increasing regularly from 0 to 1 and $\langle x_w\rangle$ being averaged over $5.0\times {10}^7$ successive steps. The two plots run almost parallel to each other for values of γ in the range $0.35\le \gamma \le 1.0$ (not shown).

Figure 12

Evolution, as a function of the damping coefficient γ, of the mean relative position of the mobile wall $\langle x_w\rangle /L_x$ for $N=50$ active dumbbells obeying Eq. (3) with $v_0=2$ and $v_{\mathrm{B}}=0$. The solid red line was obtained for $h=4$ in the expression of $V_s$ in Eq. (2), as all other simulations discussed in this paper, while the dashed blue line was obtained for $h=40$. Each plot was computed from the average of eight different simulations integrated for ${10}^{10}$ steps, with γ increasing regularly from 0 to 1 and $\langle x_w\rangle$ being averaged over ${10}^8$ successive steps.

Figure 13

Evolution, as a function of the damping coefficient γ, of the mean relative position of the mobile wall $\langle x_w\rangle /L_x$ for $N=50$ active dumbbells obeying Eq. (3) with $v_0=2$ and $v_{\mathrm{B}}=0$. The solid red line was obtained for $L_x=L_y=100$, as all other simulations discussed in this paper, while the dashed blue line was obtained for $L_x=50$ and $L_y=200$. Each plot was computed from the average of eight different simulations integrated for ${10}^{10}$ steps, with γ increasing regularly from 0 to 1 and $\langle x_w\rangle$ being averaged over ${10}^8$ successive steps.

Figure 14

Same as Fig. 5, except that $\langle x_w\rangle /L_x$ is plotted as a function of $\gamma /v_B^2$ instead of γ. The horizontal axis is consequently proportional to the Péclet number $\mathrm{Pe}=a{v}_0/D=2\gamma a{v}_0/v_B^2$, that is, the dimensionless swimming speed. It is natural to use $\gamma /v_B^2$ instead of γ as the horizontal axis when considering that the model actually consists of Brownian dumbbells perturbed by self-propulsion, rather than self-propelled dumbbells perturbed by Brownian noise. For a given value of $\gamma /v_B^2$, the effects of inertia increase with decreasing value of $v_B$.