Unified geometric formalism for dissipation and its fluctuations in finite-time microscopic heat engines

Abstract

Microscopic heat engines operate in regimes where thermodynamic quantities fluctuate strongly, making stochastic effects an essential aspect of their performance. However, existing geometric formulations of finite-time thermodynamics primarily characterize average dissipation and do not systematically capture its fluctuations. Here, we develop a unified geometric framework that consistently describes both the mean dissipated availability and its fluctuations. In the linear-response regime, we show that these quantities are governed by metric tensors constructed from equilibrium correlation functions, providing a common geometric structure for dissipation and its fluctuations. This framework yields geometric bounds on both the mean and variance of the dissipated availability, and thereby on the efficiency and its fluctuations. The formalism applies broadly to stochastic systems, including Markov jump processes and overdamped and underdamped Brownian dynamics, establishing a unified geometric description across microscopic heat engines.

Figures (4)

• Figure 1: Prefactor $f_n$ as a function of $n$. The horizontal dotted line indicates the asymptotic value $f_{n\rightarrow \infty} = 1/3$. The red dots represent the numerically obtained values of $f_n$, while the black solid line shows the functional form given in Eq. (\ref{['eq:fn']}), which is in agreement with the numerical data.
• Figure 2: Path and protocols for the Brownian Carnot cycle. (a) The path on the $T$-$\lambda_w$ plane in the experiment of Ref. Martinez16. The hot and cold temperatures of the environment in the isothermal strokes are $T_h=525$K and $T_c=300$K, respectively. The minimum and the maximum values of $\lambda_w$ are $2.0$pN $\mu$m$^{-1}$ (point 1) and $20.0$pN $\mu$m$^{-1}$ (point 3), respectively. Right panels: Protocols of (b) $\lambda_w(t)$ and (c) $T(t)$ for one cycle. The red solid lines represent the protocol optimizing $\langle A \rangle$ (protocol 1), the orange dot-dashed line the one optimizing $\langle \Delta A^2 \rangle$ (protocol 2), and the black dashed lines the protocol in the experiment Martinez16.
• Figure 3: Output power $\langle W \rangle/\tau$ of the Brownian Carnot cycle as a function of the inverse of the cycle period. The lines show theoretical results obtained by numerically solving the full Fokker--Planck equation (\ref{['eq:fp_udb']}): the red solid line corresponds to protocol 1, and the black dashed line to the experimental protocol. The gray dots represent experimental data taken from Ref. Martinez16.
• Figure 4: Efficiency and its fluctuation of the Brownian Carnot cycle as functions of the inverse cycle period. The three decreasing curves with $1/\tau$ show the efficiency $\epsilon$: the red solid line corresponds to protocol 1, the orange dot-dashed line to the geometric bound $\epsilon_{\mathrm{geo}}$, and the black dashed line to the experimental protocol. The three increasing curves with $1/\tau$ show the fluctuation $\sqrt{\langle \Delta\mathcal{E}^2 \rangle}$ of the stochastic efficiency: the pink solid line corresponds to protocol 1, the blue double-dot-dashed line to the geometric bound $\sqrt{\langle \Delta\mathcal{E}^2 \rangle^{\mathrm{geo}}}$, and the black dotted line to the experimental protocol. The gray dots represent experimental data taken from Ref. Martinez16.