Energy versus Information Based Estimations of Dissipation Using a Pair of Magnetic Colloidal Particles

Using the framework of stochastic thermodynamics, we present an experimental study of a doublet of magnetic colloidal particles that is manipulated by a time-dependent magnetic field. Because of hydrodynamic interactions, each bead experiences a state-dependent friction, which we characterize using a hydrodynamic model. In this work, we compare two estimates of the dissipation in this system: the first one is energy based since it relies on the measured interaction potential, while the second one is information based since it uses only the information content of the trajectories. While the latter only offers a lower bound of the former, we find it to be simple to implement and of general applicability to more complex systems.

Figure 1

Sketch of the experiment showing the two magnetic beads (top) and the time dependence of the applied magnetic field $B(t)$ (bottom). The protocol is composed of driving phases of duration $\tau$ followed by pauses of duration ${\tau }_{\text{eq}}$. The open circles represent two symmetric points at times $t$ and $\tau -t$ in this protocol.

Figure 2

Probability distribution function (pdf) of the relative distance between the beads for two values of the magnetic field, namely $B=0.3\mathrm{mT}$ (circles) and $B=0.45\mathrm{mT}$ (squares). In inset (i), the pdf of the angle $\theta$ is shown for these two magnetic fields with corresponding symbols, and in (ii), the measurements of the radial diffusion coefficient $D_r(r)$ (symbols) are shown as a function of the distance between the beads $r$ (in unit $\mu \mathrm{m}$), together with the theoretical prediction using Eq. (3) (solid line).

Figure 3

Probability distribution of the work $P(W)$, constructed from an experiment using 460 cycles (histogram) and from simulations of Eqs. (2) and (3) with state-dependent diffusion coefficient (red solid line) and with a constant diffusion coefficient $D=0.03\mu {\mathrm{m}}^2{\mathrm{s}}^{-1}$ (blue dashed line). In this experiment, $\tau =2\mathrm{s}$, ${\tau }_{\text{eq}}=4\mathrm{s}$ and the sampling frequency is 40 Hz. The inset shows the verification of the Crooks relation, namely the straight solid line, together with the experimental data (squares) and the simulation results (circles).

Figure 4

(a) Energy-based estimate of dissipation expressed in units of $k_{B}T$ (horizontal solid line) compared with an information-based estimate based on Eq. (7) (symbols) versus time $t$. The three estimators correspond to discarding the points where a log-divergence occurs (blue diamonds), using an adapted binning (black circles), or adding a small bias equal to $1/460$ to prevent log divergences (red squares). (b) Energy-based estimate of the dissipated work (black solid line) compared with an information-based estimate based on Eq. (8) (red squares).