One-particle engine with a porous piston

Abstract

We propose a variation of the classical Szilard engine that uses a porous piston. Such an engine requires neither information about the position of the particle, nor the removal and subsequent insertion of the piston when resetting the engine to continue doing work by lifting a mass against a gravitational field. Though the engine operates in contact with a single thermal reservoir, the reset mechanism acts as a second reservoir, dissipating energy when a mass that has been lifted by the engine is removed to initiate a new operation cycle.

Figure 1

Schematic of the engine with porous piston. (a) The piston is initially connected to the right load mass and the particle bounces between the right thermal wall and the piston. (b) The particle pushes the piston to the left. (c) Once the piston gets to position a, the high energy load mass hanging from its right is removed and a new low energy one is placed on the right end of the container. (d) As the piston gets to point b it couples with the left load mass. Once the particle crosses to the left side through the pore it will push the piston to the right and the cycle repeats.

Figure 2

Example of the system dynamics. (a) Position along the container of the particle and the piston as a function of time. (b) Position of the piston, left and right hooks as a function of time, during a simulation of the engine where $p=0.2$ and $M_l=1.0$. The arrows in (a) point to some events where the particle crosses through the pore, while the arrows in (b) point to events where one of the hooks is pushed out of the system and reset.

Figure 3

Work and heat vs. time. Work done to lift the load mass (W) and heat exchanged between the particle and the thermal walls (Q) during a simulation of the engine where $p=0.2$ and $M_l=1.0$. (a) Single simulation. (b) Ensemble average.

Figure 4

Cycle duration $t_c$ pdf for systems with $M_l=1$ and several values of p. The inset shows the logarithm of average cycle duration time ${\overline{t}}_c$ as a function of p.

Figure 5

Cycle duration $t_c$ pdf for systems with $p=0.2$ and several values of $M_l$. The inset shows the average cycle duration time ${\overline{t}}_c$ as a function of $M_l$.

Figure 6

Work distribution at fixed load mass. (a) Pdf of the work performed by the engine on the load mass during a time window $t=40{\overline{t}}_c$ for systems with $M_l=1$ and several values of p. (b) Logarithm of the local minima of the curves shown in (a). Inset: Variance of the distribution computed from a Gaussian fit to the data around its maximum (squares) and from the numerical computation using all the work data (circles).

Figure 7

Work distribution at fixed porosity. (a) Pdf of the work performed by the engine on the load mass during a time window $t=40{\overline{t}}_c$ for systems with $p=0.2$ and several values of $M_l$. (b) Logarithm of the local minima of the curves shown in (a). Inset: Variance of the distribution computed from a Gaussian fit to the data around its maximum (squares) and from the numerical computation using all the work data (circles).

Figure 8

Heat distribution at fixed load mass. (a) Pdf of the heat exchanged between the particle and the thermal walls during a time window $t=40{\overline{t}}_c$ for systems with $M_l=1$ and several values of p. (b) Logarithm of the data shown in (a). Inset: Variance of the distribution computed from a Gaussian fit to the data around its maximum (squares) and from the numerical computation using all the heat data (circles).

Figure 9

Heat distribution at fixed porosity. (a) Pdf of the heat exchanged between the particle and the thermal walls during a time window $t=40{\overline{t}}_c$ for systems with $p=0.2$ and several values of $M_l$. (b) Logarithm of the data shown in (a). Inset: Variance of the distribution computed from a Gaussian fit to the data around its maximum (squares) and from the numerical computation using all the heat data (circles).

Figure 10

Pdf of the efficiency obtained from an ensemble of systems with $p=0.2$ and $M_l=1$. The bar plots refer to the actual efficiency per cycle of the engine, while the lines represent the efficiency pdf obtained by sampling several time windows of size t.

Figure 11

Estimation of the large deviation function. Behavior of the function $-ln(P({\eta }_t))/t$ in Eq. (11) computed at $p=0.2$ and $M_l=1.0$, using several time window sizes.

Figure 12

Macroscopic efficiency as a function of p for ensembles of systems with $M_l=1$. The solid line plots the average power as a function of p using a time window of $t=320{\overline{t}}_c$ (note the corresponding scale on the right side of the figure).

Figure 13

Macroscopic efficiency as a function of $M_l$ for ensembles of systems with $p=0.2$. The solid line plots the average power as a function of $M_l$ using a time window of $t=320{\overline{t}}_c$ (note the corresponding scale on the right side of the figure).

Figure 14

Estimation of the efficacy parameter ($\gamma$) as a function of time for a system with $p=0.2$ and $M_l=1$.