Rectification of thermal fluctuations in ideal gases

We calculate the systematic average speed of the adiabatic piston and a thermal Brownian motor, introduced by C. Van den Broeck, R. Kawai and P. Meurs [Phys. Rev. Lett. 93, 090601 (2004)], by an expansion of the Boltzmann equation and compare with the exact numerical solution.

Figure 1

The adiabatic piston.

Figure 2

Stationary average speed of the adiabatic piston according to Eq. (21). Top: For ${\rho }_1=0.25$ and $T_1=2.0$ fixed, the speed is shown as a function of $T_2$. Note that ${\rho }_2$ is determined by the condition of mechanical equilibrium $({\rho }_1{T}_1={\rho }_2{T}_2)$ and that the piston always moves to the higher temperature region. Bottom: the speed of the piston as a function of $T_1$ and $T_2$, for ${\rho }_1=0.25$ and ${\rho }_2={\rho }_1{T}_1∕T_2$. The velocity vanishes when $T_1=T_2$ and is maximal for a large temperature difference. The following parameter values were used: mass of the gas particles $m=1$, mass of the piston $M=100$ and $k_B=1$ by choice of units.

Figure 3

A closed and convex object with total circumference $S$. The length of the surface with an orientation between $\theta$ and $\theta +d\theta$ is $SF\left(\theta \right)d\theta$, defining the form factor $F\left(\theta \right)$.

Figure 4

Four different realizations of the thermal Brownian motor, each consisting of two rigidly linked units. The units are simple convex objects: a bar of length $L$, a disk of radius $R$ or an isosceles triangle with apex angle $2{\theta }_0$ and base $L$. (a) was introduced in [16] and will be referred to here as Triangula.

Figure 5

Average speed of Triangula according to Eq. (B4). Top: the dependence of the velocity on the densities ($T_1=1.0$ and $T_2=5.0$). Bottom: The stationary speed increases with the temperature difference $({\rho }_1={\rho }_2=0.0022)$. The following parameter values were used: mass of the gas particles $m=1$, total mass of the motor $M=100$, apex angle of the triangles $2{\theta }_0=\pi ∕18$ and $k_B=1$ by choice of units.