Fluctuation theorems with optical tweezers: theory and practice

TL;DR

This work addresses the challenge of implementing and validating fluctuation theorems in real experiments by providing a complete, open-access tutorial that uses a single-beam optical tweezer to measure non-equilibrium work and extract equilibrium free-energy differences. It combines theory (equilibrium and stochastic thermodynamics) with practical calibration (PSD-based trap stiffness and position sensitivity), finite-time forward/reverse drives, and robust estimators for $\Delta F$ and dissipated work, accounting for finite-sampling and rare-event effects. The authors demonstrate experimental verification of the Jarzynski equality and Crooks fluctuation theorem over various protocol durations and amplitudes, supported by an extensive dataset and Python code. The work serves as an educational resource and a transferable framework for soft-matter and mesoscopic systems beyond optical tweezers, promoting reproducibility and adoption of stochastic thermodynamics in curricula and new research groups.

Abstract

Fluctuation theorems, such as the Jarzynski equality and the Crooks relation, are effective tools connecting non-equilibrium work statistics and equilibrium free energy differences. However, detailed hands-on, reproducible protocols for implementing and analyzing these relations in real experiments remain scarce. This tutorial provides an end-to-end workflow for measuring, validating, and applying fluctuation theorems using a single-beam optical tweezers setup. It introduces the foundational ideas and consolidates practical calibration (PSD-based trap stiffness and position sensitivity), protocol design (forward/reverse finite-time drives over multiple amplitudes and durations), and robust estimators for free-energy difference and dissipated work, highlighting finite-sampling and rare-event effects. We demonstrate the procedures using an extensive set of measured trajectories under different conditions and provide openly accessible datasets and Python code, enabling new researchers or educators to reproduce the results with minimal effort. Beyond pedagogical validation, we discuss how these recipes translate to broader soft-matter and mesoscopic contexts. By combining user-friendly instruments with clear and transparent analysis, this work promotes the education and reliable adoption of stochastic thermodynamic methods in the curricula of physics and chemistry, as well as among emerging research teams.

Figures (18)

• Figure 1: Schematic representation comparing the work probability density for mesoscopic and macroscopic systems, with mean values indicated by dashed lines. Fluctuations are negligible in macroscopic systems, and uncertainty is limited only by measurements, while for mesoscopic systems, fluctuations are significant, with values comparable to the average work $\left\langle W\right\rangle$.
• Figure 2: Simulated trajectories of a water-immersed particle trapped at room temperature in a harmonic potential. (Top) Trajectories for a trap stiffness $\kappa_1 = 1 \ \mathrm{pN/\mu m}$ over $10 \ \mathrm{ms}$. (Bottom) Comparison of trajectories for trap stiffnesses $\kappa_1$ and $\kappa_2 = 10 \ \kappa_1$ over $1 \ \mathrm{s}$. The simulations use a time interval of $10 \ \mathrm{\mu s}$ between points with parameters: $T = 300 \ \mathrm{K}$, $\gamma = 6\pi \eta R$, $\eta = 0.001 \ \mathrm{N \cdot s \cdot m^{-2}}$, and $R = 1 \ \mathrm{\mu m}$.
• Figure 3: The basic scheme of the beam's trajectory (black lines) during the reflection and refraction, and the resulting forces are radiation pressure (orange) and trapping force (green), respectively. The gravitational force (purple) is also represented but usually is much smaller.
• Figure 4: A schematic representation of a piston showing two different volumes, analogous to a particle trapped in a harmonic potential with two corresponding trap stiffnesses.
• Figure 5: Measured harmonic potentials for different force constants $\kappa$ (acting as the control parameter $\lambda$), with $\kappa_{1}>\kappa_{2}>\kappa_{3}$. Results were obtained from position histograms of a trapped $2 \ \mathrm{\mu m}$ silica bead. The value of $\kappa$ is proportional to the laser intensity. In experiments, the compression and expansion protocols are executed by modulating the intensity.