Figure 1

(a) When the mean velocity of the piston $\overline{V}$ is zero, the net momentum transfer by the incoming particles, $\sum _{}^{}m{v}_{\mathrm{in}}$, and by the outgoing particles, $\sum _{}^{}(-m{v}_{\mathrm{out}})$, per unit time sum up to give the force on the piston.Left: Equilibrium state where no net energy is transferred to the piston. Right: Nonequilibrium case where the energy is dissipated at a rate $J_{\mathrm{diss}}^{(e)}$. When $J_{\mathrm{diss}}^{(e)}>0[<0]$, additional force $F_{\mathrm{MDD}}<0[>0]$ is exerted on the piston. (b) Cooled or warmed Brownian piston: The piston (thick bar) and object (diamond) are tightly bound and are held by a spring. The gas environments have temperatures $T$ and $T^{\prime }$ and pressures $p$ and $p^{\prime }$, respectively.Reuse & Permissions

Figure 2

(a) A microscopic “adiabatic” piston of mass $M$ (vertical bar) separates two semi-infinite gases of pointlike particles with mass $m(\ll M)$. The two gases have the same pressure $p$ but different temperatures, $T$ and $T^{\prime }$. (b) A macroscopic inelastic piston with restitution coefficients, $e$ (left surface) and $e^{\prime }$ (right surface), is in a gas.Reuse & Permissions

Figure 3

A Brownian ratchet proposed in 7 (top) consists of two triangles (total mass $M$) that translate along the horizontal axis. This model can be mapped to the adiabatic piston (bottom) in the limit, $\theta \to 0$.Reuse & Permissions