Thermodynamic Uncertainty Relations from Exchange Fluctuation Theorems

Thermodynamic uncertainty relations (TURs) place strict bounds on the fluctuations of thermodynamic quantities in terms of the associated entropy production. In this Letter, we identify the tightest (and saturable) matrix-valued TUR that can be derived from the exchange fluctuation theorems describing the statistics of heat and particle flow between multiple systems of arbitrary dimensions. Our result holds for both quantum and classical systems, undergoing general finite-time nonstationary processes. Moreover, it provides bounds not only for the variances, but also for the correlations between thermodynamic quantities. To demonstrate the relevance of TURs to the design of nanoscale machines, we consider the operation of a 2-qubit swap engine undergoing an Otto cycle and show how our results can be used to place strict bounds on the correlations between heat and work.

Figure 1

(a) Exchange fluctuation theorem scenario: A system consisting of $M$ (here $M=3$) subsystems is allowed to interact by means of a unitary $U$. As a result, the subsystems will exchange energy and particles, amounting to net transferred charges (integrated currents) of energy ${\mathcal{Q}}_{{\mathcal{E}}_i}$ and particles ${\mathcal{Q}}_{{\mathcal{N}}_i}$. (b) For microscopic systems, any generic ${\mathcal{Q}}_i$ will be a stochastic variable and fluctuate from one repetition of the experiment to the other, represented pictorially by the jagged gray curve. This is to be contrasted with the average charge $⟨{\mathcal{Q}}_i⟩$ shown as a dashed line. The fluctuations in ${\mathcal{Q}}_i$ are represented by the variance $\text{Var}({\mathcal{Q}}_i)$, which we illustrate here by the red interval. (Inset) A plot of the function $f(x)$ on the right-hand side of Eq. (3), compared with the traditional bound $2/x$ that appears in Eq. (1).

Figure 2

Fluctuations of heat and work in a 2-qubit swap engine. (a) Schematic operation of the engine: 2 qubits thermalize with two baths at different temperatures. Then they are uncoupled from the baths and allowed to interact with each other by means of a swap operation, which produces a certain amount of work $W$. Repeating this procedure sequentially allows the device to operate as either a refrigerator, an engine, or an accelerator. (b) Averages of the work $⟨W⟩$, heat to the hot bath $⟨Q_H⟩$, and entropy production $⟨\mathrm{\Sigma }⟩$ as a function of ${\epsilon }_B/{\epsilon }_A$ for ${\beta }_A/{\beta }_B=1/2$. The different regimes of operation of the engine are separated by dashed vertical lines. (c) Fluctuations in the heat to the hot bath $\text{Var}(Q_H)$. The orange line represents our bound (3) and the green-dashed line represents the bound (1), included for comparison. For small values of the detuning ${\epsilon }_B/{\epsilon }_A$, (i.e., in the refrigerator regime), one can see that the bound (1) is violated (the black line lies below the green-dashed one), while (3) is always valid. (d) Same but for the fluctuations in the work $\text{Var}(W)$. (e) The correlations between heat and work, as measured by the covariance $\text{Cov}(W,Q)$. The two orange lines represent the bounds in Eq. (11).