= Dragged-bead fluctuation theorem experiment
The 2002 experiment trapped a $6.3\,\mathrm{\mu m}$ bead in <water> with a $980\,\mathrm{nm}$ <laser>, a $100\times$, <numerical aperture> $1.3$ objective and a piezo-driven microscope stage. A quadrant <photodiode> tracked bead position at $1\,\mathrm{kHz}$ with approximately $500\,\mathrm{Hz}$ bandwidth and reported $15\,\mathrm{nm}$ resolution. Stage translation drove water past the fixed <optical trap> at $1.25\,\mathrm{\mu m/s}$. Weak trapping prolonged the transient; stationary recording before translation established the position reference. <Stokes drag law> calibration was checked against the bead's measured <diffusion coefficient>. <Experimental Demonstration of Violations of the Second Law of Thermodynamics for Small Systems and Short Time Scales by G. M. Wang and others>[Wang and colleagues, 2002, p. 050601-2: apparatus, calibration and trajectory protocol]{ref}
From calibrated <displacement> and calibrated <harmonic optical trap> <force>, the experimenters evaluated the dimensionless driven-bead dissipation functional
$$
\Sigma_t^{\rm drive}=\frac1{k_BT}\int_0^t\mathbf v_{\rm opt}\cdot\mathbf F_{\rm opt}(u)\,du.
$$
Among 540 trajectories, negative values became less frequent with duration. Figure 2 compared the negative/positive count ratio with $\langle e^{-\Sigma_t^{\rm drive}}\rangle_{\Sigma_t^{\rm drive}>0}$; its agreement tested the paper's integrated transient relation. The detailed log-ratio plot in Figure 4 used simulated particle <sample trajectories>, not bead measurements. Near-zero sign discrimination and stage-start transients limited the earliest points. <Experimental Demonstration of Violations of the Second Law of Thermodynamics for Small Systems and Short Time Scales by G. M. Wang and others>[Wang and colleagues, 2002, Eq. (2), Eq. (3), pp. 050601-2–4 and Figs. 1, 2 and 4]{ref}
For any continuous <probability distribution> satisfying the same-law relation, integration over $s>0$ yields
$$
\frac{\Pr(\Sigma<0)}{\Pr(\Sigma>0)}
=\left\langle e^{-\Sigma}\right\rangle_{\Sigma>0}.
$$
This conditional relation explains the comparison without equating it to the unconditional <integral fluctuation theorem>. The experiment supports a specified driven-system relation; it does not establish a universal same-law symmetry for any definition of <heat> or <entropy production>. In particular, a transient work-like functional should not silently acquire the actual-distribution endpoint term in <finite trajectory entropy production>. Negative realizations are compatible with positive mean dissipation and the <Second law of thermodynamics>; they do not provide a cyclic machine with reliable <work> output from a single <thermal reservoir>.
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