Figure 1

(a) Schematic figure of Feynman’s ratchet composed of a sawtooth-shaped ratchet wheel and a pawl. The system is attached to two heat baths, the wheel to $T_x$ and the pawl to $T_y$. The temperature difference between $T_x$ and $T_y$ produces the nonvanishing steady rotation $J_p$, which can be used to execute external work. The variable $x$ denotes the angle of the wheel. There are degenerate stable positions of $x$ corresponding to the ratchet potential wells (there are eight in this figure). Heat flows $J_x$ and $J_y$ are defined as positive when the system absorbs energy from each heat bath. (b) Typical trajectories of $x$ in a NESS. The system exhibits directed motions, i.e., unidirectional rotations on average. The solid lines and the broken lines show the data for the conditions $2T_x=T_y$ and $T_x=2T_y$, respectively, in model II. We clearly see that the direction of the motion depends on the temperature difference.Reuse & Permissions

Figure 2

Time sequences of the averaged heat flow $J_y\left(t\right)$ over conditioned ensembles ($3.8\times {10}^5$ samples) for (a) leftward and (b) rightward step in equilibrium $(T_x=T_y=0.05)$ for model II, where the energy barrier height is about $3.1T_x$. For each step, time is shifted so that the particle crosses basin boundaries at $t=0$. Heat transfers $Q_{*}\left(t\right)$ for (c) leftward and (d) rightward steps, respectively, are obtained by the integration of $J_{*}\left(t\right)$ from $t_0=-20$ [see Eq. (3)].Reuse & Permissions

Figure 3

Hidden heat transfer $q^{\mathrm{eq}}$ at equilibrium and $l{\chi }_{\mathrm{p}}∕D_{\mathrm{p}}^{\mathrm{eq}}$. This reveals that there is a nontrivial relation (6) between the motion of the particle in NESSs and hidden heat transfer in equilibrium. Results for the two models, model I (filled square) at $\beta =3$ and model II (open circles) at $\beta =20$, are shown. For the latter model, the stiffness parameter of the model is varied to explore the model space broadly so that we have many points for the same temperature conditions.Reuse & Permissions

Figure 4

Profile of excess heat transfer in the NESS of model II. In the upper figures with $T_y=2T_x$, particle flow is leftward. In the lower figures with $2T_y=T_x$, particle flow is rightward.Reuse & Permissions