Work fluctuation relation of an active Brownian particle in a viscoelastic fluid

We experimentally investigate the work fluctuations of an active Brownian particle (ABP) during its self-propelled motion in a viscoelastic medium. Under such conditions, ABPs display a persistent circular motion which allows the determination of the orientational work fluctuations along its trajectory. Due to the nonlinear coupling to the non-Markovian bath, we find strong deviations from the work fluctuation theorem (WFT) due to observed increased rotational ABP dynamics. Taking this enhanced rotational diffusion into account, the orientational work distributions can be recasted to be in accordance with the WFT by considering an effective temperature of about two orders of magnitude larger than $k_{\mathrm{B}}T$. This approach is confirmed by the good agreement of the torque exerted by the viscoelastic bath on the ABP obtained from the WFT with the value obtained from the mean angular velocity and the friction coefficient of the ABP.

Figure 1

(a) Typical trajectory of a spherical ABP in a viscoelastic fluid displaying a persistent circular motion with angular velocity $\omega =0.025\mathrm{rad}{\mathrm{s}}^{-1}$. The red arrows denote the particle orientation. Inset: optical image of a single ABP with the dark side corresponding to the carbon coating. The particle orientation is defined by the angle between the vector $\mathbf{n}$, which points from capped to the uncapped side, and the $x$ axis. (b) Time dependence of the accumulated particle orientation showing a linear behavior $\omega t$. Inset: Angular mean squared displacements $\langle \mathrm{\Delta }\theta {\left(t\right)}^2\rangle$ of an ABP (dash-dotted line) and of a Brownian particle (dotted line). Below the stress relaxation time ${\tau }_0$ the behavior becomes subdiffusive due to the elastic properties of the solvent.

Figure 2

Probability density functions of the work $W_{\tau }$ for spherical active particles moving in viscoelastic fluid with different angular velocities (a) $\omega =0.018\mathrm{rad} {\mathrm{s}}^{-1}$ and (b) $\omega =0.095\mathrm{rad} {\mathrm{s}}^{-1}$. The symbols correspond to the integration times $\tau =0.06\mathrm{s}$ (circles), 0.59 s (diamonds), 2.94 s (triangles), and 5.88 s (squares). (c) $\text{ln}[P(+W_{\tau })/P(-W_{\tau })]$ as a function of $W_{\tau }$ computed for integration time $\tau =5.88\mathrm{s}$ for the active particles moving in circles in the viscoelastic fluid at angular velocities $\omega =0.018\mathrm{rad}{\mathrm{s}}^{-1}$ (diamonds), $0.025\mathrm{rad}{\mathrm{s}}^{-1}$ (circles), and $0.095\mathrm{rad}{\mathrm{s}}^{-1}$ (squares). The dash-dotted line corresponds to the work fluctuation theorem.

Figure 3

(a) The angular mean squared displacement $\langle \delta \mathrm{\Delta }\theta {\left(t\right)}^2\rangle$ of the angular fluctuations $\delta \theta$ (solid curve) obtained by subtracting the angular drift $\omega t$ from the time evolution of angular coordinate $\theta$ for an ABP rotating with an angular velocity $\omega =0.025\mathrm{rad}{\mathrm{s}}^{-1}$ and the corresponding angular mean squared displacement of a Brownian particle in the same viscoelastic fluid (dotted curve). Inset: The time evolution of the angular fluctuations $\delta \theta$ around the mean circular path for the ABP rotating with an angular velocity $\omega =0.025\mathrm{rad}{\mathrm{s}}^{-1}$. (b) $\text{ln}[P(+W_{\tau }^{*})/P(-W_{\tau }^{*})]$ vs $W_{\tau }^{*}$ of the PDF of the normalized work $W_{\tau }^{*}$ for different angular velocities: $\omega =0.018\mathrm{rad}{\mathrm{s}}^{-1}$ (diamonds), $0.025\mathrm{rad}{\mathrm{s}}^{-1}$ (circles), and $0.095\mathrm{rad}{\mathrm{s}}^{-1}$ (squares). The dash-dotted line corresponds to the work fluctuation theorem.

Figure 4

(a) $\text{ln}[P\left(\mathrm{\Delta }{\theta }_{\tau }\right)/P(-\mathrm{\Delta }{\theta }_{\tau })]$ of the PDFs of angular change $\mathrm{\Delta }{\theta }_{\tau }/k_{\mathrm{B}}{T}^{*}$ for the ABP in the viscoelastic fluid, calculated for the integration time $\tau =5.88\mathrm{s}$ for different angular velocities $\omega =0.018\mathrm{rad}{\mathrm{s}}^{-1}$ (diamonds), $0.025\mathrm{rad}{\mathrm{s}}^{-1}$ (circles), and $0.095\mathrm{rad}{\mathrm{s}}^{-1}$ (squares). The solid lines are the linear fits to the corresponding curves. (b) The measured torque acting on spherical Janus particles using the normalized fluctuation relation as a function of angular velocity $\omega$. (c) Comparison of torques measured using fluctuation relation $T_{\omega }$ with the mean estimated values $\langle T_{\omega }\rangle$ computed using Eq. (3). The dashed line corresponds to line with slope unity.