Optimizing optimal transport: Role of final distributions in finite-time thermodynamics

Performing thermodynamic tasks within finite time while minimizing thermodynamic costs is a central challenge in stochastic thermodynamics. Here, we develop a unified framework for optimizing the thermodynamic cost of performing various tasks in finite time for overdamped Langevin systems. Conventional optimization of thermodynamic cost based on optimal transport theory leaves room for varying the final distributions according to the intended task, enabling further optimization. Taking advantage of this freedom, we use Lagrange multipliers to derive the optimal final distribution that minimizes the thermodynamic cost. Our framework applies to a wide range of thermodynamic tasks, including particle transport, thermal squeezing, and information processing such as information erasure, measurement, and feedback. Our results are expected to provide design principles for information-processing devices and thermodynamic machines that operate at high speed with low energetic costs.