Geodesic bound of the minimum energy expense to achieve membrane separation within finite time

Abstract

To accomplish a task within limited operation time typically requires an excess expense of energy, whose minimum is of practical importance for the optimal design in various applications, especially in the industrial separation of mixtures for purification of components. Technological progress has been made to achieve better purification with lower energy expense, yet little is known about the fundamental limit on the least excess energy expense in finite operation time. We derive such a limit and show its proportionality to the square of a geometric distance between the initial and final states and inverse proportionality to the operation time $\tau$. Our result demonstrates that optimizing the separation protocol is equivalent to finding the geodesic curve in a geometric space. Interestingly, we show the optimal control with the minimum energy expense is achieved by a symmetry-breaking protocol, where the two membranes are moved toward each other with different speeds.

arXiv comment6+5 pages, 2+1 figures, comments are welcome

References in corpus (8)

• Measuring thermodynamic length
• Thermodynamic metrics and optimal paths
• Thermodynamic length in open quantum systems
• Experimental validation of the $1/\tau$ -scaling entropy generation in finite-time thermodynamics with dry air
• Geodesic path for the minimal energy cost in shortcuts to isothermality
• Extrapolating the thermodynamic length with finite-time measurements
• Spatial enantioseparation of gaseous chiral molecules
• Optimizing Brownian heat engine with shortcut strategy

Cited by in corpus (3)

• Engineering Ratchet-Based Particle Separation via Shortcuts to Isothermality
• Unified approach to power-efficiency trade-off relations of generic thermal machines
• Global thermodynamic manifold for conservative control of stochastic systems