Consistent thermodynamics reconstructed from transitions between nonequilibrium steady-states

Abstract

Constructing a thermodynamic framework for nonequilibrium systems remains a major challenge, as quantities such as temperature and free energy often become ambiguous when inferred solely from steady-state properties. Here we take a transformation-based approach and experimentally examine transitions between nonequilibrium steady states (NESS). Using an optically trapped microparticle driven by a tunable correlated stochastic force, we generate active-like steady states with controllable noise statistics. By abruptly changing the trap stiffness, we measure the stochastic work, heat, and entropy produced during NESS-to-NESS transformations. We identify a state-dependent effective temperature that restores the second law for these transitions, enabling the definition of a generalized work that incorporates the consequence of the nonequilibrium fluctuations. With this quantity, we derive and experimentally verify a Crooks-like fluctuation relation linking work distributions to a nonequilibrium free-energy difference defined through the effective temperature. Finally, we establish a fluctuation-response relation for the positional variance following stiffness changes. We demonstrate that this relation is key to distinguishing systems that can be described by a unique effective temperature (i.e., those under equilibrium or white-noise conditions) from those under colored-noise, where an equilibrium-like response cannot be restored. These results delineate the applicability and limits of effective-temperature thermodynamics in driven systems.

Figures (12)

• Figure 1: Colloidal particle driven out of equilibrium by colored noise in an optical trap. (a) Schematic of the experimental setup (see Appendix). (b) Vanishing external driving ($\eta = 0$) and (c) corresponding segment of an equilibrium Brownian trajectory $x(t)$ in a harmonic trap. (d) White-noise driving force (up to cut-off frequency at $2^{14}$ Hz) and (e) the associated trajectory. (f) Colored-noise driving force and (g) the corresponding trajectory. (h) Position probability distribution $P(x)$ at equilibrium (blue), under white-noise driving (green), and under colored-noise driving (red), together with the corresponding Gaussian Boltzmann-like distributions (black dashed lines). Inset: kurtosis $K=\langle x^4\rangle/\langle x^2\rangle^2$ as a function of noise correlation time; $K=3$ indicates Gaussian statistics. (i) Effective temperature extracted from the Boltzmann factor as a function of the noise correlation time $\omega_{\mathrm{c}}^{-1}$, for initial ($\kappa_{\mathrm{i}}$; orange circles and line) and final ($\kappa_{\mathrm{f}}$; dark red triangles line) trap stiffnesses. Lines show the analytical prediction [Eq. (\ref{['Eq:Teff']})] with experimental parameters.
• Figure 2: Stiffness change triggers NESS-to-NESS transformations. (a) An illustration of a step-like change of stiffness $\kappa_\mathrm{i} \rightarrow \kappa_\mathrm{f}$, along with a schematic of the resulting potential, and a conceptual analogy to a compression experiment of an ideal gas. (b) Measured time-dependent ensemble-variance $\langle x^2(t) \rangle$ (red solid line), under a stationary noise $\eta(t)$ with correlation time $\omega_\mathrm{c}^{-1} = 1$ ms; numerical simulation results (blue solid line) and analytical evolution of the variance (black dashed line; see complete expression in the appendix) using the measured values of $\kappa$.
• Figure 3: Fluctuation theorem for the generalized work. (a) Generalized work distributions for both forward transformation (compression, dark blue circles) and backward transformation (expansion, light blue triangles) for a system driven by colored noise. Solid lines corresponds to the exact results (see Appendix) using the stiffness $\kappa_{\rm i}$ and $\kappa_{\rm f}$ measured in the absence of noise (at thermal equilibrium) and temperatures $T_{\rm eff}(\kappa_\mathrm{i})$ and $T_{\rm eff}(\kappa_\mathrm{f})$. (b) Fluctuation theorem for the generalized work confirmed using the experimental data (red circles) on top of the exact result Eq. (\ref{['Eq:CFT']}) (red solid line). The expected normalized free-energy difference $\Delta (\mathcal{F}/k_{\rm B}T_{\rm eff})$ is underlined (dot-dashed vertical line). It agrees with the value of work for which the log-ratio crosses $0$, demonstrating that finite-time measurement of $\bar{w}$ allows to probe the free-energy differences (see Fig. \ref{['fig:DeltaF']} in Appendix).
• Figure 4: Fluctuation-response relation under stiffness change. (a) Verification of the variance-FRR at thermal equilibrium, showing excellent agreement between the response function (experimental data, blue solid line; analytical prediction, black dotted line) and the squared correlation function multiplied by the prefactor in Eq. (\ref{['Eq:FRR']}) (experiment, red solid line; analytical prediction, black dashed line). The time-dependent second moment and the correlation function are given by Eqs.(\ref{['eq:NeqVar']}) and (\ref{['eq:NeqCxx']}) in the Appendix, respectively. Shaded regions around the experimental curves indicate a $\chi^2$ test with a $3\sigma$ confidence interval and include the propagation of calibration uncertainties (see Appendix). Shaded regions around the analytical curves reflect uncertainty propagation from experimentally measured parameters. (b) Verification of a generalized variance-FRR under white-noise driving, using the effective temperature $T_{\rm eff}$ in the prefactor of the correlation function. (c) Violation of the variance-FRR under colored-noise driving with $\omega_{\mathrm{c}}^{-1}=2.5~\mathrm{ms}$, using $T_{\rm eff}(\kappa_{\mathrm{f}})$ in the prefactor.
• Figure 5: Simplified view of the optical trapping setup. The sphere is suspended in water inside the Sample cell inserted between the two objectives Obj1 and Obj2. The $785$ nm trapping beam is drawn in red. The $800$ nm beam used to apply radiation pressure is shown in purple. The intensity of this beam is controlled by the acousto-optic modulator (AOM). The instantaneous position of the trapped bead is probed using the auxiliary $639$ nm laser beam, drawn in orange, whose scattered signal is sent to a high-frequency photodiode.