Exact chemo--thermal Metropolis Brownian engine: chemical leverage, temperature-neutral stall, power optimization, and multicyclic dissipation

Abstract

We develop an exactly solvable chemo--thermal extension of the three-state Metropolis Brownian heat engine. The particle moves through the periodic energy sequence $0\to E\to 2E\to0$, performs mechanical work against a load $f$ on every forward step, interacts with two cold links and one hot link, and consumes one fuel molecule of free-energy drop $\muu$ on the hot transition. Local detailed balance gives an exact cycle affinity \begin{equation*} \mathcal A=E\left(\Tc^{-1}-\Th^{-1}\right)+\muu/\Th-f\left(2/\Tc+1/\Th\right), \end{equation*} and the full stationary probabilities and current are obtained without linear-response, weak-driving, or high-barrier approximations. Several results follow. First, the exact stall force is \begin{equation*} \fs=\frac{E(\Th-\Tc)+\Tc\muu}{2\Th+\Tc}. \end{equation*} Second, there is a temperature-neutral chemical compensation point $\muu_*=3E/2$ at which $\fs=E/2$ for every $\Th>\Tc$ and the hot and cold heats both vanish at reversible stall. Third, in both Metropolis branches the stationary current is a strictly increasing function of $\muu$ at fixed mechanical parameters, but approac