Strong Violation of the Thermodynamic Uncertainty Relation in a Minimal Autonomous Heat Engine

Abstract

Thermodynamic uncertainty relations (TURs) impose a universal trade-off between current precision and entropy production in autonomous steady states, constraining in particular the power, efficiency, and constancy of heat engines. We demonstrate strong violations of the long-time TUR in a minimal autonomous heat engine composed of a discrete ratchet generating work against a constant bias and an underdamped harmonic oscillator acting as an internal stochastic control. In the regime of time-scale separation, the model becomes exactly solvable and yields a closed analytical expression for the TUR ratio, where the influence of the continuous degree of freedom is fully captured by the Fano factor of oscillator zero crossings. We show that increasingly deterministic internal control drives the TUR ratio arbitrarily close to zero while the engine operates near maximal current and efficiency. In an appropriate limit, the model reduces to the classical pendulum-clock system of Pietzonka, Phys. Rev. Lett. 128, 130606 (2022).

Figures (6)

• Figure 1: Sketch of the model. The underdamped harmonic oscillator (left) controls the temperature of the bath and the transition rates in the discrete subsystem (right). Specifically, when the oscillator coordinate is positive (negative), the discrete subsystem is in contact with the hot (cold) bath at temperature $T_h$ ($T_c$), and the red (blue) transitions are allowed. The signs $\mp$ indicate the direction of the positive current. The dashed energy levels are $G$-fold degenerate, and the labels below the levels denote their energies.
• Figure 2: Fano factor $x$ of the harmonic oscillator, defined in Eq. \ref{['eq:x_def_fullmodel']}, as a function of the reduced friction coefficient $\gamma$.
• Figure 3: Average displacement per half-oscillation, $j_0$ (left), and TUR ratio, $\mathcal{T}$ (right), as functions of $k_0/\omega$ for $E=3$, $F=2$, $G=150$, $T_h=1000$, $T_c=0.1$, and $\gamma=10^{-3}$ ($x = 0.009$) in reduced units. Circles (solid lines) are obtained from Brownian dynamics simulations of the full model, and squares (dashed lines) from the approximate analytical formulas in Eqs. \ref{['eq:curr_swing']} and \ref{['eq:T_general_label']}.
• Figure 4: Average displacement per half-oscillation, $j_0^\star$, and the corresponding entropy production, $\sigma_0^\star$, together with the TUR ratio, $\mathcal{T}^\star$, and the HE efficiency, $\eta$, as functions of the cycle affinity $\Delta$ and the control precision (Fano factor) $x$ in the limit $T_h \to \infty$ and for $G = G^\star$ as defined in Eq. \ref{['eq:Gstar_jmax']}. Below the white dashed line, the TUR is violated ($\mathcal{T}^\star < 1$). Parameters: $E=3$ and $T_c=0.1$ in reduced units.
• Figure 5: TUR ratio, $\mathcal{T}^{\star}$, as a function of the cycle affinity $\Delta$ in the limit $T_h \to \infty$ and for $G = G^\star$ as defined in Eq. \ref{['eq:Gstar_jmax']}, shown for six values of the control precision (Fano factor) $x$.