Tightening the thermodynamic uncertainty relations with null-entropy events: What we learn when nothing happens

TL;DR

This work investigates null-entropy events, where no entropy is produced ($\Sigma=0$), and shows that knowing their probability $p_0$ tightens finite-time thermodynamic uncertainty bounds beyond conventional TURs. By decomposing trajectory statistics into active and null subensembles and formulating a modified fluctuation theorem for the active part, the authors derive analytically tighter TUR bounds that apply to both classical and quantum systems obeying the FT. The concepts are validated with a minimal three-point model and a qudit SWAP engine, demonstrating that larger $p_0$ yields substantially tighter, saturable bounds. The results reveal a structured precision–dissipation trade-off shaped by inactivity events, with implications for thermodynamic inference and the design of quantum thermal machines.

Abstract

Fluctuation theorems establish that thermodynamic processes at the microscale can occasionally result in negative entropy production. At the microscale, another distinct possibility becomes more likely: processes where no entropy is produced overall. In this work, we explore the constraints imposed by such null-entropy events on the fluctuations of thermodynamic currents. By incorporating the probability of null-entropy events, we obtain tighter bounds on finite-time thermodynamic uncertainty relations derived from fluctuation theorems. We validate this framework using an example of a qudit SWAP engine.

Figures (4)

• Figure 1: (a) Average entropy production $\langle \Sigma\rangle$ and (b) noise-to-signal ratio ${\rm Var}(\Sigma)/\langle \Sigma\rangle^2$ for the minimal model discussed in Sec. \ref{['sec:minimal_example']} [Eqs. \ref{['eq:ave_ep']} and \ref{['eq:nsr_ep']}]. The quantities are plotted as a function of the null-entropy probability $p_0$ for different values of $\sigma$, as shown in (a). Significant fluctuations arise precisely in regimes dominated by null-entropy events. Inset of (b): the functions $[\tanh^2(\sigma/2)-1]^{-1}$ (solid black) and $[\tanh^2(\sigma/2)]^{-1}$ (orange dashed).
• Figure 2: Modified TUR-de-force bounds from Eqs. \ref{['eq:new_f']} and \ref{['eq:tur-de-force']} shown for different values of the probability $p_0$ associated with zero integrated current and entropy production. The bound gets tighter as $p_0$ increases.
• Figure 3: (a) -- (c): Analysis of fluctuations in heat extracted for a qudit SWAP engine. (a) Schematic of a $d$-level qudit SWAP engine. Two equally spaced $d$-level qudits act as the working medium and are connected to a hot and cold reservoirs. Qudits initiate in a thermalized state, after which a SWAP unitary extracts work from them, ending the cycle with re-thermalization. Fluctuations in the SWAP engine (solid red), and TUR-de-force bounds with (dashed blue) and without (dot-dashed green) the knowledge of null-entropy values $p_0$ [Eqs. \ref{['eq:new_f']}--\ref{['eq:old_f']}], are presented for (b) qubits with $d = 2$, and (c) qutrits with $d = 3$. The null-entropy values are particularly useful for qubits, as they directly determine the variance, whereas for qutrits, they serve as a means to enhance existing bounds. These plots assume values $\beta_A = 1$, $\beta_B = 2$.
• Figure 4: Extending the analysis done for the heat fluctuations in qudit SWAP engine [see Fig. \ref{['fig:SWAP_engine']}] with (a)$d = 4$, (b)$d = 5$, (c)$d = 6$, (d)$d = 8$. Across all dimensions, the TUR bound tightens by supplying the null-entropy probability $p_0$. The extent to which $p_0$ varies is also plotted (in dashed light blue). These plots also assume the same parameters as in Fig. \ref{['fig:SWAP_engine']}.