Thermodynamic optimization of thermal landscapes and energy barriers in a Brownian heat engine

Abstract

Spatial temperature fields in Brownian heat engines are commonly prescribed \emph{a priori}, and the resulting transport and thermodynamic properties are then calculated. Here we formulate the complementary inverse-design problem: determining the temperature profile and barrier height that optimize a chosen thermodynamic objective. We consider an overdamped Brownian particle in a symmetric triangular periodic potential under a constant opposing load and derive the exact stationary current and probability density for an arbitrary bounded temperature field, $\Tc\le T(x)\le\Th$. In the quasistatic limit, the efficiency becomes an exact functional of two inverse-temperature integrals over the uphill and downhill branches. Its rigorous global maximum under the pointwise temperature bounds is $η_{\max}=1-\Tc/\Th$, attained uniquely, up to sets of measure zero, by the hot-uphill/cold-downhill piecewise-constant profile. At finite current, however, the optimization changes qualitatively because the current is determined jointly by the cycle affinity and a nonlocal transport resistance. We derive the exact functional gradient and the corresponding box-constrained optimality conditions, showing that the current- or power-maximizing profile generally differs from the quasistatic efficiency optimum. For any prescribed temperature field, the current-maximizing barrier satisfies an exact balance between the marginal gain in thermal rectification and the marginal increase in transport resistance, with the characteristic estimate $U_0^*\simeq T_{\rm act}$, where $T_{\rm act}^{-1}=(2/L)\int_0^{L/2}\dd x/T(x)$.