Driven two-state rotary motor ($\beta =k_0/\omega$)

A minimal bacterial flagellar motor model combines angular velocity $\omega >0$ with bound and detached probabilities $P_A+P_B=1$. With detachment rate $k_0$ in the first half-turn and attachment rate $k_0$ in the second, the periodic stationary bound probability is $P_A=\alpha e^{-\beta \theta }$ on $0\le \theta \le \pi$, where $\alpha =(1+e^{-\pi \beta }{)}^{-1}$, and $P_A(\theta +\pi )=1-P_A(\theta )$. The one-way switching windows describe externally driven kinetics rather than detailed balance with a passive heat bath.

For the driven two-state rotary motor with a triangular bound potential of height $U_0$, the torque scale is $T_0=U_0/\pi$. Differentiating the phase-averaged torque gives ${\overline{T}}^{\prime }(s)=T_0[2P_A(s)-1]/\pi$. The decreasing crossing of $P_A=1/2$ therefore gives $s_{\max }=\ln (2\alpha )/\beta$ and ${\overline{T}}_{\max }=T_0\ln \cosh (\pi \beta /2)/(\pi \beta )$. The slow-speed limit is $T_0/2$; the fast-speed limit vanishes as $T_0\pi k_0/(8\omega )$.