Fluctuation-response relation for a nonequilibrium system with resolved Markovian embedding

Abstract

Fluctuation-response relations must be modified to describe nonequilibrium systems with non-Markovian dynamics. Here, we experimentally demonstrate that such relation is quantitatively recovered when the appropriate Markovian embedding of the dynamics is explicitly resolved. Using a colloidal particle optically trapped in a harmonic potential and driven out of equilibrium by a controlled colored noise, we study the response to a perturbation of the stiffness of the confining potential. While the reduced dynamics violates equilibrium fluctuation-response relations, we show that the dynamical response to the stiffness perturbation is fully determined by steady-state correlations involving the exact conjugate observable in the Markovian embedding.

Figures (5)

• Figure 1: (a) Schematic view of the experimental setup: a micron-sized particle suspended in water is optically trapped by a focused laser beam. It is subjected to a time-dependent external forcing $f_{\rm ext}(t)$, imposed by the radiation pressure of a secondary laser beam. The position $x(t)$ along the optical axis is recorded via the diffraction of an third low-power laser beam (not represented). Further details are provided in Appendix \ref{['App:Exp']} and in Refs. goerlich2022harvestingGoerlich2025FT. (b) Experimentally measured positional probability density functions $P(x)$ in thermal equilibrium (blue) and in the presence of colored noise driving (red), together with the respective analytical Gaussian distributions (black dashed line) (c) Experimentally measured bivariate distribution $p(x, \eta)$ maintained out of equilibrium by the non-reciprocal coupling between both variables, as sketched in inset.
• Figure 2: (a) Schematic representation of the perturbation applied: a sudden change of the stiffness $\kappa$ of the optical potential $U(x) = \kappa x^2/2$ trapping the particle. (c) Time-dependent second moment $\langle x^2(t) \rangle$ following a sudden increase in $\kappa$ in the absence of driving noise, together with the analytical simple exponential decay (black dashed line). (c) Time-dependent second moment in the presence of exponentially correlated noise with constant variance and correlation time, together with the analytical bi-exponential decay (black dashed line).
• Figure 3: (a) FRR relation Eq.(\ref{['Eq:EqFRR']}) for a system in thermal equilibrium, probed via a stiffness-perturbation. The response function $\mathscr{R}(t)$ (blue line) agrees with the scaled squared correlation function (dark blue dashed line). (b) The same FRR breaks in the presence of colored-noise driving as shown in Ref. Goerlich2025FT. (c) Generalized FRR Eq.(\ref{['Eq:GFRR']}) taking into account both position and noise variable: the response function $\mathscr{R}(t)$ (red line) agrees with the cross-correlation between the observable $x^2$ and its exact conjugate variable $Y$, multiplied by $\kappa_{\rm f}$ (dark red dashed line).
• Figure 4: Schematic view of the calibration of the optical potential. The trapping laser is sent through an acousto-optic modulator (AOM). The AOM is fed with a tension $V_{\rm AOM}$ which controls the intensity of the first order diffracted beam, that is sent into an high NA objective to form the optical potential. Modifying $V_{\rm AOM}$ leads to a change in the trap stiffness $\kappa$. The linear relation between $\kappa$ and $V_{\rm AOM}$ is calibrated (lower right inset), allowing to control precisely the stiffness of the potential.
• Figure 5: Experimental results (red lines) and results of numerical simulation with the experimental parameters (purple lines) in $\rm{m^2}$ for (a) the response function $\mathscr{R}(t)$, (b) the scaled autocorrelation $\langle x^2(0) x^2(t) \rangle_{\rm f}$ (c) the scaled cross-correlation $\langle x(0) \eta(0) x^2(t) \rangle_{\rm f}$ and (d) the scaled cross-correlation $\langle \eta^2(0) x^2(t) \rangle_{\rm f}$, the latter three being evaluated in the final steady-state.