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.

We study the statistics of heat exchange of a quantum system that collides sequentially with an arbitrary number of ancillas. This can describe, for instance, an accelerated particle going through a bubble chamber. Unlike other approaches in the literature, our focus is on the joint probability distribution that heat $Q_1$ is exchanged with ancilla 1, heat $Q_2$ is exchanged with ancilla 2, and so on. This allows us to address questions concerning the correlations between the collisional events. For instance, if in a given realization a large amount of heat is exchanged with the first ancilla, then there is a natural tendency for the second exchange to be smaller. The joint distribution is found to satisfy a Fluctuation theorem of the Jarzynski–Wójcik type. Rather surprisingly, this fluctuation theorem links the statistics of multiple collisions with that of independent single collisions, even though the heat exchanges are statistically correlated.

In classical mechanics, external constraints on the dynamical variables can be easily implemented within the Lagrangian formulation. Conversely, the extension of this idea to the quantum realm, which dates back to Dirac, has proven notoriously difficult due to the noncommutativity of observables. Motivated by recent progress in the experimental control of quantum systems, we propose a framework for the implementation of quantum constraints based on the idea of work protocols, which are dynamically engineered to enforce the constraints. As a proof of principle, we consider the dynamical mean-field approach of the many-body quantum spherical model, which takes the form of a quantum harmonic oscillator plus constraints on the first and second moments of one of its quadratures. The constraints of the model are implemented by the combination of two work protocols, coupling together the first and second moments of the quadrature operators. We find that such constraints affect the equations of motion in a highly nontrivial way, inducing nonlinear behavior and even classical chaos. Interestingly, Gaussianity is preserved at all times. A discussion concerning the robustness of this approach to possible experimental errors is also presented.

A plethora of natural and socio-economic phenomena share a striking statistical regularity, that is the magnitude of elements decreases with a power law as a function of their position in a ranking of magnitude. Such regularity is known as Zipf–Mandelbrot (ZM) law, and plenty of problem-specific explanations for its emergence have been provided in different fields. Yet, an explanation for ZM ubiquity is currently lacking. In this paper we first provide an analytical expression for the cumulants of any ranked sample of independently and identically distributed (i.i.d.) random variables once sorted in decreasing order. Then we make use of this result to rigorously demonstrate that, whenever a small fraction of such a ranked dataset is considered, it becomes statistically indistinguishable from ZM law. We finally validate our results against several relevant examples.

Fluctuation theorems establish that thermodynamic processes at the microscale can occasionally result in negative entropy production. At such scales, another distinct possibility becomes more likely: processes in which 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. Our results are directly applicable to both quantum and classical systems that satisfy the fluctuation theorem. We validate our framework using an example of a qudit SWAP engine.