**3. Results and Discussion**

In this section, we present the main results of our work. We verify the RRs and the FDT for a complete understanding of the impact of stalling currents in coupled conductors.

### *Roots of the Drag Current and Equilibrium-Like Behavior*

The aim of our study is to verify the generalized non-equilibrium reciprocity relations and the fluctuation–dissipation relations. As discussed above, they require that the involved currents be at stall in order to hold arbitrarily far from equilibrium. We exclusively focus on situations where the stalling currents are those between the upper dot and the first lead, i.e., the ones in the drag system. Since *I*1 = −*I*2, it is enough for our purposes to seek for roots of *I*1. We also only look for roots of *J*1, even though *J*1 = *J*2. For all the out-of-equilibrium calculations, we consider the isothermal case *T* = *Ti*, with *i* = 1, ... , 4. Since we are only interested in the responses of the currents to small temperature fluctuations in one of the leads (with the rest held constant), we must formally treat the temperatures in each lead as independent of each other for computational means. However, in the end, all derivatives are evaluated at temperature *T*.

The electric current *I*1 [Equation (17)] is a highly nonlinear function of the biases *V*12, *V*13 and *V*34. Consequently, the solutions to *I*1 = 0 must be found by means of numerical analysis in order to verify Onsager's relations and the FDT (further justifications below). To this purpose, we set Γ*i* = *γi* = Γ except for *γ*1 = 0.1Γ, *kBT* = 5*h*¯ Γ, *q*2/*Ci* = 20*h*¯ Γ, *q*2/*<sup>C</sup>* = 50*h*¯ Γ and *εu* = *εd* = 0. Furthermore, we consider natural units where *h*¯ = −*q* = *kB* = Γ = 1. Unless otherwise mentioned, these parameters are used in the rest of this work.

We remark that our analysis is purely numerical since the solutions for *I*1 = 0 require large values of *V*12 at a given set of voltages *V*13 and *V*34. This fact prevents us from employing a perturbative scheme in terms of the dc voltages. The charge current through the upper dot is composed of the current directly induced by the bias *V*12 and the contribution due to the charge fluctuations caused by the transport in the lower dot. The latter contribution is precisely the drag effect, which is much less significant to the creation of a charge flow through the *up* dot than the effect of a voltage directly applied between the upper terminals. The need for a numerical analysis of this system is hereby justified. To find the roots of the currents for a given set of parameters, we implemented a bisection algorithm (see Appendix A).

Since there is no magnetic field present in our system, its dynamical evolution is time-reversible. Accordingly, microreversibility ensures that the RRs should be satisfied for stalling currents far from equilibrium, as discussed in Reference [15]. In this section, we analyze both the case when the charge and heat currents stall at the same time, as well as the scenario when they do not necessarily vanish simultaneously for the same voltage configuration. The Onsager matrix for our two dot system with four leads should be of dimension 8 × 8 with elements denoted by *Lij*,*mn*. In the absence of a magnetic field, Onsager's relations imply *Lij*,*mn* = *Lji*,*mn*. Furthermore, charge conservation laws imply relations such as *I*2 = −*I*<sup>1</sup> and therefore more elements of the Onsager matrix are related. At the stalling configuration, we thus check for the fulfilment of the particular relation *<sup>L</sup>*12,11 = *<sup>L</sup>*21,11, with

$$L\_{12,11} \equiv L\_{12} = \frac{\partial I\_1}{\partial T\_1}, \quad L\_{21,11} \equiv L\_{21} = \frac{1}{T\_1} \frac{\partial I\_1}{\partial V\_1} \tag{24}$$

As we can see, *I*1 = *Icharge* and *J*1 = *Jheat* in terms of the example in Equation (22). Here, we consider as conjugate forces the *absolute* potentials and temperatures. This is justified since the thermodynamic variables of the quantum dots do not show up in the currents, and therefore differentiating them with respect to the gradients Ω*i* − Ωdot yields the same result as differentiating

with respect to Ω*i* (where Ω represents either a voltage or a temperature). A summary of our first results is presented in Figure 3. We show the coefficients for a given *V*12 as a function of *V*13. It is understood that the value of *V*34 at each point corresponds to the one where stalling has been numerically found. We consider four cases: (i) the globally stalled configuration depicted in Figure 3a; (ii) the locally charge-stalled case shown in Figure 3b; (iii) the locally heat-stalled scenario in Figure 3c; and (iv) a configuration where none of the currents vanish, as shown in Figure 3d. Firstly, we notice that the RRs are satisfied at the configurations where the current *I*1 stalls [cases shown in Figure 3a,b] with *J*1 being either zero or not. On the other hand, considering the stalling points of *J*1 [see Figure 3c], in general, we do not observe an equality between *L*12 and *L*21. Even so, there are some exceptions (not shown here) in which the RR are satisfied despite having *I*1 = 0 and *J*1 = 0. For these cases, however, we checked that they do not follow the FDT.

We now move on to study the validity of the fluctuation–dissipation theorem. In this case, we only consider the FDT for the charge currents. Firstly, we give explicit expressions for the relations between the transport coefficients, i.e., the FDRs. They have been established for the non-equilibrium case. Here, it is instructive to first consider the FDT *near equilibrium*. We consider the following voltage expansion of the currents around the equilibrium point *Vi* = 0:

$$I\_{\mathfrak{a}} = \sum\_{\mathfrak{f}} G\_{\mathfrak{a},\mathfrak{f}}^{\mathfrak{e}\mathfrak{q}} V\_{\mathfrak{f}} + \sum\_{\mathfrak{f},\gamma} G\_{\mathfrak{a},\mathfrak{f}\gamma}^{\mathfrak{e}\mathfrak{q}} V\_{\mathfrak{f}} V\_{\gamma} + \mathcal{O}\left(V^{2}\right) \tag{25}$$

where the *n*th order conductances *Geq μ*,*ν*1...*ν<sup>n</sup>* = *∂n <sup>I</sup>μ*/*∂V<sup>ν</sup>*1 ... *V<sup>ν</sup>n Vi*=0 are related to *n*th order FDRs. For instance, at second-order equilibrium FDRs lead to the FDT

$$S\_{a\beta}^{c\eta} = k\_B T \left( G\_{a,\beta}^{c\eta} + G\_{\beta,a}^{c\eta} \right) \tag{26}$$

The non-equilibrium FDT is then established to have the same form replacing the equilibrium condition by the stalling condition.

**Figure 3.** Onsager coefficients *L*12 and *L*21 versus the *V*13 bias voltage at the indicated *V*12 biases: (**a**) strong coupling configuration (Γ2 = *γ*1 = 0) with *I*1 = 0 and *J*1 = 0; (**b**) *I*1 = 0 and *J*1 = 0; (**c**) *I*1 = 0 and *J*1 = 0; and (**d**) *I*1 = 0, and *J*1 = 0. Γ*i* = *γi* = Γ except for *γ*1 = 0.1Γ, *kBT* = 5*h*¯ Γ, *q*2/*Ci* = 20*h*¯ Γ, *q*2/*<sup>C</sup>* = 50¯*h*Γ and *εu* = *<sup>ε</sup>d* = 0. Furthermore, we consider natural units where ¯*h* = −*q* = *kB* = Γ = 1.

*Entropy* **2020**, *22*, 8

For our particular device, we investigated the non-equilibrium FDT for two cases. We tested the FDT only for the upper dot charge current, i.e., *I*1 = −*I*2. The results are shown in Figure 4, where we check the FDT for the drag current, i.e.,

$$S\_{11} = 2k\_B T G\_{1,1} \tag{27}$$

as well as the FDT involving the cross-correlations between the drag current (*I*1) and the drive current (*I*3) contributions, i.e.,

$$S\_{1\heartsuit} = k\_B T \left( G\_{1,\heartsuit} + G\_{3,1} \right) \tag{28}$$

where in both cases the noise *<sup>S</sup>αβ* was computed by applying the Full Counting Statistics (FCS) formalism described in Appendix B. We observe that only the former relation for the drag current is satisfied (Figure 4a,b) since *I*1 vanishes but *I*3 does not. The FDT involving cross-correlations between *I*1 and *I*2 also holds since both of the currents stall (not included). These results are independent of whether the heat current vanishes (see Figure 4a for *J*1 = 0, i.e., the strong coupling regime) or not (see Figure 4b with *J*1 = 0). In the two remaining cases (Figure 4c,d), the fact that the drive current *I*3 does not vanish prevents the fulfilment of the FDT for the cross-correlations *S*13.

To make a complete description of the transport under stalling conditions we now discuss a remarkable result involving the third cumulants of the current. In this case, we talk about fluctuation–dissipation relations instead of the FDT. As mentioned in the Introduction, the FDRs were originally formulated by adding to the transition rates and the excess of entropy production the external potential that drives the system out of equilibrium [8]. In that sense, it is possible to establish relations between the transport coefficients such as nonlinear conductances, non-equilibrium noises, and the third cumulant. In all these cases, the transport coefficients are computed at the non-equilibrium configuration. In particular, López et al. [26] found that the following FDR is satisfied under equilibrium conditions:

$$\mathcal{L}\_{\mathfrak{a}\mathfrak{f}\gamma} = \left(k \boxtimes T\right)^{\mathfrak{a}} \left(\mathcal{G}\_{\mathfrak{a},\mathfrak{f}\gamma} + \mathcal{G}\_{\mathfrak{f}\mathfrak{f},\mathfrak{a}\mathfrak{a}} + \mathcal{G}\_{\mathfrak{f},\mathfrak{a}\mathfrak{f}\mathfrak{f}}\right) \tag{29}$$

where *<sup>C</sup>αβγ* = *Iα Iβ <sup>I</sup>γ* are the third-order cumulants. We ignored the indices referring to the spin degree of freedom appearing in the original paper as in our system we have spin degeneracy due to the absence of a magnetic field. Here, we checked for the fulfilment of the previous relation at stalling conditions far from equilibrium, where all the nonlinear transport coefficients *<sup>G</sup><sup>α</sup>*,*βγ* are computed under non-equilibrium conditions. Note that this can be rewritten as

$$\mathcal{L}\_{\mathfrak{a}\mathfrak{F}\gamma} = \mathfrak{Z} \left(k\_B T\right)^2 G\_{\left(\mathfrak{a}, \mathfrak{F}\gamma\right)} \tag{30}$$

where we understand *<sup>G</sup>*(*<sup>α</sup>*,*βγ*) as the symmetrization with respect to the three indices, *<sup>G</sup>*(*<sup>α</sup>*,*βγ*) = (1/3!) *<sup>G</sup><sup>α</sup>*,*βγ* + *<sup>G</sup><sup>α</sup>*,*γβ* + *<sup>G</sup>β*,*γα* + *<sup>G</sup>β*,*αγ* + *<sup>G</sup>γ*,*αβ* + *<sup>G</sup>γ*,*βα*.

We explored the fulfilment of Equation (29) when stalling currents are present in the system, with *<sup>G</sup>αβγ* again computed with help of FCS (see Appendix B). Figure 5 represents the third cumulant fluctuation relations. The case in which *I*1 = 0 is shown in Figure 5a when *J*1 = 0 and in Figure 5b when *J*1 = 0. In these two scenarios, the FDRs are fulfilled. However, when the cumulant relation involves currents from both the drive (either *I*3 or *I*4) and the drag (either *I*1 or *I*2) subsystems, then the corresponding FDR is no longer satisfied. Finally, for completeness, our last result is shown in Figure 6, where the FDT and third-order cumulant relations are displayed for cases where the system is not in a stalling configuration. As can be seen, none of these relations hold, as expected.

**Figure 4.** Fluctuation–dissipation theorem *<sup>S</sup>αβ* = <sup>2</sup>*kBTG*(*<sup>α</sup>*,*β*) for: (**a**) the strong coupling regime for the drag current, *I*1 = 0 and *J*1 = 0; (**b**) the locally charge-stalled configuration, *I*1 = 0 and *J*1 = 0; and (**<sup>c</sup>**,**d**) the drag and drive currents with *I*1 = 0 and *J*1 = 0, and *I*1 = 0 and *J*1 = 0, respectively. The rest of the parameters are those of Figure 3.

**Figure 5.** Third-order fluctuation–dissipation relations *<sup>C</sup>αβγ* = 3 (*kBT*)<sup>2</sup> *<sup>G</sup>*(*<sup>α</sup>*,*βγ*) for: (**a**) the strong coupling regime for the drag current *I*1 = 0 and *J*1 = 0; (**b**) the locally charge-stalled configuration *I*1 = 0 and *J*1 = 0; and (**<sup>c</sup>**,**d**) the drag and drive currents for *I*1 = 0 and *J*1 = 0, and *I*1 = 0 and *J*1 = 0, respectively. The rest of the parameters are those of Figure 3.

**Figure 6.** Fluctuation–dissipation theorem in a non-stalled configuration *I*1 = 0 and *J*1 = 0 for: (**a**) the drag current; and (**b**) the drag and drive currents. Fluctuation–dissipation relations in a non-stalled configuration *I*1 = 0 and *J*1 = 0 for: (**c**) the drag current; and (**d**) the drag and drive currents. The rest of the parameters are those of Figure 3.
