**1. Introduction**

The two paradigms of statistical mechanics for systems that are close to equilibrium are: (i) the Onsager–Casimir reciprocity relations [1]; and (ii) the fluctuation–dissipation theorem (FDT) [2–4]. Both relations are not only inherent to classical systems but are also applicable to the quantum regime. The Onsager–Casimir reciprocity relations state that the Onsager matrix that relates physical fluxes and their conjugate forces is symmetric. For example, considering as forces the electrostatic and thermal gradients, and their associated currents being the electrical and heat fluxes, these relations set an identity between the thermoelectrical conductance (electrical response to a thermal gradient) and the electrothermal conductance (response of the heat current to an electrical bias). On the other hand, the FDT establishes that statistical fluctuations occurring in a system at equilibrium behave similarly to the dissipation that takes place under the action of an external perturbation. Major examples of manifestations of the FDT are found in Einstein's treatment of Brownian motion where the diffusion constant is found to be proportional to the mobility [5] or the Johnson–Nyquist formula for electronic white noise [6]. In the context of quantum transport through electronic nanodevices, the FDT allows us to relate the dissipative response of one current with respect to a variation of its affinity or conjugate force with its spontaneous fluctuations. This property of equilibrium systems is a very important topic when we are interested in controlling dissipation due to currents induced through quantum conductors by external forces.

As mentioned above, the range of validity of the FDT is limited to the linear response regime, i.e., for sufficiently small perturbations. Going beyond this regime requires generalizing this formulation to non-equilibrium conditions. This has been done by introducing additional correlations involving the activity, a magnitude related to the transition rates and the excess of entropy production that is modified antisymmetrically by the external potential that drives the system out of equilibrium [7–10]. In this view, these extensions to the FDT are indeed fluctuation–dissipation relations (FDR) that establish the frequency at which a system produces entropy to the environment between forward and backward processes. The interest of the FDR has been highlighted in the field of quantum transport [1,11–13].

However, here we adopt a different perspective, reported in the work of B. Altaner, M. Polettini, and M. Esposito [14], in which the concept of *stalling currents* is introduced in the context of stochastic thermodynamics. A current that traverses a system can be nullified because of the cancellation of a set of distinct internal processes, and is then called a *stalling current*. Under these conditions, if the perturbative force solely affects the microscopic transitions that contribute to this current, the FDT is restored [14,15]. In addition, we test numerically that Onsager reciprocity relations are additionally satisfied at stalling conditions. We speculate that this property is attained due to the lack of entropy production at stalling conditions forced by the tight coupling between the charge and heat currents (see below). The conclusion is that all contributing elemental transitions being internally equilibrated is equivalent to them being microscopically reversible. One interesting application to the stalling configuration is that, even though correlations are usually difficult to access experimentally, the fact that the FDT is applicable makes it rather easy to obtain such correlations by means of a response function instead.

Our purpose in this work is to implement these conclusions in a nanodevice consisting of two interacting conductors. Such setup was previously investigated by R. Sánchez et al. [16] to analyze the drag effect. The device consists of a parallel double quantum dot system in which the quantum dots interact electrostatically via a mutual capacitance. Besides, each quantum dot is tunnel-connected to two electronic reservoirs. A drag current is encountered in one of the dots, which is unbiased, due to the charge fluctuations provoked by the electrical current driven through the other dot. The detection of this drag current has been demonstrated experimentally [17] showing that high-order tunneling events such as cotunneling have a significant contribution. Besides, a drag current control has been proposed by attaching to the dots different materials with nontrivial energy-dispersion relations [18]. This system has additionally been proposed for the implementation of a Maxwell demon [19], in which one of the dots (the demon) acquires information from the other one, allowing a current to flow opposite to the applied bias voltage in the other dot.

Our goal in this article is to explore the transport properties in an out-of-equilibrium configuration that drives the system into an effective equilibrium in which both the Onsager relations and the FDT are recovered. For this purpose, we compute the electrical and heat currents through each quantum dot. By a numerical search of stalling currents in one of the dots, we check whether or not Onsager relations and the FDT are satisfied. We consider different situations. Firstly, we consider the case where both the electrical and heat flows are cancelled simultaneously under non-equilibrium configurations. This can be achieved only in the so-called strong coupling regime. For this case, we demonstrate that the system indeed behaves as at equilibrium. We also analyze the scenario where only one of the currents vanishes (either the charge or the heat flow), while the other one is kept finite. Finally, we show that the absence of stalling currents prevents the fulfillment of the Onsager relations and the FDT as expected. To conclude, we go beyond the FDT and additionally check the FDRs for the third cumulant in the presence of stalling currents.
