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The excess work performed in a heat-engine process with given finiteoperation time \tau is bounded by the thermodynamic length, which measures thedistance during the relaxation along a path in the space of the thermodynamicstate. Unfortunately, the thermodynamic length, as a guidance for the heatengine optimization, is beyond the experimental measurement. We propose tomeasure the thermodynamic length \mathcal{L} through the extrapolation offinite-time measurements\mathcal{L}(\tau)=\int_{0}^{\tau}[P_{\mathrm{ex}}(t)]^{1/2}dt via the excesspower P_{\mathrm{ex}}(t). The current proposal allows to measure thethermodynamic length for a single control parameter without requiring extraeffort to find the optimal control scheme. We illustrate the measurementstrategy via examples of the quantum harmonic oscillator with tuning frequencyand the classical ideal gas with changing volume.

Extrapolating the thermodynamic length with finite-time measurements