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To accomplish a task within limited operation time typically requires anexcess expense of energy, whose minimum is of practical importance for theoptimal design in various applications, especially in the industrial separationof mixtures for purification of components. Technological progress has beenmade to achieve better purification with lower energy expense, yet little isknown about the fundamental limit on the least excess energy expense in finiteoperation time. We derive such a limit and show its proportionality to thesquare of a geometric distance between the initial and final states and inverseproportionality to the operation time $\tau$. Our result demonstrates thatoptimizing the separation protocol is equivalent to finding the geodesic curvein a geometric space. Interestingly, we show the optimal control with theminimum energy expense is achieved by a symmetry-breaking protocol, where thetwo membranes are moved toward each other with different speeds.

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