Theoretical description of effective heat transfer between two viscously coupled beads

We analytically study the role of nonconservative forces, namely viscous couplings, on the statistical properties of the energy flux between two Brownian particles kept at different temperatures. From the dynamical model describing the system, we identify an energy flow that satisfies a fluctuation theorem both in the stationary and in transient states. In particular, for the specific case of a linear nonconservative interaction, we derive an exact fluctuation theorem that holds for any measurement time in the transient regime, and which involves the energy flux alone. Moreover, in this regime the system presents an interesting asymmetry between the hot and cold particles. The theoretical predictions are in good agreement with the experimental results already presented in our previous article [Imparato et al., Phys. Rev. Lett. 116, 068301 (2016)], where we investigated the thermodynamic properties of two Brownian particles, trapped with optical tweezers, interacting through a dissipative hydrodynamic coupling.

Figure 1

Schematic representation of the two particles trapped by optical tweezers, separated by a distance $d$ along the $x$ axis.

Figure 2

Experimental heat rates $q_1$ and $q_2$. (a) In the case $k_1=k_2$ ($\approx 3.35\mathrm{pN}/\mu \mathrm{m}$); (b) in the case $k_1\ne k_2$ ($k_1\approx 4.20\mathrm{pN}/\mu \mathrm{m}$ and $k_2\approx 2.55\mathrm{pN}/\mu \mathrm{m}$). The dashed lines are the theoretical predictions obtained by substituting the corresponding experimental values in Eqs. (24) and (25).