Experimental test of a new equality: Measuring heat dissipation in an optically driven colloidal system

Measurement of energy dissipation in small nonequilibrium systems is generally a difficult task. Recently, Harada and Sasa [Phys. Rev. Lett. 95, 130602 (2005)] derived an equality relating the energy dissipation rate to experimentally accessible quantities in nonequilibrium steady states described by the Langevin equation. Here, we show an experimental test of this new relation in an optically driven colloidal system. We find that this equality is validated to a fairly good extent, thus the irreversible work of a small system is estimated from readily obtainable quantities.

Figure 1

Experimental setup.

Figure 2

Switching: (a) To achieve nonequilibrium steady states (NESSs), position of the optical tweezer was switched between two sites $(\pm L∕2)$ using a piezo mounted mirror at random with Poissonian process. In the equilibrium state (EQ), the position of the tweezer was fixed. In NESS, we decided whether to switch or not at random every $4\mathrm{ms}$. One switching takes $5\mathrm{ms}$ due to the mechanical response of the mirror unit. (b) Typical trajectory for EQ and NESS $(\lambda =1.0)$. Black line is the trajectory of the optical tweezer. Gray line is the trajectory of the particle. $L=460.8\pm 0.14\mathrm{nm}$.

Figure 3

Measurement of response function: (a) We shifted the tweezer position with a distance of $d=190.2\pm 0.11\mathrm{nm}$ at $t=0$ and observed the relaxation processes to the new laser position. We repeated this and obtained an ensemble average. (b) Similarly in NESSs, we added a bias $d$ and observed the relaxation processes.

Figure 4

(Color) response function: The trajectories of the relaxation processes in EQ [equilibrium state (no switching)] and NESSs (nonequilibrium steady states) with mean switching rates $\lambda =1.0$, 2.0, 3.0, and $4.0\mathrm{Hz}$. Dashed line is the fitting curve: $⟨x\left(t\right)⟩∕d=1-\exp (-{\omega }_{c}t)$ for EQ. $d$ was $190.2\pm 0.11\mathrm{nm}$ and the cutoff frequency ${\omega }_c$ was $27.61\pm 0.05\mathrm{rad}∕\sec$. We averaged 7854 trajectories for EQ, $\lambda =1.0$, 3.0, and 4.0, respectively, and 8085 trajectories for $\lambda =2.0$.

Figure 5

(Color) $\tilde{C}\left(\omega \right)$ and $2k_{\mathrm{B}}T{\tilde{R}}^{\prime }\left(\omega \right)$: Dashed line is the response function $2k_{\mathrm{B}}T{\tilde{R}}^{\prime }\left(\omega \right)=2k_{\mathrm{B}}T{\omega }^2∕({\omega }^2+{\omega }_c^2)\gamma$. $52\text{runs}\times 69.17\mathrm{s}$ (1 703 884 points) in EQ and NESS at $\lambda =3.0$, 4.0, respectively, and $53\text{runs}\times 69.17\mathrm{s}$ (1 736 651 points) in NESS at $\lambda =1.0$ and 2.0, respectively, were averaged for $\tilde{C}\left(\omega \right)$. Moving averages were taken in $\omega$ direction.

Figure 6

(Color) (a) The left hand side (LHS) and the right hand side (RHS) of the equality (1) were plotted against the mean switching rate $\lambda$ Hz. For the RHS, we integrated only in the region $\omega <850$ because the difference between the LHS and RHS should vanish at sufficiently high frequencies. Dashed line is the theoretical curve [22]. The standard errors of the RHS were around $1k_{\mathrm{B}}T∕\mathrm{s}$ at all switching rates. (b) Validation of the equality for each $\omega$ [Eq. (3)]; $\tilde{I}\left(\omega \right)$ (blue line) vs $\gamma [\tilde{C}\left(\omega \right)-2k_{\mathrm{B}}T{\tilde{R}}^{\prime }\left(\omega \right)]$ (black line). We used the same data for $\tilde{C}\left(\omega \right)$ and ${\tilde{R}}^{\prime }\left(\omega \right)$ in Fig. 5 to calculate $\gamma [\tilde{C}\left(\omega \right)-2k_{\mathrm{B}}T{\tilde{R}}^{\prime }\left(\omega \right)]$.