Microscopic Analysis of a Thermal Brownian Motor

We study a genuine Brownian motor by hard disk molecular dynamics and calculate analytically its properties, including its drift speed and thermal conductivity, from microscopic theory.

Figure 1

(a) Schematic representation of the Smoluchowski-Feynman ratchet. (b) Similar construction without a pawl and a spring. (c) Two-dimensional analog referred to as the $AB$ motor. The motor is constrained to move along the horizontal $x$ direction (without rotation or vertical displacement). The host gas consists of hard disks whose centers collide elastically with the engine parts. The shape of the arrow is determined by the apex angle $2{\theta }_0$ and the vertical cross section $S$. Periodic boundary conditions are used in the computer simulations. (d) A symmetric construction referred to as the $AA$ motor.

Figure 2

Probability density $P(x,t)$ for the position $x$ of motor $AB$ at times $t=1000$ (shaded) and $t=4000$ (open). Left inset: mean square displacement versus time. Right inset: velocity correlation function.

Figure 3

Upper panel: position of the motor as a function of time. The thin solid curve shows a typical trajectory. All other curves represent the average $\langle x(t)\rangle$. The thick solid line is the equilibrium case ($T_1=T_2=1$) for motor $AB$. The dotted and dashed curves correspond to the nonequilibrium situation for, respectively, motor $AA$ and $AB$ ($T_1=1.9$, $T_2=0.1$). The situation with switched temperatures for $AB$ is the dashed curve with a negative velocity. Lower panel: average velocity of motor $AB$ as a function of its mass $M$. Inset: average speed (average over 2000 runs) of motors $AA$ and $AB$ as a function of the initial temperature difference $T_1-T_2$ ($T_1+T_2=2$ fixed). The theoretical results (7, 8) predict lower speeds than the simulations. However, when the magnitude is scaled, the theoretical curves (dotted lines) fit well with the simulations.

Figure 4

Exponential decay of the temperatures to a final common value ($T_{\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}}=(T_1+T_2)/2=1$.) in motor $AB$ and concomitant disappearance of the average drift speed. To enlarge the conductivity, a small mass $M=1$, a large vertical cross section $S=10$, and a large apex angle ${\theta }_0=\pi /9$ are used.