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≤ T(x)≤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 ηmax=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 U0*≃ Tact, where Tact−1=(2/L)∫0L/2dd x/T(x).
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BibTeX
@misc{taye2026thermodynamic,
title = {{Thermodynamic optimization of thermal landscapes and energy barriers in a Brownian heat engine}},
author = {Mesfin Taye},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02613},
eprint = {2609.02613},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02613},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Thermodynamic optimization of thermal landscapes and energy barriers in a Brownian heat engine
AU - Mesfin Taye
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02613
UR - https://arxiv.org/abs/2609.02613
AB - 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≤ T(x)≤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^*≃ T_act, where T_act^−1=(2/L)∫_0^L/2dd x/T(x).
ER -