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Geodesic path for the optimal nonequilibrium transition: Momentum-independent protocol

Accelerating controlled thermodynamic processes requires an auxiliary Hamiltonian to steer the system into instantaneous equilibrium states. An extra energy cost is inevitably needed in such finite-time operation. We recently develop a geodesic approach to minimize such energy cost for the shortcut to isothermal process. The auxiliary control typically contains momentum-dependent terms, which are hard to be experimentally implemented due to the requirement of constantly monitoring the speed. In this work, we employ a variational auxiliary control without the momentum-dependent force to approximate the exact control. Following the geometric approach, we obtain the optimal control protocol with variational minimum energy cost. We demonstrate the construction of such protocol via an example of Brownian motion with a controllable harmonic potential.