Mechanical Autonomous Stochastic Heat Engine

Stochastic heat engines are devices that generate work from random thermal motion using a small number of highly fluctuating degrees of freedom. Proposals for such devices have existed for more than a century and include the Maxwell demon and the Feynman ratchet. Only recently have they been demonstrated experimentally, using, e.g., thermal cycles implemented in optical traps. However, recent experimental demonstrations of classical stochastic heat engines are nonautonomous, since they require an external control system that prescribes a heating and cooling cycle and consume more energy than they produce. We present a heat engine consisting of three coupled mechanical resonators (two ribbons and a cantilever) subject to a stochastic drive. The engine uses geometric nonlinearities in the resonating ribbons to autonomously convert a random excitation into a low-entropy, nonpassive oscillation of the cantilever. The engine presents the anomalous heat transport property of negative thermal conductivity, consisting in the ability to passively transfer energy from a cold reservoir to a hot reservoir.

Figure 1

Cyclic thermal engines. (a) Stirling heat engine. The engine uses a piston to cyclically compress and expand a gas. A secondary piston displaces the gas and regulates the coupling to the hot and cold reservoirs. (b) Thermal cycle for the Stirling engine. The difference in pressure during expansion and contraction causes the gas to perform net work over a cycle (green shaded area). (c) Proposed mechanical autonomous stochastic engine, consisting of two ribbons, main and secondary (displacements $x_M$ and $x_H$, respectively), and a cantilever (displacement $x_W$). (d) Thermal cycle for the proposed engine consisting of four steps: (i) $x_w$ is at its leftmost position and energy flows from $x_H$ to $x_M$, (ii) $M_w$ moves to the right (${\dot{x}}_w>0$), while $x_M$ stays in a high-energy state, (iii) $x_w$ is at its rightmost position, and energy flows from $x_M$ to the cold bath, (iv) $x_w$ moves back to the initial position while $x_M$ stays in a low-energy state.

Figure 2

Thermal engine operation. (a) Theoretical uncoupled frequency response of the main ribbon ($x_M$), for cantilever displacements $x_W$ of $-50\mu \mathrm{m}$ (blue, solid curve), $0\mu \mathrm{m}$ (green, dashed curve), and $50\mu \mathrm{m}$ (red, dotted curve). The uncoupled frequency response of the secondary ribbon $x_H$ (thick purple curve) is shown for comparison. (b) Theoretical energy transfer $Q_H$ between the hot bath (applied to the secondary ribbon $x_H$) and the main ribbon ($x_M$), as a function of the cantilever displacement. The colored dots correspond to the curves in (a). The roman numerals indicate the step of the thermal cycle associated to each displacement and energy transfer. (c) Experimental probability distribution of $x_M$ as a function of the cantilever’s oscillation phase, $\theta ({\dot{x}}_W,x_W)$. (d) Experimental force acting on the cantilever as a function of the cantilever displacement. (e) Theoretical (blue line) and experimental (black dots) power transfer from the main ribbon to the cantilever as a function of the effective temperature of $x_H$. (f) Experimental time evolution of the cantilever (dark red) and ribbon (light blue). The green circle indicates the case $T_H=4.2\times {10}^{18}\mathrm{K}$, corresponding to the experimental conditions depicted in (c), (d), and (f).

Figure 3

Properties of the cantilever motion. (a) Experimental phase space probability distribution corresponding to the case when the frequency of the ribbons is tuned to achieve thermal engine operation at $T_H=4.2\times {10}^{18}\mathrm{K}$. (b) Theoretical phase space probability distribution for the experimental case in (a). (c) Phase space probability density function for the cantilever in the detuned system (where $f_M>f_H$; see Supplemental Material [29] for the exact values) at $T_H=2\times {10}^{18}\mathrm{K}$. (d) Theoretical prediction for the system in (c). (e) Experimental entropy of the cantilever motion as a function of the energy (blue crosses), compared to a theoretical prediction (red line) and to the maximal entropy for the given energy (green dotted line). The green circle indicates the experimental conditions used in (a), (b), and (f), as well as in Figs. 2, 2, and 2. (f) Fourier transform of the experimental cantilever motion.

Figure 4

Theoretical investigation. (a) Mass-spring model for the system. (b) Energy transfer as a function of the cantilever temperature $T_W$. (c) Efficiency of the thermal machine (blue, solid line) and comparison with the Carnot efficiency (red, dashed line). Here, the cantilever motion has been prescribed to be $x_W=(50\mu \mathrm{m})\cos ({\omega }_{W}t)$ to prevent incoherent energy transfer between the ribbons and the cantilever. (d) Refrigerator coefficient of performance ($\mathrm{C}.\mathrm{O}.\mathrm{P}.=Q_C/W$) when the cantilever is forced to oscillate at $A_W=50\mu \mathrm{m}$ (blue, solid line) and when driven by noise at $T_W=k_W{A}_W^2/k_B$ (green, dotted line). The red dashed line is the Carnot maximal C.O.P.