Non-monotonic Irreversibility in Polytropic Steering

TL;DR

The paper develops finite-time polytropic steering for Brownian particles, providing an exact bridge between isothermal ($\xi=0$) and adiabatic ($\xi=-1$) driving via the invariant $\theta(t)\lambda^{\xi}(t)=\mathrm{const}$ and explicit control of $\lambda(t)$ for arbitrary durations. It reveals a non-monotonic dependence of irreversible entropy generation on driving time, with a most-irreversible timescale beyond which rapid driving suppresses dissipation. By mapping to Brownian polytropic cycles, the authors derive power–efficiency trade-offs and show that the polytropic index $\xi$ acts as a thermodynamic knob to tailor work-heat partition. The framework provides a versatile blueprint for high-speed, high-performance micro-thermodynamic machines and has potential extensions to quantum, active-matter, and information-thermodynamics settings.

Abstract

The efficient manipulation of thermodynamic states within the finite time is fundamentally constrained by the intrinsic dissipative cost. While the slow-driving regime is well-characterized by a universal $1/τ$-scaling of irreversibility, the physics governing fast, non-adiabatic transitions remains elusive. Here, we propose the polytropic steering protocols that provide an exact analytical bridge between the isothermal and adiabatic limits for Brownian particles far-from-equilibrium. We demonstrate that for any protocol duration $τ$, the system can be precisely steered along a prescribed polytropic trajectory, revealing a striking non-monotonic dependence of irreversibility on the driving rate. Contrary to the near-equilibrium paradigm where faster driving necessitates higher energetic costs, we identify a most-irreversible timescale, beyond which dissipation is anomalously suppressed by rapid driving. By mapping these protocols onto a broad class of controllable thermodynamic cycle, we establish power-efficiency tradeoffs and position the polytropic index as a genuine thermodynamic control knob for the rational design of high-speed, high-performance microscopic thermal machines.

Figures (3)

• Figure 1: (a) The finite-time evolution of the thermodynamic state $(\lambda,\theta)$ as a function of normalized time $t/\tau$ for polytropic index $\xi=-0.3$ (red), $\xi=-0.5$ (blue), and $\xi=-0.7$ (purple). The dashed curves projected on the shaded gray plane are given by the polytropic constraint $\theta\lambda^{\xi}={\rm{const.}}$ The dimensionless process duration $\tilde{\tau}$ as a function of $\xi$ for underdamped regime (b) and overdamped regime (c), where simulation data are represented by pink dots. In this figure, $u=1.5$ and $\delta=1 \times 10^{-3}$ are used.
• Figure 2: (a) IEG $\Delta S_{\mathrm{ir}}$ as a function of the dimensionless process duration $\tilde{\tau}$ (solid red curve). $\Delta S_{\mathrm{ir}}$ versus $\tilde{\tau}$ for different $\delta$ (b) and $u$ (c). In these two plots,the maximum process durations $\tilde{\tau}_{\mathrm{max}}$ associated with $\xi=0$ are marked with different markers. (d) $\Delta S_{\mathrm{ir}}$ as a function of the process duration $\tau$ for endoreversible isothermal processes (solid curve, $\xi=0$). The left (right) axis is associated with with $u=3/2$($u=2/3$). The dashed line represent the predicted asymptotic scaling. (e) Normalized work $W$ (pink solid curve) and heat $-Q$ (purple dashed curve) as a function of process duration, where $W_{\mathrm{iso}}=\theta_0 \ln u$. To plot (a) and (e), we set $u=1.5$ and $\delta=1 \times 10^{-2}$.
• Figure 3: The Brownian polytropic engine. (a) The time evolution of the polytropic protocol. (b) Thermodynamic diagrams for the transition of $\xi=-1$, $-0.5$, $-0.25$, and $0$. (c) Normalized power $\tilde{P} \equiv P/P_{\mathrm{max}}$ (orange solid curve), and efficiency $\eta \equiv \eta/\eta_{\mathrm{C}}$ (blue dashed curve), versus $\tilde{\tau}$. (d) The power–efficiency trade-off (pink solid curve). The Simulation data are shown as blue dots and the red line indicates the Carnot efficiency.