Power output for a nonlinear Brownian machine

We propose a method that makes use of the nonlinear properties of some hypothetical microscopic solid material as the working substance for a microscopic machine. The protocols used are simple (step and elliptic) and allow us to obtain the work and heat exchanged between machine and reservoirs. We calculate the work for a nonlinear single-particle machine that can be treated perturbingly. We obtain the instantaneous work and heat for the machine undergoing cycles that mimic the Carnot and multireservoir protocols. The work calculations are then extended to high values of the nonlinear parameter yielding the quasistatic limit, which is verified numerically. The model we propose is fluctuation driven and we can study in detail its thermostatistics, namely, the work distribution both per cycle and instantaneous and the corresponding fluctuation relations.

Figure 1

Brownian machine: gas and piston.

Figure 2

Brief representation of the cycle: (a) step cycle and (b) elliptic cycle. For the step cycle, compression phase $L_2\to L_1$ is performed while the system is in contact with the hot heat bath $T_H$. We then instantly switch to a cold bath $T_C$ following by an expansion phase $L_1\to L_2$. At the end we switch the heat bath back to the hot bath. For the elliptic cycle both $T$ and $L$ change continuously, with the system absorbing heat when the temperature increases (labeled 1) and rejecting heat when the temperature decreases (labeled 2).

Figure 3

The black solid line represents the numerical integration (17), using our analytical expansions for the step cycle. The blue and violet dotted lines and the orange and red dot-dashed lines depict the expansions of Eq. (17) for large and small $k_4$ using different numbers of terms, respectively. The values used are $\mathrm{\Delta }T=\mathrm{\Delta }L=m=k_2=k_L=1$ and $L_m=T_m=1.5$.

Figure 4

Average work performed by the machine $-\mathcal{W}$ vs $k_4$ with $m=k_2=k_L=\gamma =1, L_m=3/2, \mathrm{\Delta }L=1$, and $\mathrm{\Omega }=\pi /100$. The blue solid line corresponds to the step cycle with $T_H=2$ and $T_C=1$, whereas the green dot-dashed line depicts the analytical results assuming the local equilibrium approach, which yields a maximum at $k_4=1.47$. The red dashed line corresponds to the elliptic analog of the step cycle with $T_m=3/2$ and $\mathrm{\Delta }T=\pi /2$. The orange dotted line corresponds to the step cycle, but with $T_H=3/2+\pi /4$ and $T_C=3/2-\pi /4$.

Figure 5

Step cycle with $T_H=2, T_C=1, m=k_2=k_L=\gamma =1, k_4=3/2, L_m=3/2, \mathrm{\Delta }L=1$, and $\mathrm{\Omega }=\pi /100$: (a) distribution of the instantaneous work $p\left(w\right)$ vs $w$ in log-lin scale and in lin-lin scale in the inset and (b) fluctuation relation $p(-w)/p\left(w\right)$ vs $w$ in log-lin scale.

Figure 6

Distribution of the instantaneous work $p\left(w\right)$ vs $w$ in log-lin scale and in lin-lin in the inset for the elliptic cycle with $T_m=3/2, \mathrm{\Delta }T=\pi /2, m=k_2=k_L=\gamma =1, k_4=3/2, L_m=3/2, \mathrm{\Delta }L=1$, and $\mathrm{\Omega }=\pi /100$.

Figure 7

Standard deviation for the work ${\sigma }_W$ vs $k_4$ for the step cycle (blue solid line) and for the elliptic cycle (red dashed line). For both cases, the curves have a maximum at $k_4=0$ and tend to zero as the nonlinear spring stiffens, as predicted by the theory.

Figure 8

Distribution of the work per cycle $p\left(W\right)$ vs $W$ in log-lin scale for $m=k_2=k_L=\gamma =1, k_4=3/2, L_m=3/2, \mathrm{\Delta }L=1, \mathrm{\Omega }=\pi /100$ and (a) the step cycle with $T_H=2$ and $T_C=1$ and (b) the elliptic cycle with $T_m=3/2$ and $\mathrm{\Delta }T=\pi /2$.

Figure 9

Fluctuation relations $p(-W)/p\left(W\right)$ vs $W$ of the distributions presented in Fig. 8 for (a) the step cycle and (b) the elliptic cycle. The red solid lines test Eq. (28) using the values of $\mathcal{W}$ and ${\sigma }_W$ obtained by the numerical calculations.