Efficiency at optimal performance of Brownian heat engines under double tangent constraint

Abstract

Abstract The efficiency of a steady-state thermochemical Brownian heat engine is considered, where the heat flux from the hot to the cold reservoir drives particles from the lower chemical potential of the hot reservoir to the higher chemical potential of the cold reservoir, converting heat into chemical or potential energy. This model system is used to investigate the optimal performance of a Brownian heat engine under a new constraint based on a double tangent construction between the occupation number distributions of the particles in the two heat reservoirs with temperatures {T_2}$?> T 1 > T 2 and chemical potentials μ 1 < μ 2 . This constraint allows an operation mode that forces the heat engine to attain Carnot efficiency η C = 1 − T 2 / T 1 at nonvanishing power output when approaching thermal equilibrium. For classical particles, the efficiency at maximum power under double tangent constraint results to η m p t = η C 2 / η C 2 − 1 − η C ln 1 − η C with the asymptotic series expansion η m p t =   η C − η C 2 2 + 5 η C 3 12 + O η C 4 , which shows Carnot behavior for η C toward zero. This efficiency is significantly higher than the corresponding efficiency at unconstrained maximum power η m p = η C 2 / η C − 1 − η C ln 1 − η C derived by Tu (2008 J. Phys. A Math. Theor. 41 312003) for the classical Feynman ratchet and by van den Broeck and Lindenberg (2012 Phys. Rev. E 86 041144) for classical particle transport, which has the expansion η m p = 1 2 η C + 1 8 η C 2 + 7 96 η C 3 + O η C 4 showing the Curzon–Ahlborn η C / 2 behavior for η C toward zero. The applicability of the double tangent constraint to other optimization criteria is demonstrated using the ecological criterion as an example. The method can also be applied to quantum distributions, for which, however, only a numerical solution is possible.