Extrapolating the thermodynamic length with finite-time measurements

TL;DR

This paper introduces a method to experimentally estimate the thermodynamic length in finite-time processes by extrapolating from measurements of excess power, facilitating heat engine optimization without needing optimal control schemes.

Contribution

It proposes a novel extrapolation technique to measure thermodynamic length from finite-time data, applicable to single control parameters and demonstrated with quantum and classical systems.

Findings

Thermodynamic length can be estimated from finite-time excess power measurements.

The method applies to both quantum harmonic oscillators and classical ideal gases.

It simplifies the experimental determination of thermodynamic length for heat engine optimization.

Abstract

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