Inertia Tames Fluctuations in Autonomous Stationary Heat Engines

Abstract

Thermodynamic uncertainty relations (TURs) provide fundamental constraints on the interplay between power fluctuations, entropy production, and efficiency in overdamped stationary autonomous heat engines. However, their validity in underdamped regimes remains limited and less explored. Here, we analytically and numerically study a physically realizable autonomous heat engine composed of two underdamped continuous degrees of freedom coupled to a two-level system. We show that this nonlinear setup can robustly violate TUR-based trade-offs by exploiting resonant coupling, effectively using one underdamped mode as an internal periodic drive. When this coupling is suppressed, the system recovers TUR-like bounds consistent with overdamped theory. Importantly, we demonstrate that the strongest suppression of current fluctuations occurs in a resonance regime that can be directly inferred from mean current measurements - a quantity typically much easier to access experimentally than fluctuations. Our results reveal new pathways to circumvent classical TUR constraints in underdamped systems and provide practical guidelines for designing efficient, precise microscopic engines and autonomous clocks.

Figures (7)

• Figure 1: Operational principle of the heat engine. Red and blue solid lines indicate regions of the angular coordinate $\alpha$ where the temperature \ref{['eq:T']} is hot ($T = T_+$) or cold ($T = T_-$), respectively. When the system parameters are tuned as described in the main text, the two-level system occupies the excited state $+$ with near-unit probability in the hot region and the ground state $-$ in the cold region (see Figs. \ref{['fig:1Dseries']}c and \ref{['fig:1Dseries']}f). As a result, the potential $U_d$ is, with high probability, given by the solid line, and the corresponding force—determined by the negative slope of this potential—is positive, generating a net current in $\alpha$-space (see Figs. \ref{['fig:1Dseries']}a and \ref{['fig:1Dseries']}b). Dashed lines indicate alternative potential configurations that are rarely realized. The sharp solid lines correspond to the piecewise linear potential \ref{['eq:Ud']} used in our theoretical analysis, while the transparent curves illustrate a possible experimental realization, proposed in Sec. \ref{['sec:experiment']}, based on the sinusoidal potential \ref{['eq:Udexp']}. Our numerical results indicate that the TUR and PECT can be violated for the linear profile, but remain valid for the sinusoidal one.
• Figure 2: Dynamics of the system for $\partial_\alpha U = 0$. (a) Current $j(t)$ in Eq. \ref{['eq:current_def']}, (b) the corresponding angular velocity $\omega(t)$, defined in Eq. \ref{['eq:om']}, (c) the state of the two-level system, expressed as $\tau_d(t)/\tau_0$, all shown as functions of time. (d) Probability density of the angular coordinate $\alpha(t)$, corresponding to $\omega(t)$ in (b). (e) Probability density of $\omega(t)$. (f) Probability density of $\tau_d \, \text{sgn}[\sin(\alpha)]/\tau_0$, illustrating the deterministic dynamics of the two-level system in the given parameter regime. Solid blue lines represent results from Brownian dynamics simulations, while black dashed lines show theoretical predictions: (a–b) $j(t)\, t = \omega_t\, t$ and $\omega(t) = \omega_t$, with $\omega_t$ given in Eq. \ref{['eq:omT']}; (c) $\tau_d(t)/\tau_0 = \text{sgn}[\sin(\alpha(t))]$; (d) $P(\alpha) = 1/(2\pi)$; (e) $P(\omega)$ is given by Eq. \ref{['eq:Pomega']}, with the mean shifted to $\omega_t$. In panel (d), $t_0 = t_{max} - 3 \times 2\pi / \omega_t$. All parameters, with the exception of $\kappa_{\alpha x} = 0$, are listed in Tab. \ref{['tab:parameters']}.
• Figure 3: Time-resolved TUR for $\partial_\alpha U = 0$: (a) Current variance and (b) $\mathcal{T}$ from Eq. \ref{['eq:T']} as functions of the measurement duration $t$. Blue solid lines correspond to Brownian dynamics simulations, while black dashed lines show theoretical predictions from Eqs. \ref{['eq:varj']} (upper bound) and \ref{['eq:TFree']}. The grey dot-dashed line in (b) depicts $\mathcal{T}$ for free diffusion under constant drift, given by Eq. \ref{['eq:TFree']} with $\sigma_{2L} = 0$. The TUR in Eq. \ref{['eq:T']} is violated when $\mathcal{T}$ in panel (b) falls below the grey dotted line marking the threshold $\mathcal{T} = 1$. The breakdown of the TUR at short times is expected, as it is known to be invalid in this regime Fischer2020.
• Figure 4: Dynamics of the full system. (a) Current $j(t)$ in Eq. \ref{['eq:current_def']}; (b) corresponding angular velocity $\omega(t)$, defined in Eq. \ref{['eq:om']}; (c) position $x(t)$; and (d) velocity $v(t)$, defined in Eq. \ref{['eq:v']}, all shown as functions of time. (e) Probability density of the angular coordinate $\alpha(t)$, corresponding to $\omega(t)$ in (b). (f–h) Probability densities of $\omega(t)$, $x(t)$, and $v(t)$, corresponding to panels (b), (c), and (d), respectively. Solid blue lines represent results from Brownian dynamics simulations, while black dashed lines denote theoretical predictions: (a–b) $j(t)\, t = \omega_t\, t$ and $\omega(t) = \omega_t$, with $\omega_t$ given in Eq. \ref{['eq:omT']}; (c–d) $x(t)$ and $v(t)$ given by Eqs. \ref{['eq:xt']} and \ref{['eq:vt']}; (e) $P(\alpha) = 1/(2\pi)$; (f) $P(\omega)$ is given by Eq. \ref{['eq:Pomega']}, with the mean shifted to $\langle \omega \rangle$ [dot–dashed grey line in (b)]; additionally, the red dot–dashed line shows the same distribution with the variance rescaled as $T_- \to 2.35$; (g–h) $P(x)$ and $P(v)$ are given by Eqs. \ref{['eq:px']}, \ref{['eq:pv']}, and \ref{['eq:PConv']}. Additionally, the grey dot-dashed lines show the distributions from Eqs. \ref{['eq:px']} and \ref{['eq:pv']} without the convolution in Eq. \ref{['eq:PConv']} with the noise kernel. In panels (c–d), $t_0 = t_{\mathrm{max}} - 3 \times 2\pi / \omega_t$. All parameters are listed in Tab. \ref{['tab:parameters']}.
• Figure 5: Time-resolved TUR in the full system ($\partial_\alpha U \neq 0$): (a) Current variance and (b) $\mathcal{T}$ from Eq. \ref{['eq:T']} as functions of the measurement duration $t$. Blue solid lines are obtained from Brownian dynamics simulations, while black dashed lines represent theoretical predictions from Eqs. \ref{['eq:varj']} (upper bound) and \ref{['eq:TFree']}, which apply to $\partial_\alpha U$ when the $\alpha$-dynamics is independent of $x$. The TUR in Eq. \ref{['eq:T']} is violated when $\mathcal{T}$ in panel (b) drops below the grey dotted line, which marks the threshold $\mathcal{T} = 1$. In this case, the expected breakdown of the TUR at short times Fischer2020 extends to arbitrary times, similarly to the classical pendulum clocks considered in Ref. Pietzonka2022.