Direct verification of the fluctuation-dissipation relation in viscously coupled oscillators

The fluctuation-dissipation relation, a central result in nonequilibrium statistical physics, relates equilibrium fluctuations in a system to its linear response to external forces. Here we provide a direct experimental verification of this relation for viscously coupled oscillators, as realized by a pair of optically trapped colloidal particles. A theoretical analysis, in which interactions mediated by slow viscous flow are represented by nonlocal friction tensors, matches experimental results and reveals a frequency maximum in the amplitude of the mutual response which is a sensitive function of the trap stiffnesses and the friction tensors. This allows for its location and width to be tuned and suggests the utility of the trap setup for accurate two-point microrheology.

Figure 1

A schematic diagram of the experimental setup. SC: Sample chamber; TL1: trapping laser for driving particle B1; TL2: trapping laser for driven particle B2; DL: detection laser; PBS: polarizing beam splitter cube; $\frac{\lambda }{2}$: half wave plate; DC: dichroic mirror; MO: microscope objective; CS: coverslip; BD1 and BD2: balanced detection systems based on Thorlabs photodiodes PD-EC2; M: mirror; EM: edge mirror; LIA1 and LIA2: lock-in amplifiers for B1 and B2, respectively.

Figure 2

Verification of the fluctuation-response relation $C_{ij}=(k_{B}T/\pi f){\chi }_{ij}^{\prime \prime }$ for a pair of viscously coupled colloidal particles in optical traps. The first and third panels compare the self-response and the position autocorrelation of, respectively, the driving and driven colloid, while the second panel compares their mutual response and cross-correlation. Theoretically computed correlation functions, assuming overdamped motion of the colloids and slow viscous flow in the fluid, are shown as solid lines.

Figure 3

Amplitude and phase response for driven (B2) bead for different trap stiffness ratios with the interparticle separation $0.67a$. (a) and (b) demonstrate amplitude and phase responses (with respect to driving frequency) of B2, respectively, for trap stiffness ratios of 2.5:1 (black), 5.7:1 (red), and 14.5:1 (blue) for (a) and (b). The resonance frequency in (a) is 20 Hz (black), 35 Hz (red), and 111 Hz (blue). The solid spheres denote experimental data points while the solid lines are corresponding theoretical fits.