Extrapolating the thermodynamic length with finite-time measurements

The thermodynamic length, though providing a lower bound for the excess work required in a finite-time thermodynamic process, is determined by the properties of the equilibrium states reached by the quasistatic process and is thus beyond the direct experimental measurement. We propose an experimental strategy to measure the thermodynamic length of an open classical or quantum system by extrapolating finite-time measurements. The current proposal enables the measurement of the thermodynamic length for a single control parameter without requiring extra effort to find the optimal control scheme, and is illustrated with examples of the quantum harmonic oscillator with tuning frequency and the classical ideal gas with changing volume. Such a strategy shall shed light on the experimental design of the lacking platforms to measure the thermodynamic length.

Figure 1

Flowchart for measuring the thermodynamic length $\mathcal{L}$ with finite-time extrapolation. The excess power is evaluated with the performed work rate $\dot{W}\left(t\right)$ and the quasistatic work rate ${\dot{W}}_{\left(0\right)}\left(t\right)$ at each moment of the finite-time process.

Figure 2

Finite-time extrapolation to measure the thermodynamic length for the quantum Brownian motion in a tuned harmonic potential. With the current chosen parameters, the black dotted line presents the thermodynamic length $\mathcal{L}=0.864$. It is approached by the extrapolated functions with $N=3$ for both the linear (blue dashed curve) and the optimal (green solid curve) protocols. The inset shows the excess work in a slow process is bounded as $W_{\mathrm{ex}}\left(\tau \right)\ge {\mathcal{L}}^2/\tau$ (black dotted line), which is saturated by the optimal protocol at the long-time limit.

Figure 3

Finite-time extrapolation to measure the thermodynamic length for the compression of the classical ideal gas in a cylinder. We adopt the linear protocol (blue dashed curve) and the optimal protocol (green solid curve) to compress the volume of the cylinder to one half. With the current chosen parameters, the length extrapolated with the two protocols match the exact thermodynamic length $\mathcal{L}=0.566$ (black dotted line) by Eq. (14). The thin curves show the effect of the calibration error on the finite-time thermodynamic length, where we consider ${10}^{-5}$ calibration error to exist in the measured pressure.

Figure 4

The optimal protocols of tuning the energy spacing $\tilde{E}\left(s\right)$ of the two-level system with different bath spectral $\alpha =0.5,1$, and 2. The optimal protocols are obtained by numerically solving Eq. (B15) with the parameters ${\beta }_{\mathrm{b}}=1$ and ${\gamma }_0=1$.

Figure 5

Finite-time extrapolation for the two-level system with the tuned energy spacing. (a) Finite-time thermodynamic lengths for the linear and the optimal protocols. Through measuring the finite-time thermodynamic length with short duration (markers), the thermodynamic length can be approached by the extrapolated functions of both protocols. (b) The ${\tau }^{-1}$ scaling of the excess work. For a long-time process, the optimal protocol (green solid line) consumes less excess work and saturates the lower bound given by the thermodynamic length (black dotted line).

Figure 6

The ${\tau }^{-1}$ scaling of the excess work for tuning the frequency of the harmonic oscillator. We compare the excess work in the linear protocol (blue dashed curve) and the optimal protocol (green solid curve). The frequency is tuned from ${\omega }_0=1$ to ${\omega }_{\tau }=2$ with the temperature $k_{\mathrm{B}}{T}_{\mathrm{b}}=1$, and the dissipation rate is set as $\kappa =1$. For the slow process, the lower bound of the excess work ${\mathcal{L}}^2/\tau$ is given by the thermodynamic length with $\mathcal{L}=0.864$.

Figure 7

The ${\tau }^{-1}$ scaling of the excess work for the finite-time compression of the ideal gas. The linear protocol ${\tilde{V}}_{\mathrm{li}}\left(s\right)=V_0+(V_{\tau }-V_0)s$ and the exponential protocol ${\tilde{V}}_{\mathrm{op}}\left(s\right)=V_0{(V_{\tau }/V_0)}^s$ are considered. The parameters are set as $nR=T_{\mathrm{b}}=\gamma =1$ and $C_V=1.5$, and the volume is tuned from $V_0=1$ to $V_{\tau }=0.5$. The black dotted line shows the lower bound of the excess work $W_{\mathrm{ex}}\ge {\mathcal{L}}^2/\tau$ given by the thermodynamic length $\mathcal{L}=0.566$, saturated by the exponential protocol.

Figure 8

The measured finite-time thermodynamic length $\mathcal{L}\left(t\right)$ with the random error. The standard deviation of the random error is $\sigma ={10}^{-3}$. The solid and the dashed curves present the extrapolation functions of the optimal and the linear protocols, while the markers show the corresponding simulated results with the random errors.