Figure 1

Classification of particle trajectories on the basis of the initial position $x_0$ and the initial speed $v_0$. For either $v_0<v_0^{\left(1\right)}$ or $v_0>v_0^{\left(3\right)}$ the particle reaches the bottom, $x=0$, of the cylinder [red (dark gray) sectors]. For $v_0^{\left(1\right)}<v_0<v_0^{\left(2\right)}$ it neither reaches the bottom nor the piston [yellow (light gray) sector], whereas for $v_0^{\left(2\right)}<v_0<v_0^{\left(3\right)}$ it reaches the piston but not the bottom [orange (middle gray) sector].Reuse & Permissions

Figure 2

Time evolution of the distribution of work $p(W;\tau )$ for different durations $T=L_0(e^{\tau }-1)/u$, for both compressions (a) and expansions (b). The system starts in equilibrium at temperature $1/\beta =1$, the initial length of the cylinder is set $L_0=1$, the piston speed is $u=\pm 0.2$. Shown are the analytic expressions [lines, Eq. (6)] together with results of molecular dynamics simulations (circles) of ${10}^6$ trajectories each. The $\delta$ peaks at $W=0$ are not displayed.Reuse & Permissions

Figure 3

Distribution of work $p(W;\tau )$ for compressions (a) and expansions (b) starting from equilibrium at temperature $1/\beta =1$ for different speeds $u$ of the piston. The length of the cylinder is changed from $L_0=1$ to $L_{\mathrm{f}}=2$ and vice versa respectively. Shown are the analytic expressions [lines, Eq. (6)] together with results of molecular dynamics simulations (circles) of ${10}^6$ trajectories each. The $\delta$ peaks at $W=0$ are not displayed.Reuse & Permissions

Figure 4

Verification of the Jarzynski equality. The mean $\langle e^{-\beta W}\rangle$ for compressions and expansions starting from $L_0=1$ in equilibrium at temperature 1 is determined for different speeds of the piston and different final lengths of the cylinder. The results are plotted against $e^{-\beta \Delta F}$ which is proportional to the final length $L_{\mathrm{f}}$.Reuse & Permissions

Figure 5

Verification of the Crooks relation. The log-ratio of the work distribution $p\left(W\right)$ of an isothermal compression from $L_0=2$ to $L_{\mathrm{f}}=1$ and $\tilde{p}(-W)$ of the corresponding reverse process is plotted for different values of $u$. All points lie on the line $W-\Delta F$ as required by (21). For $\left|u\right|=0.02$ and $W>3$, $p\left(W\right)$ and $\tilde{p}(-W)$ are less than ${10}^{-16}$ and dominated by round-off errors. Hence these values are not shown.Reuse & Permissions

Figure 6

Variables used to describe the full cyclic process discussed in Sec. 4. $x_i$ and $v_i$ denote the position and the speed of the particle at the end of the $i$th stroke, $L_i$ is the length of the cylinder at this moment, and $W_i$ and $Q_i$ denote the work and heat respectively exchanged during the $i$th stroke.Reuse & Permissions

Figure 7

Comparison of the analytical solutions for the quasistatic case with simulations of ${10}^5$ cycles with a piston speed $u=\pm {10}^{-5}$. In the subplots, the respective joint pdfs ${\mu }_{\mathrm{cold}}(W,Q_{\mathrm{cold}})$ (a) and ${\mu }_{\mathrm{hot}}(W,Q_{\mathrm{hot}})$ (b) of the work $W$ and the heat $Q_i$ are shown in the center. The analytical solution is depicted by lines of equal probability, while each simulated cycle is represented by one black dot. The marginal probability $p_W\left(W\right)$ is shown on the top of each subplot whereas the marginal probabilities $q_i\left(Q_i\right)$ can be seen at the right-hand sides of (a) and (b). Here lines denote the analytical solutions, while the simulations are depicted as circles. The inverse temperatures are ${\beta }_{\mathrm{hot}}^{-1}=2$ and ${\beta }_{\mathrm{cold}}^{-1}=1$, and in the isothermal stroke at the higher temperature the cylinder length changes from $L_2=1$ to $L_3=2$.Reuse & Permissions

Figure 8

Distribution of work $W$ and heat $Q_{\mathrm{cold}}$ and $Q_{\mathrm{hot}}$ exchanged with the cold and the hot bath respectively during one cycle for different piston speeds $u_{\mathrm{e}}=-u_{\mathrm{c}}$. In each subplot the center plot displays the joint probability ${\mu }_{\mathrm{cold}}(W,Q_{\mathrm{cold}})$ or ${\mu }_{\mathrm{hot}}(W,Q_{\mathrm{hot}})$ respectively. The marginal work distribution $p_{\mathrm{W}}\left(W\right)$ is always shown on the top whereas the marginal distributions $q_{\mathrm{cold}}\left(Q_{\mathrm{cold}}\right)$ and $q_{\mathrm{hot}}\left(Q_{\mathrm{hot}}\right)$ are given at the right. Every black dot represents one of ${10}^5$ simulated cycles. Parameter values are ${\beta }_{\mathrm{cold}}^{-1}=1$, ${\beta }_{\mathrm{hot}}^{-1}=2$, and $u_{\mathrm{e}}=-u_{\mathrm{c}}=0.01$ (a), (b), $u_{\mathrm{e}}=-u_{\mathrm{c}}=0.1$ (c), (d), and $u_{\mathrm{e}}=-u_{\mathrm{c}}=1$ (e), (f). In all cases the length of the cylinder changes in the isothermal stroke at the higher temperature from $L_2=1$ to $L_3=2$. The solid line in (b), (d), and (f) indicates the quasistatic limit.Reuse & Permissions

Figure 9

(a) Mean total power $\langle \dot{W}\rangle$ and mean heat fluxes $\langle {\dot{Q}}_{\mathrm{cold},\mathrm{hot}}\rangle$ during the hot and the cold isothermal stroke respectively in dependence of the piston speed $u$. The simulations are conducted for $2\times {10}^5$ cycles each, with ${\beta }_{\mathrm{cold}}^{-1}=1$ and ${\beta }_{\mathrm{hot}}^{-1}=2$, and the position of the piston changing from $L_2=1$ to $L_3=2$ during the isothermal expansion at ${\beta }_{\mathrm{hot}}^{-1}$. (b) Average efficiency obtained from the same simulations. There are three regimes of operation: $\langle \dot{W}\rangle <0$ (A), $\langle \dot{W}\rangle >0$ and $\langle {\dot{Q}}_{\mathrm{hot}}\rangle >0$ (B), $\langle {\dot{Q}}_{\mathrm{hot}}\rangle <0$ (C). (c) Regimes of operations as obtained from simulations of ${10}^4$ combinations of piston speeds $u_{\mathrm{c}}$ and $u_{\mathrm{e}}$.Reuse & Permissions

Figure 10

Comparison of the efficiency at maximum power ${\eta }^{*}$ (circles) for different temperature combinations with the Curzon-Ahlborn efficiency ${\eta }_{\mathrm{CA}}$ (solid line) and the lower and upper bounds ${\eta }_{-}$ (dashed-dotted line) and ${\eta }_{+}$ (dashed line) obtained in Refs. [19, 20]. The power was optimized using simulations of ${10}^5$ cycles each for which the cylinder length again changes from $L_2=1$ to $L_3=2$ during the isothermal expansion at the higher temperature.Reuse & Permissions