Stationary and Transient Fluctuation Theorems for Effective Heat Fluxes between Hydrodynamically Coupled Particles in Optical Traps

We experimentally study the statistical properties of the energy fluxes between two trapped Brownian particles, interacting through dissipative hydrodynamic coupling, and submitted to an effective temperature difference $\mathrm{\Delta }T$, obtained by random forcing the position of one trap. We identify effective heat fluxes between the two particles and show that they satisfy an exchange fluctuation theorem in the stationary state. We also show that after the sudden application of a temperature gradient $\mathrm{\Delta }T$, the total hot-cold flux satisfies a transient exchange fluctuation theorem for any integration time, whereas the total cold-hot flux only does it asymptotically for long times.

Figure 1

(a) Average value of the four $Q_{ij}$ in the stationary regime, integrated over $\tau =0.2\mathrm{s}$, for different values of $\mathrm{\Delta }T$. (b) Average value of $Q_{11}$ and $Q_{12}$ at $\mathrm{\Delta }T=1000\mathrm{K}$ for different integration times $\tau$. For these data $k_1=3.6\mathrm{pN}/\mu \mathrm{m}$ and $k_2=3.7\mathrm{pN}/\mu \mathrm{m}$.

Figure 2

Experimental probability distribution functions (PDF) of the four $Q_{ij}$ in the stationary regime, integrated over $\tau =0.25\mathrm{s}$ for different values of $\mathrm{\Delta }T$.

Figure 3

(a) Symmetry function $\mathrm{\Sigma }(Q_{21})$ as a function of $Q_{21}$ (in $k_{B}T$ units) for different values of $\mathrm{\Delta }T$, in the stationary regime. (b) Slopes of the symmetry function as a function of $\mathrm{\Delta }T$, in the stationary regime. The main figure is for the case where $k_1=k_2=3.35\mathrm{pN}/\mu \mathrm{m}$. The inset is for the case where $k_1=4.20\mathrm{pN}/\mu \mathrm{m}$ and $k_2=2.55\mathrm{pN}/\mu \mathrm{m}$. The error bars are calculated from the uncertainties on $k_1$, $k_2$, and $\epsilon$.

Figure 4

(a) Symmetry function $\mathrm{\Sigma }(Q_1)$ as a function of $Q_1$ (in $k_{B}T$ units) for different values of the integration time $\tau$, in transient regime. The prediction is a linear function of $Q_1$ with a slope equal to $T/(T+\mathrm{\Delta }T)-1$. (b) Symmetry function $\mathrm{\Sigma }(Q_2)$ as a function of $Q_2$ (in $k_{B}T$ units) for different values of the integration time $\tau$, in the transient regime.