Dragged-bead fluctuation theorem experiment

The 2002 experiment trapped a $6.3\mu \mathrm{m}$ 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 $1.25\mu m/s$. 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. ^[ref]

From calibrated displacement and calibrated harmonic optical trap force, the experimenters evaluated the dimensionless driven-bead dissipation functionalAmong 540 trajectories, negative values became less frequent with duration. Figure 2 compared the negative/positive count ratio with $⟨e^{-{\Sigma }_t^{\mathrm{drive}}}{⟩}_{{\Sigma }_t^{\mathrm{drive}}>0}$; 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. ^[ref]

For any continuous probability distribution satisfying the same-law relation, integration over $s>0$ yieldsThis 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.