# A post-selection algorithm for improving dynamic ensemble selection methods

Paulo R.G. Cordeiro1, George D.C. Cavalcanti2 and Rafael M.O. Cruz3 1Paulo
R.G. Cordeiro is with Centro de Informática, Universidade Federal de
Pernambuco, Recife, Brazil (prgc@cin.ufpe.br)2George D.C. Cavalcanti is with
Centro de Informática, Universidade Federal de Pernambuco, Recife, Brazil
(gdcc@cin.ufpe.br)3Rafael M.O. Cruz is with Département de génie Logiciel et
des TI, École de Technologie Supérieure, Montreal, Canada (rafael.menelau-
cruz@etsmtl.ca)4https://github.com/prgc/ps-des

###### Abstract

Dynamic Ensemble Selection (DES) is a Multiple Classifier Systems (MCS)
approach that aims to select an ensemble for each query sample during the
selection phase. Even with the proposal of several DES approaches, no
particular DES technique is the best choice for different problems. Thus, we
hypothesize that selecting the best DES approach per query instance can lead
to better accuracy. To evaluate this idea, we introduce the Post-Selection
Dynamic Ensemble Selection (PS-DES) approach, a post-selection scheme that
evaluates ensembles selected by several DES techniques using different
metrics. Experimental results show that using accuracy as a metric to select
the ensembles, PS-DES performs better than individual DES techniques. PS-DES
source code is available in a GitHub repository4.

##  I Introduction

Multiple Classifier Systems (MCS) are often used to improve the accuracy and
reliability of machine learning models [1]. MCS has three main phases:
generation, selection, and combination. In the generation phase, a pool of
classifiers is created by training base classifiers with techniques such as
Bagging [2], different models, and variations of learning algorithms [3].

The selection phase selects the most competent classifiers in the pool to
predict a given query sample. Two main approaches for selecting classifiers
are static selection [4] and dynamic selection (DS) [1]. In the static
approach, the selection is performed during training and works on the
classifiers’ overall performance on a validation set. In contrast, the DS
approach selects the classifiers on the fly based on their competence in
predicting a specific query sample. When only one classifier is selected, it
is called Dynamic Classifier Selection (DCS), and when more than one
classifier is selected, it is called Dynamic Ensemble Selection (DES).
Examples of DES techniques include META-DES [5], Dynamic Ensemble Selection
Performance (DES-P) [6], and K-Nearest Oracles Union (KNORA-U) [4]. The last
phase of an MCS is combination, also called integration. In this phase, the
output of all the classifiers selected is combined to produce a final
prediction. The combination or integration of the predictions can be done in
various ways, including voting and weighting [7].

Currently, research efforts in DES are focused on proposing new methods to
improve phases, such as generation [8], selection [9], and combination [10].
Additionally, there have been attempts to apply DES to other areas of
knowledge [11]. Despite the progress that has been made, no DES technique is
suitable for all problems. This is in line with the statistical rationale for
MCS [12], which suggests that combining multiple classifiers increases the
likelihood of finding the optimal result for any given problem. However, to
the authors’ knowledge, the field still lacks techniques that work on
evaluating the ensembles selected by DES methods and explores the advantages
of pre-selected ensembles to obtain better performance.

Aiming to evaluate this gap in DES’ research field, we pose the following
research question: “How to analyze ensembles selected by different DES
techniques and choose the one having the highest correct prediction
potential?” To investigate this question, we propose the Post-Selection
Dynamic Ensemble Selection (PS-DES) approach. PS-DES is based on the
assumption that different selection criteria may lead to different selected
ensembles, and the best criteria used to select an ensemble may differ on an
instance-to-instance basis. PS-DES aims to analyze and choose the best
ensemble from a set of ensembles generated by various DES techniques to obtain
more reliable predictions. Therefore, our proposal work as a post-selection
scheme, i.e., it performs after the selection phase of different DES methods
and before the combination phase.

Moreover, the best ensemble is selected based on a new concept of ensemble
potential proposed in this work. In contrast to the selection criteria
employed in many DES methods such as META-DES [5] that work by estimating the
quality or competence of each model, the proposed ensemble potential evaluates
whether the final selected ensemble of classifiers is reliable. We propose
three approaches based on classical performance estimation metrics for
measuring the ensemble potential: Accuracy, F-score, and Matthew’s Correlation
Coefficient.

Experiments over 20 classification datasets and considering three different
performance evaluation metrics demonstrate that the post-selection scheme
based on the ensemble potential leads to systematic improvement in
classification performance over state-of-the-art DES methods. Thus, the
evaluation of the pre-selected ensemble capabilities should not be neglected.
The rest of the paper is organized as follows: Section II shows a literature
review on DES. Section III presents our proposal. Section IV shows the
experimental setup. The results are discussed in Section V, and Section VI
presents the conclusions.

##  II Literature review

Typically, the development of a DES involves three stages. Firstly, in the
generation phase, a pool of classifiers is generated. Secondly, in the
selection phase, a subset of classifiers (ensemble) is chosen from the pool
created in the generation phase. Finally, the classifiers in the ensemble are
combined to classify a given query sample in the combination, or integration,
phase.

During DES’ generation phase, a pool of classifiers is created, denoted as
$P=\\{C_{1},C_{2},\ldots,C_{M}\\}$, where $M$ represents the number of
classifiers in the pool. The classifiers in the pool must exhibit both
diversity and accuracy. Diversity [12] refers to the property that the
classifiers should not make the same prediction mistakes, as this is crucial
to cover the feature space adequately. Several approaches can be used to
generate a pool. These approaches include using different distributions of the
training set, such as Bagging [2], using different parameters for the same
base classifier (e.g., variations in the number of neighbors in a k-Nearest
Neighbors algorithm), or using different base classifiers altogether, which
are called heterogeneous ensembles [3]. Heterogeneous ensembles tend to be
more diverse than homogeneous ones due to their different mathematical
formulations, which typically result in different classification results [13].

The second phase of developing a DES is selection, which aims to choose a
subset of classifiers ($P^{\prime}\subseteq P$), also known as an ensemble.
There are two approaches to selection, namely static and dynamic [1]. A fixed
subset of classifiers is chosen for all test samples in the static approach.
In contrast, the dynamic approach, called Dynamic Ensemble Selection (DES),
involves selecting a subset of the pool for each query sample
$\mathbf{x}_{q}$. In dynamic selection, classifiers are chosen based on some
criteria, given the pool created in the previous phase. Among the criteria
found in the literature are the Oracle approach, as seen in KNORA-E, KNORA-U
[4], and K-Nearest Output Profile (KNOP) [14], accuracy-based methods, such as
DES Performance (DES-P) [6], and meta-learning, as in the case of META-DES
[5]. These criteria are typically computed from the Region of Competence
(RoC), a local region for a query sample ($\mathbf{x}_{q}$), denoted as
$\theta_{\mathbf{x}_{q}}$, which is a fundamental concept in dynamic selection
approaches. The RoC is usually obtained by applying k-NN or clustering methods
to a validation set (DSEL) or the training set itself, such that
$\theta_{\mathbf{x}_{q}}=\\{\mathbf{x}_{1},\ldots,\mathbf{x}_{k}\\}$, where
$k$ is the size of the ROC.

The final phase of a DES is integration, also called aggregation or
combination, which involves combining the classifiers selected in the
selection phase when multiple classifiers are chosen. Techniques used in this
phase include majority vote, product rule, and sum rule [12].

It is worth noting that research papers related to DES do not focus on
assessing ensembles that DES techniques have already generated. Elmi and
Eftekhari [9] presented a solution by utilizing the selection phase of DES
approaches. However, their proposed approach only allowed for layer-by-layer
ensemble selection and did not allow the evaluation of the collective output
of all ensembles generated by DES methods.

##  III Post-Selection Dynamic Ensemble Selection

The proposed Post-Selection Dynamic Ensemble Selection (PS-DES) is based on
the notion of potential, the capability of an ensemble selected by a given DS
technique to make a correct prediction. Consequently, it works as a post-
processing scheme for ensembles chosen according to different criteria (e.g.,
meta-learning, Oracle, accuracy). In addition, this proposal aims to evaluate
the quality or potential of a selected ensemble, which contrasts with the
current DES methods that build an ensemble by selecting multiple competent
classifiers individually without trying to characterize the selected dynamic
ensemble. This approach consists of three phases: (1) pool generation and
setup, (2) post-selection, and (3) combination.

###  III-A Phase 1: Pool generation and DES’ setup

The initial stage of PS-DES involves generating a pool of classifiers and
configuring the DES techniques. First, Bagging generates multiple bootstraps
($T^{b}$) from the original dataset ($T$), where $b$ is the number of
bootstraps. Then, a pool of $b\times m$ classifiers, denoted as
$P=\\{{C_{1}^{1},C_{1}^{2},\ldots,C_{m}^{b}\\}}$, is constructed by training
each of the $m$ classifiers ($C_{1},\ldots,C_{m}$) on each of the $b$
bootstraps generated by Bagging. Finally, any DES techniques specified by the
user, including META-DES, KNORA-U, and DES-P, are initialized using the same
pool $P$ as input. These techniques can be used to select an intermediate
ensemble later validated by our post-processing scheme to obtain the optimal
one. All DES approaches, $DES_{set}=\\{des_{1},des_{2},\ldots,des_{n}\\}$, are
consolidated into the set $DES_{set}$.

###  III-B Phase 2: Selection

This phase seeks to identify the optimal ensemble from a set of ensembles
generated by different DES techniques, and Algorithm 1 shows its pseudo-code.
Given a query sample ($\mathbf{x}_{q}$), the validation dataset $DSEL$, and a
set of DES techniques ($DES_{set}$), this phase involves selecting several
dynamic ensembles $P^{\prime}$, each one created using a different DES method,
and assessing their effectiveness in order to determine which ensemble is most
likely to perform well for the given $\mathbf{x}_{q}$.

Algorithm 1 PS-DES selection

1:procedure PS-DES_Selection($\mathbf{x_{q}},DSEL,DES_{set}$)

2: $pot_{max}\leftarrow 0$

3: $P^{sel}\leftarrow\emptyset$

4: $\theta_{\mathbf{x}_{q}}\leftarrow calculate\\_{ROC}(\mathbf{x}_{q},DSEL)$

5: for $des$ in $DES_{set}$ do

6: $P^{\prime}\leftarrow get\\_ensemble(des,\theta_{\mathbf{x}_{q}})$

7: $pot_{des}\leftarrow calculate\\_pot(P^{\prime})$

8: if $pot_{des}\geq pot_{max}$ then

9: $pot_{max}\leftarrow pot_{des}$

10: $P^{sel}\leftarrow P^{\prime}$

11: end if

12: end for

13: return $P^{sel}$

14:end procedure

This phase begins by computing the Region on Competence (RoC)
($\theta_{\mathbf{x}_{q}}$) for the query sample $\mathbf{x}_{q}$ using the
k-NN algorithm. It is essential to note that all $DES_{n}$ utilize the same
RoC ($\theta_{\mathbf{x}_{q}}$) based on the k-NN, thereby reducing the
computational burden of implementing multiple selection criteria.

Then, for each technique in $DES_{set}$, a set of classifiers is selected
according to its competence estimation and selection criterion, forming the
ensemble $P^{\prime}$ (lines 5 and 6). Subsequently, the potential of the
generated ensemble $P^{\prime}$ is evaluated (line 7). As the class label of
$\mathbf{x}_{q}$ is unknown, the potential assumes that the output class of
$\mathbf{x}_{q}$ corresponds to the majority vote of the ensemble.
Consequently, it computes the potential of this ensemble by assessing the
proportion of methods in it that contribute to this decision. For instance,
given a binary classification problem and an ensemble with seven base
classifiers, $P^{\prime}=\\{C_{1},\ldots,C_{7}\\}$ selected by a given DES
technique, and let $y_{P^{\prime}}=\\{0,1,0,0,1,1,1\\}$ be the predictions of
the base classifier for the given $\mathbf{x}_{q}$. The majority vote would
give the class $1$ as the answer. The potential is then estimated based on a
classical performance metric comparing the ensemble majority vote and the
votes of each classifier using a performance metric. If accuracy is used to
calculate $P^{\prime}$ potential, the value would be $pot_{des}=0.57$. If the
F-score is chosen as the potential metric, the $P^{\prime}$ potential is
$pot_{des}=0.73$.

After evaluating the potential of all possible ensembles, the one that
obtained the highest value, $P^{sel}$, is returned as the selected one for the
combination step.

###  III-C Phase 3: Combination

Once the $P^{sel}$ selection is complete, Phase 3 begins, which is accountable
for combining the classifiers into $P^{sel}$, using techniques such as
majority vote or sum rule.

##  IV Experimental setup

Datasets. The experiments were conducted using 20 datasets from the UCI
Machine Learning Repository [15], which vary in sample size, dimensions,
number of classes, and Imbalance Ratio (IR) (Table I). Each dataset $T$ is
split into three parts: training (50%), $DSEL$ (25%), and testing (25%). This
split is stratified, meaning that the proportions of the classes between the
three datasets are maintained. For each dataset, we run 30 replications,
changing the distribution of the sets (holdout) to obtain the average values
for the evaluated metrics. The data is scaled using the Standard Scaler (also
known as Z-score normalization [16]).

TABLE I: Datasets main characteristics. The number of samples, dimensions (Dim), classes, and Imbalance Ratio (IR). Datasets | Examples | Dim | Classes | IR
---|---|---|---|---
appendicitis | 106 | 7 | 2 | 2.52
australian | 690 | 14 | 2 | 1.12
balance | 625 | 4 | 3 | 2.63
cmc | 1473 | 9 | 3 | 1.30
column_3C | 310 | 6 | 3 | 2.50
diabetes | 768 | 8 | 2 | 1.86
glass1 | 214 | 9 | 2 | 1.82
glass6 | 214 | 9 | 2 | 6.38
haberman | 306 | 3 | 2 | 2.78
hayes | 160 | 4 | 3 | 3.40
heart | 270 | 13 | 2 | 1.25
led7digit | 500 | 7 | 10 | 1.54
mammographic | 830 | 5 | 2 | 1.15
musk | 476 | 166 | 2 | 1.29
pima | 768 | 8 | 2 | 1.90
sonar | 208 | 60 | 2 | 1.14
vehicle | 846 | 18 | 4 | 1.10
vehicle2 | 846 | 18 | 2 | 2.88
vowel | 990 | 13 | 11 | 1.00
wdbc | 683 | 9 | 2 | 1.85

Phase 1. First, Bagging, 100 bootstraps were used for all experiments,
consistent with previous studies [1, 4, 5]. For the pool generation, three
base classifiers (Perceptron, Logistic Regression, and Naive Bayes) were
selected for the experiments. As they have different mathematical foundations
and low computational costs [17, 10, 1] they are suitable for building a
diverse and lightweight pool of classifiers. Thus, the classifier pool ($P$)
consisted of 300 classifiers ($\text{3 base classifiers}\times\text{100
bootstraps}$). Since the focus of the research was not on optimizing each base
model’s hyperparameters, the default hyperparameters values from scikit-learn
were used.

Four DES approaches (KNORA-U, KNOP, DES-P, and META-DES) were chosen due to
their application of various selection criteria (e.g., Oracle, accuracy, meta-
learning). These approaches showed superior performance in a recent empirical
study [1]. We applied these DES methods default hyperparameter configurations
of the DESlib 0.3 library [18] to guarantee experiment consistency. Moreover,
the same pool of classifiers was utilized to fairly compare all DES
techniques.

Phase 2. The Region of Competence (RoC) was calculated applying k-Nearest
Neighbors (k-NN) with k = 7, as suggested in [1]. To assess the performance of
the ensembles, we employed a range of evaluation metrics, including accuracy,
F-score, and Matthews Correlation Coefficient (MCC). Accuracy is a popular
metric for DES techniques, although it may not be suitable for imbalanced
datasets (i.e., high IR). Meanwhile, F-score and MCC are more suitable for
such datasets. F-score is advantageous in scenarios where there is an
appreciation for recall and precision, since these two metrics are used in its
calculation. The MCC considers false-negative rates in its formulation, which
can be of interest to specific problems.

The PS-DES variants are labeled according to the metric used to calculate the
potential: accuracy (PS-DES-acc), F-score (PS-DES-F), and Matthews Correlation
Coefficient (PS-DES-MCC). To assess whether the proposed metrics perform
better than random selection, we also conducted an experiment that randomly
selected the best ensemble (PS-DES-Random).

Phase 3. Finally, majority voting was used as a combination approach since
individual DES techniques usually apply it [1].

##  V Results and discussions

The proposed method is evaluated based on three metrics: accuracy (Table II),
F-score (Table III), and MCC (Table IV). Upon examining the tables, our
results indicate that PS-DES-acc outperforms all the other approaches in all
metrics. The PS-DES-acc obtained the best rank considering all performance
metrics, followed by the variant using the F-score metric for computing the
ensemble’s potential. These results are interesting since, even though the
final proposal may be evaluated regarding a different performance metric
(e.g., F-score or MCC), using accuracy as the metric for computing the
ensemble potential is more advantageous.

Analogously, MCC obtained the lowest ranking among all PS-DES variants even
when in the scenario that MCC is used as a performance evaluation metric to
compute the overall method performance. This result indicates no relation
between the metrics selected for calculating ensemble potential and the same
metric applied to evaluate the approaches. For accuracy, F-score, and MCC, the
chosen metric for calculating the potential does not interfere with the
approach’s evaluation. Nevertheless, according to these tables, the average
ranking of PS-DES approaches is systematically better when compared to
individual DS techniques (e.g., META-DES). Thus, the proposed post-processing
selection scheme indeed leads to more robust dynamic ensemble selection
systems.

However, to see if such a difference in performance is significant, we need to
go further and perform a more fine-grained analysis by comparing pair of
techniques over multiple datasets. Hence, we also conducted one analysis based
on the number of wins, ties, and losses (w/t/l) obtained by a control
technique and the Wilcoxon signed rank test with a confidence level of 95%.
Results of these pairwise comparisons are presented in Tables V, VI, and VII
for the PS-DES-acc, PS-DES-F and PS-DES-MCC methods, respectively.

TABLE II: Average accuracy and rankings of the evaluated methods for each dataset. The best result per dataset is presented in bold. Datasets | KNORA-U | KNOP | META-DES | DES-P | Random | PS-DES-MCC | PS-DES-F | PS-DES-acc
---|---|---|---|---|---|---|---|---
appendicitis | 0.859 | 0.862 | 0.851 | 0.864 | 0.855 | 0.859 | 0.868 | 0.869
australian | 0.847 | 0.846 | 0.848 | 0.846 | 0.846 | 0.849 | 0.849 | 0.849
balance | 0.883 | 0.886 | 0.893 | 0.883 | 0.885 | 0.893 | 0.891 | 0.889
cmc | 0.509 | 0.512 | 0.488 | 0.513 | 0.506 | 0.483 | 0.505 | 0.510
column_3C | 0.845 | 0.844 | 0.841 | 0.848 | 0.846 | 0.839 | 0.845 | 0.847
diabetes | 0.768 | 0.769 | 0.762 | 0.765 | 0.767 | 0.762 | 0.767 | 0.770
glass1 | 0.642 | 0.640 | 0.674 | 0.682 | 0.653 | 0.682 | 0.672 | 0.672
glass6 | 0.940 | 0.938 | 0.947 | 0.944 | 0.943 | 0.948 | 0.948 | 0.947
haberman | 0.731 | 0.731 | 0.726 | 0.737 | 0.734 | 0.727 | 0.737 | 0.734
hayes | 0.611 | 0.619 | 0.670 | 0.640 | 0.640 | 0.665 | 0.684 | 0.682
heart | 0.836 | 0.838 | 0.833 | 0.841 | 0.836 | 0.837 | 0.838 | 0.837
led7digit | 0.725 | 0.726 | 0.708 | 0.723 | 0.721 | 0.697 | 0.704 | 0.723
mammographic | 0.830 | 0.831 | 0.822 | 0.828 | 0.830 | 0.827 | 0.824 | 0.825
musk | 0.779 | 0.781 | 0.788 | 0.792 | 0.786 | 0.790 | 0.791 | 0.794
pima | 0.770 | 0.768 | 0.761 | 0.766 | 0.766 | 0.762 | 0.763 | 0.766
sonar | 0.771 | 0.771 | 0.796 | 0.784 | 0.787 | 0.787 | 0.796 | 0.798
vehicle | 0.750 | 0.752 | 0.761 | 0.752 | 0.750 | 0.755 | 0.758 | 0.761
vehicle2 | 0.949 | 0.947 | 0.956 | 0.951 | 0.951 | 0.955 | 0.954 | 0.955
vowel | 0.965 | 0.965 | 0.981 | 0.967 | 0.969 | 0.980 | 0.978 | 0.977
wdbc | 0.970 | 0.970 | 0.969 | 0.970 | 0.970 | 0.968 | 0.969 | 0.968
ranking | 5.45 | 5.03 | 4.90 | 3.88 | 5.23 | 4.73 | 3.70 | 3.10
TABLE III: Average F-score and rankings of the evaluated methods for each dataset. The best result per dataset is presented in bold. Datasets | KNORA-U | KNOP | META-DES | DES-P | Random | PS-DES-MCC | PS-DES-F | PS-DES-acc
---|---|---|---|---|---|---|---|---
appendicitis | 0.708 | 0.718 | 0.717 | 0.749 | 0.715 | 0.732 | 0.760 | 0.754
australian | 0.845 | 0.843 | 0.845 | 0.844 | 0.843 | 0.846 | 0.847 | 0.846
balance | 0.612 | 0.615 | 0.631 | 0.613 | 0.615 | 0.637 | 0.628 | 0.626
cmc | 0.472 | 0.477 | 0.454 | 0.484 | 0.472 | 0.451 | 0.474 | 0.478
column_3C | 0.799 | 0.796 | 0.792 | 0.803 | 0.800 | 0.790 | 0.798 | 0.801
diabetes | 0.725 | 0.725 | 0.719 | 0.723 | 0.724 | 0.719 | 0.726 | 0.729
glass1 | 0.422 | 0.415 | 0.585 | 0.577 | 0.512 | 0.584 | 0.530 | 0.535
glass6 | 0.850 | 0.847 | 0.880 | 0.868 | 0.863 | 0.879 | 0.878 | 0.873
haberman | 0.506 | 0.507 | 0.525 | 0.542 | 0.519 | 0.520 | 0.555 | 0.549
hayes | 0.627 | 0.634 | 0.688 | 0.658 | 0.654 | 0.681 | 0.697 | 0.699
heart | 0.832 | 0.834 | 0.829 | 0.837 | 0.832 | 0.833 | 0.834 | 0.833
led7digit | 0.718 | 0.719 | 0.700 | 0.715 | 0.714 | 0.687 | 0.693 | 0.719
mammographic | 0.829 | 0.830 | 0.821 | 0.826 | 0.829 | 0.825 | 0.822 | 0.824
musk | 0.770 | 0.772 | 0.781 | 0.785 | 0.778 | 0.783 | 0.784 | 0.787
pima | 0.727 | 0.724 | 0.718 | 0.724 | 0.722 | 0.718 | 0.721 | 0.725
sonar | 0.768 | 0.767 | 0.792 | 0.780 | 0.783 | 0.784 | 0.793 | 0.795
vehicle | 0.742 | 0.745 | 0.757 | 0.744 | 0.743 | 0.751 | 0.753 | 0.755
vehicle2 | 0.933 | 0.931 | 0.943 | 0.936 | 0.935 | 0.941 | 0.939 | 0.941
vowel | 0.886 | 0.886 | 0.940 | 0.894 | 0.899 | 0.936 | 0.929 | 0.929
wdbc | 0.966 | 0.966 | 0.965 | 0.967 | 0.967 | 0.965 | 0.966 | 0.964
ranking | 5.80 | 5.45 | 4.45 | 3.95 | 5.45 | 4.60 | 3.45 | 2.85
TABLE IV: Average MCC and rankings of the evaluated methods for each dataset. The best result per dataset is presented in bold. Datasets | KNORA-U | KNOP | META-DES | DES-P | Random | PS-DES-MCC | PS-DES-F | PS-DES-acc
---|---|---|---|---|---|---|---|---
appendicitis | 0.503 | 0.517 | 0.489 | 0.563 | 0.500 | 0.529 | 0.574 | 0.566
australian | 0.694 | 0.691 | 0.694 | 0.691 | 0.690 | 0.695 | 0.697 | 0.695
balance | 0.794 | 0.801 | 0.812 | 0.795 | 0.798 | 0.813 | 0.809 | 0.806
cmc | 0.227 | 0.232 | 0.196 | 0.238 | 0.224 | 0.187 | 0.223 | 0.232
column_3C | 0.755 | 0.753 | 0.748 | 0.759 | 0.756 | 0.744 | 0.755 | 0.757
diabetes | 0.466 | 0.468 | 0.454 | 0.460 | 0.464 | 0.453 | 0.466 | 0.472
glass1 | 0.019 | 0.008 | 0.226 | 0.242 | 0.134 | 0.247 | 0.189 | 0.196
glass6 | 0.729 | 0.723 | 0.770 | 0.755 | 0.749 | 0.775 | 0.774 | 0.764
haberman | 0.131 | 0.130 | 0.144 | 0.193 | 0.157 | 0.138 | 0.201 | 0.185
hayes | 0.416 | 0.427 | 0.505 | 0.467 | 0.461 | 0.499 | 0.529 | 0.524
heart | 0.673 | 0.677 | 0.667 | 0.682 | 0.673 | 0.674 | 0.677 | 0.674
led7digit | 0.696 | 0.697 | 0.678 | 0.693 | 0.692 | 0.666 | 0.674 | 0.694
mammographic | 0.660 | 0.662 | 0.644 | 0.655 | 0.660 | 0.652 | 0.647 | 0.650
musk | 0.547 | 0.551 | 0.566 | 0.575 | 0.562 | 0.572 | 0.574 | 0.579
pima | 0.471 | 0.466 | 0.451 | 0.462 | 0.461 | 0.452 | 0.455 | 0.464
sonar | 0.547 | 0.544 | 0.594 | 0.570 | 0.576 | 0.575 | 0.595 | 0.600
vehicle | 0.672 | 0.673 | 0.684 | 0.674 | 0.671 | 0.676 | 0.681 | 0.684
vehicle2 | 0.867 | 0.863 | 0.886 | 0.873 | 0.872 | 0.884 | 0.880 | 0.883
vowel | 0.784 | 0.782 | 0.885 | 0.798 | 0.808 | 0.877 | 0.864 | 0.864
wdbc | 0.933 | 0.933 | 0.931 | 0.934 | 0.933 | 0.931 | 0.932 | 0.929
ranking | 5.45 | 5.30 | 4.80 | 3.80 | 5.35 | 4.65 | 3.55 | 3.10
TABLE V: Statistical Analyses for PS-DES-Acc against state-of-the-art DS methods. The line (w/t/l) presents the number of wins, ties, and losses it obtained compared to the column-wise technique. The p-value line shows the result of applying the paired Wilcoxon statistical test. Statistically different results ($\alpha=0.5$) are highlighted in bold. Metric |  | KNORA-U | KNOP | META-DES | DES-P | Random | PS-DES-MCC | PS-DES-F
---|---|---|---|---|---|---|---|---
Accuracy | w/t/l | 16/0/4 | 14/0/6 | 13/0/7 | 13/0/7 | 17/1/2 | 12/1/7 | 12/0/8
p-value | 0.002 | 0.003 | 0.005 | 0.062 | 0.001 | 0.016 | 0.071
F-score | w/t/l | 17/0/3 | 17/0/3 | 13/0/7 | 14/0/6 | 18/0/2 | 12/0/8 | 12/0/8
p-value | 0.000 | 0.000 | 0.049 | 0.020 | 0.000 | 0.066 | 0.139
MCC | w/t/l | 16/0/4 | 14/0/6 | 13/0/7 | 13/0/7 | 18/0/2 | 13/0/7 | 11/0/9
p-value | 0.001 | 0.001 | 0.029 | 0.077 | 0.000 | 0.049 | 0.261
TABLE VI: Statistical Analyses for PS-DES-F against state-of-the-art DS methods. The line (w/t/l) presents the number of wins, ties, and losses that it obtained compared to the column-wise technique. The p-value line shows the result of applying the paired Wilcoxon statistical test. Statistically different results ($\alpha=0.5$) are highlighted in bold. Metric |  | KNORA-U | KNOP | META-DES | DES-P | Random | PS-DES-MCC | PS-DES-acc
---|---|---|---|---|---|---|---|---
Accuracy | w/t/l | 13/1/6 | 13/1/6 | 13/0/7 | 10/0/10 | 13/0/7 | 15/0/5 | 8/0/12
p-value | 0.015 | 0.032 | 0.049 | 0.237 | 0.018 | 0.024 | 0.934
F-score | w/t/l | 15/0/5 | 15/0/5 | 13/0/7 | 11/0/9 | 15/0/5 | 14/0/6 | 8/0/12
p-value | 0.003 | 0.005 | 0.174 | 0.147 | 0.003 | 0.066 | 0.869
MCC | w/t/l | 13/0/7 | 14/0/6 | 14/0/6 | 11/0/9 | 14/0/6 | 14/0/6 | 9/0/11
p-value | 0.010 | 0.013 | 0.045 | 0.139 | 0.005 | 0.041 | 0.751
TABLE VII: Statistical Analyses for PS-DES-MCC against state-of-the-art DS methods. The line (w/t/l) presents the number of wins, ties, and losses that it obtained compared to the column-wise technique. The p-value line shows the result of applying the paired Wilcoxon statistical test. Statistically different results ($\alpha=0.5$) are highlighted in bold. Metric |  | KNORA-U | KNOP | META-DES | DES-P | Random | PS-DES-F | PS-DES-acc
---|---|---|---|---|---|---|---|---
Accuracy | w/t/l | 12/0/8 | 10/0/10 | 10/0/10 | 9/0/11 | 12/0/8 | 5/0/15 | 7/1/12
p-value | 0.174 | 0.205 | 0.580 | 0.649 | 0.273 | 0.978 | 0.984
F-score | w/t/l | 13/0/7 | 12/0/8 | 7/0/13 | 9/0/11 | 13/0/7 | 6/0/14 | 8/0/12
p-value | 0.032 | 0.045 | 0.861 | 0.522 | 0.101 | 0.938 | 0.938
MCC | w/t/l | 13/0/7 | 12/0/8 | 9/0/11 | 9/0/11 | 11/0/9 | 6/0/14 | 7/0/13
p-value | 0.071 | 0.088 | 0.676 | 0.663 | 0.194 | 0.962 | 0.955

The pairwise statistical analysis of PS-DES-acc shows it outperforms KNORA-U,
KNOP, META-DES, Random, and PS-DES-MCC regarding accuracy (Table V). No
significant difference is observed between PS-DES-acc and DES-P or PS-DES-F.
However, considering the presence of datasets with $IR>1$, it is necessary to
consider F-score and MCC. The F-score analysis reveals that PS-DES-acc
outperforms all DES individual techniques and Random, with no significant
difference to PS-DES-F and PS-DES-MCC. For MCC, PS-DES-acc performs
exceptionally well and obtains significantly better results compared to all
techniques apart from DES-P and PS-DES-F. Ultimately, this variant based on
accuracy for computing the ensemble potential obtained more victories than all
other models, regardless of the performance metric used.

The statistical analysis of PS-DES-F (Table VI) indicates that it performs
better than KNORA-U, KNOP, and Random on all three metrics. However, no
statistical difference is found for MCC when compared with DES-P. However, the
win-tie-loss analysis demonstrates that the PS-DES-F systematically obtained
more wins against the state-of-the-art DES techniques and the random selection
scheme (between 13 to 15 wins over the 20 datasets). In contrast, the analysis
of PS-DES-MCC (Table VII) presents the worst results compared to PS-DES-acc
and PS-DES-F. Based on Wilcoxon’s test analysis, PS-DES-MCC scores better than
KNORA-U and KNOP only in F-score. The hypothesis that PS-DES-MCC scores better
cannot be refuted for all other metrics and comparisons.

In summary, the results indicate that PS-DES-acc and PS-DES-F yield comparable
outcomes. Still, PS-DES-acc holds a slight advantage over its competitor,
particularly when it is compared against the state-of-the-art DES methods.

##  VI Conclusion

This work proposed a new Dynamic Ensemble Selection (DES) method: Post-
Selection Dynamic Ensemble Selection (PS-DES). This method is based on the
idea that the optimal criteria for ensemble selection may differ at the
instance level leading to ensembles with different qualities or “potentials”.
To this end, the approach evaluates the potential of ensembles chosen by
various DES techniques to determine which is more suitable for labeling a
given instance.

Experiments demonstrate no direct correlation between the metrics applied for
calculating the ensemble potential and for evaluating the approaches, as the
PS-DES-acc was found to achieve the best overall results in all cases.
Additionally, PS-DES was consistently superior to the existing state-of-the-
art DES techniques, which implies that evaluating the selected ensembles as a
collective is more important than assessing and choosing each base classifier
separately, as is the trend in most DES methods. Thus, post-processing
approaches in DES are vital, and future works will explore new metrics for
measuring the ensemble’s potential.

## Acknowledgment

The authors would like to thank the Instituto Federal de Pernambuco,
Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES), Fundação
de Amparo à Ciência e Tecnologia de Pernambuco (FACEPE), and the Natural
Sciences and Engineering Research Council of Canada (NSERC).

## References

  * [1] R. M. Cruz, R. Sabourin, and G. D. Cavalcanti, “Dynamic classifier selection: Recent advances and perspectives,” _Information Fusion_ , vol. 41, pp. 195–216, 2018.
  * [2] L. Breiman, “Bagging predictors,” _Machine Learning_ , vol. 24, no. 2, pp. 123–140, 1996.
  * [3] G. Tsoumakas, L. Angelis, and I. Vlahavas, “Selective fusion of heterogeneous classifiers,” _Intelligent Data Analysis_ , vol. 9, no. 6, pp. 511–525, 2005.
  * [4] A. H. Ko, R. Sabourin, and A. S. Britto Jr, “From dynamic classifier selection to dynamic ensemble selection,” _Pattern recognition_ , vol. 41, no. 5, pp. 1718–1731, 2008.
  * [5] R. M. Cruz, R. Sabourin, G. D. Cavalcanti, and T. Ing Ren, “META-DES: A dynamic ensemble selection framework using meta-learning,” _Pattern Recognition_ , vol. 48, no. 5, p. 1925–1935, 2015.
  * [6] T. Woloszynski, M. Kurzynski, P. Podsiadlo, and G. W. Stachowiak, “A measure of competence based on random classification for dynamic ensemble selection,” _Information Fusion_ , vol. 13, no. 3, pp. 207–213, 2012.
  * [7] Z.-H. Zhou, _Ensemble Methods: Foundations and Algorithms_ , 1st ed. Chapman and Hall/CRC, 2012.
  * [8] M. Monteiro, A. S. Britto, J. P. Barddal, L. S. Oliveira, and R. Sabourin, “Exploring diversity in data complexity and classifier decision spaces for pool generation,” _Information Fusion_ , vol. 89, pp. 567–587, 2023.
  * [9] J. Elmi and M. Eftekhari, “Multi-layer selector (mls): Dynamic selection based on filtering some competence measures,” _Applied Soft Computing_ , vol. 104, p. 107257, 2021.
  * [10] V. S. Costa, A. D. S. Farias, B. Bedregal, R. H. Santiago, and A. M. d. P. Canuto, “Combining multiple algorithms in classifier ensembles using generalized mixture functions,” _Neurocomputing_ , vol. 313, pp. 402–414, 2018.
  * [11] B. Swaminathan, S. Palani, and S. Vairavasundaram, “Meta learning-based dynamic ensemble model for crop selection,” _Applied Artificial Intelligence_ , vol. 36, no. 1, p. 2145646, 2022.
  * [12] L. I. Kuncheva, _Combining Pattern Classifiers: Methods and Algorithms: Second Edition_. Wiley Publishing, 2014.
  * [13] L. Wang, T. Mo, X. Wang, W. Chen, Q. He, X. Li, S. Zhang, R. Yang, J. Wu, X. Gu _et al._ , “A hierarchical fusion framework to integrate homogeneous and heterogeneous classifiers for medical decision-making,” _Knowledge-Based Systems_ , vol. 212, p. 106517, 2021.
  * [14] P. R. Cavalin, R. Sabourin, and C. Y. Suen, “Dynamic selection approaches for multiple classifier systems,” _Neural Computing and Applications_ , vol. 22, pp. 673–688, 2013.
  * [15] D. Dua and C. Graff, “UCI machine learning repository,” 2020. [Online]. Available: http://archive.ics.uci.edu/ml
  * [16] L. B. de Amorim, G. D. Cavalcanti, and R. M. Cruz, “The choice of scaling technique matters for classification performance,” _Applied Soft Computing_ , vol. 133, p. 109924, 2023.
  * [17] T. T. Nguyen, A. V. Luong, M. T. Dang, A. W. C. Liew, and J. McCall, “Ensemble Selection based on Classifier Prediction Confidence,” _Pattern Recognition_ , vol. 100, p. 107104, 2020.
  * [18] R. M. O. Cruz, L. G. Hafemann, R. Sabourin, and G. D. C. Cavalcanti, “DESlib: A dynamic ensemble selection library in python,” _Journal of Machine Learning Research_ , vol. 21, no. 8, pp. 1–5, 2020.


# Thermoelectric cooling of a finite reservoir coupled to a quantum dot

Stephanie Matern  Pitaevskii BEC Center, CNR-INO and Dipartimento di Fisica,
Università di Trento, Trento, Italy  NanoLund and Solid State Physics, Lund
University, Box 118, 22100 Lund, Sweden  Saulo V. Moreira  School of Physics,
Trinity College Dublin, Dublin 2, Ireland  NanoLund and Mathematical Physics,
Lund University, Box 118, 22100 Lund, Sweden  Peter Samuelsson  NanoLund and
Mathematical Physics, Lund University, Box 118, 22100 Lund, Sweden  Martin
Leijnse  NanoLund and Solid State Physics, Lund University, Box 118, 22100
Lund, Sweden

###### Abstract

We investigate non-equilibrium transport of charge and heat through an
interacting quantum dot coupled to a finite electron reservoir. Both the
quantum dot and the finite reservoir are coupled to conventional electric
contacts, i.e., infinite electron reservoirs, between which a bias voltage can
be applied. We develop a phenomenological description of the system, combining
a rate equation for transport through the quantum dot with standard linear
response expressions for transport between the finite and infinite reservoirs.
The finite reservoir is assumed to be in a quasi-equilibrium state with time-
dependent chemical potential and temperature which we solve for self-
consistently. We show that the finite reservoir can have a large impact on the
stationary state transport properties, including a shift and broadening of the
Coulomb diamond edges. We also demonstrate that there is a region around the
conductance lines where a heat current flows out of the finite reservoir. Our
results reveal the dependence of the temperature that can be reached by this
thermoelectric cooling on the system parameters, in particular the coupling
between the finite and infinite reservoirs and additional heat currents
induced by electron-phonon couplings, and can thus serve as a guide to
experiments on quantum dot-enabled thermoelectric cooling of finite electron
reservoirs. Finally, we study the full dynamics of the system, with a
particular focus on the timescales involved in the thermoelectric cooling.

##  I Introduction

Experimental advances in the fabrication of nanoscale electronic devices have
made it possible to employ measurements of non-equilibrium transport to deduce
the properties of a (quantum) system. Such devices include a wide range of
systems, e.g., molecular junctions, carbon nanotubes, graphene structures and
quantum dots (QDs)[1, 2, 3, 4, 5]. They are a fundamental building block for
applications such as transistors, (thermal) diodes, nanoscale heat engines,
and sensors [6, 7, 8, 9, 10, 11, 12]. From a theoretical point of view, the
description of open quantum systems out of equilibrium remains a challenging
task [13]. In non-equilibrium transport scenarios, it is common to assume that
the system is in contact with infinite reservoirs at different temperatures
and chemical potentials. Although this is a suitable description for a number
of experimental situations, accounting for the finite aspect of reservoirs is
needed in many nanoscale, non-equilibrium scenarios [14]. Various experiments
on, e.g., superconducting [15, 16] or ultracold atom platforms [17]
demonstrate intricate control over the non-equilibrium properties of the
system, including access to the state of the reservoir [18, 19].

Therefore, finite reservoirs have recently attracted increased attention in
theoretical investigations of non-equilibrium phenomena. Efforts to understand
how finite reservoirs affect transport properties have involved the
development of more appropriate transport models [20, 21], encompassing
relaxation processes in QD systems [22] and ultracold atoms [23], many-
particle transport [24] and the full system dynamics [25, 26]. Further
theoretical works in the literature address temperature fluctuations in
metallic islands [27] and their thermodynamic implications [28]. Considering a
finite reservoir as a heat source for an engine, the optimization of such a
heat engine’s efficiency has been investigated [29, 30, 31, 32].

Particle transport also induces the exchange of energy and heat between
different parts of a system. It is natural to consider thermal properties and
thermoelectric behavior of finite reservoirs, which are the subject of ongoing
interest [33]. Generally, thermoelectric effects describe a system’s or
material’s ability to convert a temperature difference to an electric voltage
(or vice-versa). A high thermoelectric efficiency results from strong energy
asymmetries in electron transport properties [34, 35], which result from sharp
resonances in meso- and nanoscale systems [36, 37]. They readily occur in QD
devices, promoting QDs to an ideal platform for experimental studies of
thermoelectric behavior in quantum devices, see e.g. [38, 39, 40, 41, 12, 42].
In particular, an immediate use of thermoelectric properties is Peltier
cooling, where a bias voltage can be used to force heat to flow from a cold to
a hot reservoir. Such electronic cooling has been realized experimentally
using QD systems [43], normal metal-insulator-superconductor junctions [44,
45] and has been investigated in cold atomic gases [46, 47]. In this context,
given that any physical device or system being refrigerated is necessarily of
a finite size, it is essential to understand transport and thermoelectric
behaviors in non-equilibrium settings with finite reservoirs.

In this paper we set out to investigate transport and thermoelectric
properties of a system consisting of a finite electron reservoir in contact
with a QD. We assume a quasi-equilibrium state of the finite reservoir [22,
24], which can be described by a local equilibrium distribution. To allow for
stationary-state non-equilibrium transport, the system is additionally coupled
to two infinite electron reservoirs, see Fig. 1. When there is an applied
voltage and/or a temperature difference between the infinite reservoirs, we
self-consistently solve for the chemical potential and temperature of the
finite reservoir together with the exchanged heat and charge currents, which
also means that these quantities become time-dependent. We investigate the
transport properties in the stationary state, as well as the full dynamics. We
find that the finite reservoir modifies the stationary state properties
considerably compared to treating all reservoirs as infinite. Additionally, we
discover a parameter regime where heat is carried out of the finite reservoir,
resulting in a stationary-state temperature of the finite reservoir _lower_
than the temperature of the infinite reservoirs.

The paper is organized as follows. In Section II, the system is introduced,
and its non-equilibrium dynamics is modeled using a combination of a rate
equation for the QD and a mesoscopic, linear-response approach for the
interaction between the finite and infinite reservoirs. In Section III, we
solve for the non-equilibrium stationary state and analyse the impact of the
finite reservoir on the transport through the QD. The thermoelectric
refrigeration of the finite reservoir is discussed in Section IV. Finally, we
investigate the system’s full dynamics in Section V.

##  II Model

The system consists of a single level, spin-degenerate QD tunnel-coupled to an
infinite reservoir on the right and a finite reservoir on the left, see Fig.
1. The finite reservoir is additionally in contact with a second infinite
reservoir to the left. All reservoirs are described by non-interacting
electrons. For the finite reservoir we assume that the internal relaxation is
fast compared to any other time-scale in the system, such that it is in a
quasi-equilibrium state with a well-defined (but time-dependent) chemical
potential $\mu_{L}(t)$ and temperature $T_{L}(t)$, and is described by the
Fermi-Dirac distribution $f(E,\mu_{L}(t),T_{L}(t)$ with
$f(E,\mu,T)~{}=~{}\\{\exp\left[(E-\mu)/T\right]+1\\}^{-1}$. Throughout the
paper we set Boltzmann’s constant $k_{B}=1$, the elementary charge $e=1$ and
$\hbar=1$.

A phenomenological rate equation to describe the transport through the QD
system is given by

$\displaystyle\frac{d}{dt}\begin{pmatrix}p_{0}\\\
p_{1}\end{pmatrix}=\sum_{\alpha}\begin{pmatrix}-\Gamma^{\text{in}}_{\alpha}&\Gamma^{\text{out}}_{\alpha}\\\
\Gamma^{\text{in}}_{\alpha}&-\Gamma^{\text{out}}_{\alpha}\end{pmatrix}\begin{pmatrix}p_{0}\\\
p_{1}\end{pmatrix},$ (1)

where $p_{0},p_{1}$ denote the probabilities of the QD orbital with energy
$\epsilon$ being empty or occupied with either a spin-up or a spin-down
electron, and $\alpha=L,R$ indexes the left and right reservoirs. At all times
$p_{0}(t)+p_{1}(t)=1$ holds. Eq. 1 is valid for a QD system without applied
magnetic field, and in the limit of large Coulomb interaction where the state
with two electrons on the QD can be neglected. The rates for an incoming (or
outgoing) electron with respect to the QD from (or to) the left,
$\Gamma_{L}^{\text{in}}$ and $\Gamma_{L}^{\text{out}}$, are time-dependent due
to the finite nature of the reservoir. They are given by [48]

$\displaystyle\Gamma^{\text{in}}_{L}(t)$
$\displaystyle=2\Gamma_{L}f\left(\epsilon,\mu_{L}(t),T_{L}(t)\right),$ (2)
$\displaystyle\Gamma^{\text{out}}_{L}(t)$
$\displaystyle=\Gamma_{L}\left[1-f\left(\epsilon,\mu_{L}(t),T_{L}(t)\right)\right],$

where $\Gamma_{L}$ is the tunneling rate between the QD and finite reservoir.
The total rate $\Gamma_{L}^{\text{in}}\sim 2\Gamma_{L}$ in Eq. 2 takes into
account the spin degeneracy of the state with one electron on the QD. For the
exchange with the right reservoir the tunnel rates are time-independent and
given by

$\displaystyle\Gamma^{\text{in}}_{R}$
$\displaystyle=2\Gamma_{R}f\left(\epsilon,\bar{\mu}_{R},\bar{T}_{R}\right),$
(3) $\displaystyle\Gamma_{R}^{\text{out}}$
$\displaystyle=\Gamma_{R}\left[1-f\left(\epsilon,\bar{\mu}_{R},\bar{T}_{R}\right)\right],$

where $\Gamma_{R}$ is the tunneling rate between the QD and infinite
reservoir. The description of the QD transport with the rate equation (1) is
applicable in the weak coupling limit where $\Gamma_{\alpha}\ll T_{\alpha}$.
Throughout, we denote the time-independent chemical potential and temperature
of the infinite reservoirs with $\bar{\mu}_{\alpha}$ and $\bar{T}_{\alpha}$
with $\alpha=L,R$.

Figure 1: Sketch of the setup. A QD is coupled to a infinite reservoir on the
right. On the left it is coupled to a finite reservoir in a quasi-equilibrium
state described by time-dependent $\mu_{L}(t)$ and $T_{L}(t)$.
$\Gamma_{\alpha}^{\text{in,out}}$ with $\alpha=L,R$ are the rates for
tunneling between the QD and the left (finite) and right (infinite)
reservoirs. The finite reservoir is in turn in contact with a infinite one.
The exchange between them is characterized by the electrical conductivity
$\sigma$ and thermal conductivity $\kappa_{\text{el}}$. The wavy lines
indicate the phonon contribution $\Sigma^{\text{ph}}$ to the heat current due
to electron-phonon interaction.

The electron current, $I_{L/R}^{\text{QD}}$, and energy current,
$J_{L/R}^{\text{QD}}$, flowing out from the QD to the finite (L) and infinite
(R) size reservoirs can be expressed in terms of the QD occupation
probabilities and tunnel rates as

$\displaystyle I^{\text{QD}}_{L/R}(t)$
$\displaystyle=p_{1}(t)\Gamma^{\text{out}}_{L/R}(t)-p_{0}(t)\Gamma^{\text{in}}_{L/R}(t),$
(4) $\displaystyle J^{\text{QD}}_{L/R}(t)$ $\displaystyle=\varepsilon
I^{\text{QD}}_{L/R}(t).$

If $I_{L}^{\text{QD}}<0$, the current flows out of the finite reservoir into
the QD, while for $I_{L}^{\text{QD}}>0$ it flows in the opposite direction.

Assuming the contact between the finite and infinite reservoirs on the left to
have energy independent transport properties, the charge and energy currents
flowing out of the infinite reservoir are given by [49]

$\displaystyle I_{L}(t)$ $\displaystyle=-\sigma\Delta\mu(t),$ (5)
$\displaystyle J_{L}(t)$ $\displaystyle=-\kappa_{\text{el}}\Delta
T(t)+\frac{1}{2}\left(\mu_{L}(t)+\bar{\mu}_{L}\right)I_{L}(t).$

where we introduced $\Delta\mu(t)~{}=~{}\mu_{L}(t)-\bar{\mu}_{L}$ and $\Delta
T(t)~{}=~{}T_{L}(t)-\bar{T}_{L}$. Here, $\sigma$ is the electrical conductance
and we take the temperature-dependent electronic thermal conductance to be

$\displaystyle\kappa_{\text{el}}\left(\tilde{T}(t)\right)=\frac{\pi^{2}}{3}\sigma\tilde{T}(t),$
(6)

where $\tilde{T}(t)=(T_{L}(t)+\bar{T}_{L})/2$. In the strict linear response
regime, Eq. 6 reduces to the Wiedemann-Franz law [50].

To self-consistently determine the fluctuating $T_{L}(t)$ and $\mu_{L}(t)$ of
the finite reservoir, we make use of the continuity, or conservation,
equations for the charge and energy on the reservoir. The time derivative of
the number of electrons of the reservoir, $N_{L}(t)$, is equal to the sum of
the charge currents flowing into the reservoir

$\displaystyle\dot{N}_{L}(t)=I^{\text{QD}}_{L}(t)+I_{L}(t),$ (7)

where, within our free electron assumption for the finite reservoir, we have

$\displaystyle\dot{N}_{L}(t)$ $\displaystyle=\nu_{0}\frac{d}{dt}\int
dE~{}f\left[E,\mu_{L}(t),T_{L}(t)\right]$ (8)
$\displaystyle=\nu_{0}\dot{\mu}_{L}(t).$

Here, $\nu_{0}$ is the density of states of the finite reservoir, which we
take to be energy independent for simplicity, equal to the quantum capacitance
of the reservoir. We note that, for simplicity, we neglect possible
geometrical capacitive couplings (e.g., from the other reservoirs) to the
excess electrons on the reservoir.

In the same way, the time derivative of the total energy of the finite
reservoir is equal to the sum of the energy currents flowing into the
reservoir

$\displaystyle\dot{E}_{L}(t)=J^{\text{QD}}_{L}(t)+J_{L}(t).$ (9)

where

$\displaystyle\dot{E}_{L}(t)$ $\displaystyle=\nu_{0}\frac{d}{dt}\int
dEEf\left[E,\mu_{L}(t),T_{L}(t)\right]$ (10)
$\displaystyle=\nu_{0}\mu_{L}(t)\dot{\mu}_{L}+C_{\text{el}}[T_{L}(t)]\dot{T}_{L}(t),$

where $C_{\text{el}}(T)=\nu_{0}\pi^{2}T/3$ is the specific heat of a free
Fermi gas. Combining Eqs. 7, 9, 8 and 10 we can write

$\displaystyle C_{\text{el}}[T_{L}(t)]\dot{T}_{L}(t)$ (11)
$\displaystyle=J^{\text{QD}}_{L}(t)+J_{L}(t)-\mu_{L}(t)\left[I^{\text{QD}}_{L}(t)+I_{L}(t)\right]$
$\displaystyle\equiv\dot{Q}^{\text{QD}}_{L}(t)+\dot{Q}_{L}(t).$

Here we introduced the heat currents flowing into the finite reservoir

$\displaystyle\dot{Q}^{\text{QD}}_{L}(t)$
$\displaystyle=(\epsilon-\mu_{L}(t))I_{L}^{\text{QD}}(t),$ (12)
$\displaystyle\dot{Q}_{L}(t)$ $\displaystyle=J_{L}(t)-\mu_{L}(t)I_{L}(t).$

This shows that the dynamics of the reservoir temperature $T_{L}(t)$ is
governed by the heat flows into the reservoir, as one would expect. Taken
together, Eqs. 1, 7, 8, 9, 10 and 11 provide a full description of the system
dynamics, where the evolution of $\mu_{L}(t)$ and $T_{L}(t)$ of the finite
reservoir is governed by

$\displaystyle\dot{\mu}_{L}(t)$
$\displaystyle=\frac{1}{\nu_{0}}\left[I^{\textrm{QD}}_{L}(t)+I_{L}(t)\right],$
(13) $\displaystyle\dot{T}_{L}(t)$
$\displaystyle=\frac{\dot{Q}_{L}^{\textrm{QD}}+\dot{Q}_{L}(t)}{C_{\text{el}}\left(T_{L}(t)\right)}.$

A strict linear response approach for a small applied bias reveals two
relevant dimensionless parameters governing the stationary transport
properties, see Appendix A for details. The first one is $T\sigma/\Gamma_{L}$.
The ratio $\sigma/\Gamma_{L}$ (in units of inverse temperature) characterizes
the deviations from a system with only infinite reservoirs in the stationary
state, which is apparent from Eqs. 4 and 5. The second parameter,
$\epsilon/T$, describes how far from resonance the QD level is.

##  III Stationary state transport properties

In the following, we determine the transport properties of the set-up shown in
Fig. 1 in a non-equilibrium setting. A symmetrically applied bias voltage
$V_{b}$ across the two infinite reservoirs sets their chemical potentials
$\bar{\mu}_{L}=-\bar{\mu}_{R}=V_{b}/2$. The energy level $\epsilon$ of the QD
can be tuned by an additional gate voltage $V_{g}=-\epsilon$ (for simplicity
we set the gate lever arm to unity). Finally, we assume equal temperatures of
the infinite reservoirs on the left and right, $\bar{T}_{L}=\bar{T}_{R}=T$, as
well as equal tunneling rates, $\Gamma_{L}=\Gamma_{R}=\Gamma$. We analyse the
stationary electron current
$I_{L}^{\text{QD,s}}~{}=~{}\lim_{t\to\infty}I_{L}^{\text{QD}}(t)$; we use the
superscript s for the stationary state throughout.

To find the stationary state we solve the coupled equations of motions defined
by Eqs. 1, 7, 8, 9, 10 and 11 for $\dot{\mu}_{L}=\dot{T}_{L}=\dot{p}_{0,1}=0$.
The stationary state solution for the probabilities is given by

$\displaystyle p_{0}^{s}$ $\displaystyle=1-p_{1}^{s},$ (14) $\displaystyle
p_{1}^{s}$
$\displaystyle=\sum_{\alpha=L,R}\frac{\Gamma_{\alpha}^{\text{in},s}}{\Gamma_{\alpha}^{\text{in},s}+\Gamma_{\alpha}^{\text{out},s}}.$

A general closed-form expression for the stationary state values for
$\mu_{L}^{s}$ and $T_{L}^{s}$ of the finite reservoir is unattainable and we
rely on a numerical solution for the transport properties.

Figure 2:  (a) Stability diagrams showing $dI^{\text{QD},s}_{L}/dV_{b}$ as a
function of $V_{b}$ and $V_{g}$ for two different values of $\sigma/\Gamma$.
For $\sigma/\Gamma=10/T$ the transport looks qualitatively similar to a single
level QD coupled to infinite source and drain leads. For $\sigma/\Gamma=0.1/T$
the visible effects of the finite reservoir include an overall suppression of
$I_{L}^{\text{QD},s}$ and a broadening and shift of one conductance peak. (b)
Heat current $\dot{Q}_{L}^{\text{QD},s}/\Gamma^{2}$ (purple, left axis) and
electron current $I_{L}^{\text{QD},s}/\Gamma$ (orange, right axis) as a
function of $\sigma/\Gamma$ at fixed $V_{b}/T=10$ and $V_{g}/T=0$. The dotted
lines show the analytical stationary state solution up to second order in
$\Gamma/\sigma$ according to Eq. 29. For large $\sigma/\Gamma$ both
$\dot{Q}_{L}^{\text{QD},s}$ and $I_{L}^{\text{QD},s}$ asymptotically approach
the values for the QD coupled to infinite source and drain reservoirs,
indicated by the dashed lines. For smaller $\sigma/\Gamma$ the stationary
state values deviate, and can even lead to $\dot{Q}_{L}^{\text{QD},s}<0$,
shown in the inset. In all plots $\bar{T}_{L}=\bar{T}_{R}=T=10\Gamma$.

For the stationary current $I_{L}^{\text{QD,s}}=\sigma\Delta\mu$ must hold due
to current conservation. Therefore, the magnitude of the current is ultimately
limited by $\sigma$, if the QD level does not block the transport, i.e., if
$\mu_{L}^{s}\gtrsim\epsilon\gtrsim\bar{\mu}_{R}$ for $V_{b}>0$. Consequently,
we can identify two effects of the finite reservoir on the stationary
transport properties compared to coupling of the QD to infinite reservoirs:
(i) A modification of the range where electrons can pass through the dot, and
(ii) a suppression of the current due to finite $\sigma$. There is no strict
bound on the temperature $T_{L}^{s}$ as the heat current through the system is
not conserved. Due to coupling of temperature and chemical potential, there is
also no strict bound on $\mu_{L}^{s}$. Figure 2(a) shows the differential
conductance $dI_{L}^{\text{QD},s}/dV_{b}$ for the QD system as a function of
$V_{g}$ and $V_{b}$ for two different values of $\sigma/\Gamma$. For a single
QD coupled to two infinite reservoirs, the stability diagram displays a
Coulomb diamond, or a pair of crossing conductance lines for infinite charging
energy. Considering now the finite reservoir in Fig. 2(a) with
$\sigma/\Gamma=10$, changes of $\mu_{L}^{s}$ in comparison to $\bar{\mu}_{L}$
are small and the qualitative features of the resulting stability diagram are
already similar to the case $\sigma/\Gamma\to\infty$; this limit corresponds
to effectively coupling the QD to two infinite reservoirs.

For smaller values of $\sigma/\Gamma$ (focusing on $V_{b}>0$), the conductance
lines associated with tunneling between the QD and the finite reservoir are
broadened and shifted to higher $\lvert V_{b}\rvert$. To understand this
stationary state behavior it is useful to consider the full time dependence.
If $\mu_{L}~{}=~{}\bar{\mu}_{L}$ at the initial time, electrons tunneling out
of the finite reservoir and into the QD will lower $\mu_{L}$ until the current
between the infinite and finite reservoirs (given by $\sigma\Delta\mu$)
becomes equal to $I_{L}^{\textrm{QD}}$. For small $\sigma/\Gamma$, this
happens for $\mu_{L}$ close to (or below) $\epsilon$. Thus, in the stationary
state $\mu_{L}^{s}$ is pinned close to $\epsilon$ over a range in $V_{b}$ and
the conductance peak shifts and broadens.

The width and position of the conductance peaks associated with tunneling
between the QD and right reservoir remain largely unaffected, especially for
large $V_{b},V_{g}$, but their heights are reduced as $I_{L}^{\text{QD},s}$ is
overall suppressed for small $\sigma/\Gamma$. If the QD would also be coupled
to a finite reservoir on the right this peak would broaden and shift. For
$V_{b}<0$ the explanations above hold for reversed inequalities.

We expect our results to remain qualitatively similar for finite values of the
Coulomb interaction. The stationary transport properties displayed in Fig.
2(a) would instead show a typical closed Coulomb diamond. However, the
conductance lines would be modified in the same way. Similarly, the Zeeman
splitting of the QD level due to a magnetic field would not alter the
qualitative results.

Further analytical insight can be gained in the regime of large
$\sigma/\Gamma$ by writing

$\displaystyle\mu_{L}(t)$ $\displaystyle=\bar{\mu}_{L}+\delta\mu_{L}(t),$ (15)
$\displaystyle T_{L}(t)$ $\displaystyle=T+\delta T_{L}(t).$

Assuming $\delta\mu_{L},\delta T_{L}$ to be small we expand Eqs. 1 and 13 to
first order in the small deviations. This is a good approximation when the
finite reservoir is not (significantly) depleted because $\sigma/\Gamma\gg 1$.
It does not rely on a small applied bias voltage and is therefore valid beyond
a strict linear response theory approach (see Appendix A for the linear-
response result). The stationary state solutions for $\delta\mu_{L}^{s}$ and
$\delta T_{L}^{s}$ are given in Eq. 24 in Appendix B.

In the limit $\sigma/\Gamma\to\infty$, $\mu_{L}^{s}\to\bar{\mu}_{L}$ and
$T_{L}^{s}\to T$, and we recover the results of a QD coupled to two infinite
reservoirs. This is exemplified by the stationary heat and electron currents
shown in Fig. 2(b), including both the full numerical solution (solid lines)
and the analytical solution for small $\delta\mu_{L}^{s},\delta T_{L}^{s}$ up
to second order in $\Gamma/\sigma$ (dotted lines).

We find good qualitative agreement of the analytical result with the numerics
for the parameters in Fig. 2(b). In general, $V_{b}$ and $\epsilon$ affect the
qualitative agreement, and for small $V_{b}$ the analytics capture the
behavior across the full range of $\sigma/\Gamma$. The expansion of the
analytical results in Eq. 24 to second order in $\Gamma/\sigma$ captures a
suppression of the currents $\sim\Gamma/\sigma$ (see also Eq. 29 in Appendix
B) compared to the currents expected for the system if the QD was coupled to
infinite reservoirs. However, making $\sigma/\Gamma$ smaller than $\sim 1$,
eventually leads to violating the normalization of the probabilities
$p_{0,1}$, which implies a breakdown of the assumption of small deviations.

##  IV Refrigeration of the finite reservoir

Figure 3:  (a) Temperature reduction for the finite reservoir as a function of
$V_{b}$ and $V_{g}$ for $\sigma/\Gamma=10/T$ (left) and $\sigma/\Gamma=0.1/T$
(right). Plotted is the percentage $\lvert\Delta
T^{s}\rvert/T~{}=~{}\lvert(T_{L}^{s}~{}-~{}T)\rvert~{}/~{}T$; regions with
heating where $T_{L}^{s}>T$ are not shown. (b) $T_{L}^{\text{min}}$:
$T_{L}^{s}$ minimized along cuts of fixed $V_{b}/T=10$, plotted as a function
of $\sigma/\Gamma$. The dotted and dashed lines indicate $T_{L}^{\text{min}}$
including the electron-phonon interaction $\Sigma^{\text{ph}}$ in the thermal
conductivity. All other parameters are chosen as in Fig. 2.

Under certain conditions, the stationary transport results show a negative
heat current $\dot{Q}_{L}^{\text{QD},s}$ out of the finite reservoir, see Fig.
2(b). Here we investigate under which conditions this leads to refrigeration
of the finite reservoir.

The heat current flowing out of the finite reservoir as a result of electrons
tunnelling into the QD is given by Eq. 12. For $V_{b}>0$, the heat current
$\dot{Q}_{L}^{\text{QD},s}$ is negative when $\epsilon>\mu_{L}^{s}$, because
the corresponding electron current $I_{L}^{\text{QD},s}<0$. Electrons above
the Fermi level are then removed from the finite reservoir and the temperature
$T_{L}^{s}$ is lowered. In turn, for $V_{b}<0$, $I_{L}^{\text{QD},s}>0$ and
$\dot{Q}_{L}^{\text{QD},s}~{}<~{}0$ when $\epsilon<\mu_{L}^{s}$, electrons
below the Fermi energy $\mu_{L}^{s}$ are added, again leading to a lower
temperature. Achieving cooling also requires the magnitude of the (positive)
heat current flowing into the finite reservoir from the infinite reservoir to
be smaller than the heat current flowing out to the QD, i.e.,
$\lvert\dot{Q}_{L}^{s}\rvert=\lvert-\kappa_{\textrm{el}}\Delta
T^{s}+\sigma\left(\Delta\mu^{s}\right)^{2}/2\rvert<\lvert\dot{Q}_{L}^{\text{QD},s}\rvert$.
If those requirements are fulfilled, a temperature $T_{L}^{s}~{}<~{}T$ is
induced and the finite reservoir is cooled.

The parameter regime where refrigeration is possible is shown in Fig. 3(a) for
two different choices of $\sigma/\Gamma$ as a function of $V_{b}$ and $V_{g}$.
First, the region in the $(V_{g},V_{b})$ plane where cooling of the finite
reservoir is possible increases for smaller $\sigma/\Gamma$. For $V_{b}>0$,
this correlates with a larger region where $\epsilon~{}>~{}\mu_{L}^{s}$ is
fulfilled due to the larger shift of $\mu_{L}^{s}$ compared to larger
$\sigma/\Gamma$. Similarly, for $V_{b}~{}<~{}0$ we have instead
$\epsilon<\mu_{L}^{s}$. Second, the difference between $T_{L}^{s}$ and $T$
increases (non-linearly) with decreasing $\sigma/\Gamma$, showing a reduced
temperature down to $T_{L}^{s}=0.65T$ for $\sigma/\Gamma=0.1$. The slight bias
asymmetry in the temperature reduction is related to spin degeneracy, which
causes an asymmetry between the rates for electrons tunneling in and out of
the QD.

A key question for any cooling application is what minimum temperature
$T_{L}^{\text{min}}$ of the finite reservoir can be reached. We determine
$T_{L}^{\text{min}}$ across a line cut at constant $V_{b}$ (the results only
depends very weakly on the choice as long as $V_{b}\gg T$). The
$T_{L}^{\text{min}}$ that can be reached depending on $\sigma$ is shown in
Fig. 3(b). It keeps decreasing with decreasing $\sigma/\Gamma$, but, as we
will see in Section V, the time needed to reach the stationary state increases
because the rate of cooling decreases. This also means that the cooling is
more sensitive to other sources of heat flows for small $\sigma/\Gamma$.

The phonon contribution to the heat exchange is expected to play an important
role in quantum devices [51, 52]. Assuming that the phonon temperature is
given by $T_{p}=T$, the heat current flowing from the finite reservoir into
the phonon bath can be included in Eq. 12 through a modified thermal
conductivity, where we replace

$\displaystyle\kappa_{\text{el}}\to\kappa_{\text{el}}+\Sigma^{\text{ph}}T^{4},$
(16)

with the electron-phonon coupling strength $\Sigma^{\text{ph}}$. For the
temperature dependence we choose the typical $T^{4}$ for metals [53, 54],
however, the exact dependence is not crucial and different exponents are
observed experimentally [36]. The phonon contribution becomes more prominent
for small $\sigma/\Gamma$, leading to a non-monotonic behaviour of
$T_{L}^{\text{min}}$ as a function of $\sigma/\Gamma$ with a clear minimum.
The reason is that for small $\sigma/\Gamma$ the electron current is lower
which corresponds to a lower heat current out of the finite reservoir. This
leads to a lower cooling power and therefore a larger impact of the phonons.
An additional source of heat exchange in this system, with the potential to
influence $T_{L}^{\text{min}}$ further, are higher-order tunneling processes
[55]. Within the weak coupling regime ($\Gamma\ll T$) studied here, such
contributions are small.

##  V Full time evolution

Figure 4: Full time evolution of (a) $\lvert I_{L}^{\text{QD}}(t)\rvert$, (b)
$\mu_{L}(t)$ and (c) $T_{L}(t)$ for the initial condition
$\mu_{L}(0)=\bar{\mu}_{L}$, $T_{L}(0)~{}=~{}\bar{T}_{L}$, $p_{0}(0)~{}=~{}1$,
$V_{b}=10T$, $T~{}=~{}10\Gamma$, $\epsilon~{}=~{}V_{b}/4$,
$\sigma/\Gamma~{}=~{}5~{}\cdot~{}10^{-3}/T$ and $\nu_{0}=1000/T$. The black
dashed line marks the time when $\mu_{L}(t)$ = $\epsilon$ . The dotted line in
(a) indicates $I_{L}^{\text{QD},\textrm{inf}}$ and the dotted lines in (b) and
(c) indicate the evolution according to Eq. 17.

We now investigate the full system dynamics to gain insight into the typical
timescales, including the time needed to cool the finite reservoir.

Beyond the stationary state we have access to the full time evolution by
numerically integrating the coupled equations of motions for
${p}_{0,1}(t),\,{\mu}_{L}(t)$ and ${T}_{L}(t)$ defined in Eqs. 1 and 13. For
further physical insight we focus on a distinct regime where the QD dynamics
sets the fastest time scale. We assume $\sigma/\Gamma\ll 1$ and $V_{b}\gg T$,
and we consider an initial state where the finite reservoir is in equilibrium
with the left infinite reservoir, and the QD is unoccupied, $p_{0}(t=0)=1$.
Additionally, we choose $\epsilon/T=V_{b}/4$, such that the finite reservoir
is cooled in the stationary state. The transients for $I_{L}^{\text{QD}}(t),\
\mu_{L}(t)$ and $T_{L}(t)$ are shown in Fig. 4.

Initially, the fast QD dynamics on the time scale of $\sim~{}1/\Gamma$ is
associated with establishing the QD occupation $p_{0}^{\textrm{inf}}$ and
current $I^{\text{QD},\textrm{inf}}_{L}$ that would be the stationary state
values if the QD was coupled to an infinite reservoir. The initial dynamics of
$I_{L}^{\textrm{QD}}(t)$ is largely independent of $\mu_{L}(t)$ as long as
$\mu_{L}(t)>\epsilon$ because the Fermi function determining the electron
tunneling rate is close to one. In response, $\mu_{L}(t)$ and $T_{L}(t)$
adjust for intermediate times $t>1/\Gamma$ according to

$\displaystyle\dot{\mu}_{L}(t)$
$\displaystyle\sim\frac{1}{\nu_{0}}I^{\text{QD},\textrm{inf}}_{L},$ (17)
$\displaystyle\dot{T}_{L}(t)$
$\displaystyle\sim\frac{\left[\epsilon-\mu_{L}(t)\right]}{C_{\text{el}}\left[T_{L}(t)\right]}I_{L}^{\text{QD},\textrm{inf}},$

which follows from Eq. 13 with $\Delta\mu_{L},\Delta T_{L}\ll 1$ and

$\displaystyle I^{\text{QD},\textrm{inf}}_{L}\sim-2\Gamma
p_{0}^{\textrm{inf}}.$ (18)

Consequently, the initial downward shift of $\mu_{L}(t)$ is linear, while
$T_{L}(t)$ first increases as only electrons at energy $\epsilon$ can tunnel
into the QD, and electrons from below $\mu_{L}(t)$ are removed from the finite
reservoir. As soon as $\mu_{L}(t)<\epsilon$, $\dot{Q}_{L}^{\textrm{QD}}$
becomes negative and the finite reservoir eventually starts to cool (although
$\dot{Q}_{L}$ remains positive). Simultaneously, the current decreases since
the tunneling rate is sensitive to the tail of the Fermi function. At these
later times, the feedback between $\mu_{L}(t),T_{L}(t)$ and $p_{j}(t)$ via the
Fermi function $f(\epsilon,\mu_{L}(t),T_{L}(t))$ becomes important. We can
estimate the longest possible time scale of the system, which is predominately
set by $\sigma/\nu_{0}$ as the exponential evolution according to
$\dot{\mu}_{L}\sim\sigma\Delta\mu(t)/\nu_{0}$ and
$\dot{T}_{L}\sim-\sigma(T_{L}^{2}(t)-T^{2})/C_{\textrm{el}}(T_{L}(t))$ from
Eq. 13 set in. However, the exact details of how the stationary state is
reached depends strongly on the chosen system parameters, as well as the
initial condition.

##  VI Conclusion

In this work, we investigated the non-equilibrium transport properties of a QD
coupled to a finite reservoir, as well as two infinite reservoirs. The finite
nature of the reservoir leads to (i) a shift of the chemical potential of the
finite reservoir which in turn leads to a broadening of one of the conductance
peaks in the charge stability diagram; (ii) a suppressed electron current
compared to the current that would flow in the case of coupling to infinite
reservoirs. Along the conductance peaks we find a region where a temperature
$T_{L}^{s}$ smaller than the temperature of the infinite reservoirs can be
established. Therefore, the electron transport and associated heat flow
through a finite reservoir leads to stationary-state refrigeration of that
reservoir. This result is immediately relevant for refrigeration applications,
as any physical system to be cooled is necessarily finite. The minimum
temperature $T_{L}^{\text{min}}$ that can be reached strongly depends on the
coupling to the infinite reservoir and is subject to a trade-off. Decreasing
$\sigma/\Gamma$ results in decreasing $T_{L}^{\text{min}}$, but also leads to
an increasing shift of the chemical potential and time needed to reach the
stationary state. Coupling to phonons strongly impacts $T_{L}^{\text{min}}$
for small $\sigma/\Gamma$, and leads to a minimum of $T_{L}^{\text{min}}$ for
finite $\sigma/\Gamma$. Furthermore, we studied the transient dynamics of the
system and identified three distinct regimes. An important timescale is set by
the ratio of the electrical conductivity with the density of states of the
finite reservoir, $\sigma/\nu_{0}$. Therefore, a choice of small
$\sigma/\Gamma$ for a minimal temperature also leads to slow dynamics towards
the stationary state.

Utilizing microscopic approaches [26] for finite reservoirs might provide a
promising route to address more complicated quantum systems to further study
non-equilibrium transport including fluctuations and noise. But also more
fundamental questions such as the modification of the energy distribution
[25], especially in quasi-equilibrium or even out-of-equilibrium reservoir
states [21] could be addressed.

###### Acknowledgements.

We would like to thank Viktor Svensson and Konstantin Nestmann for insightful
discussions. We acknowledge financial support from the Wallenberg Center for
Quantum Technologies (WACQT), the Swedish Research Council (Grant Agreement
No. 2020-03412 and No. 2018-03921), and from NanoLund. SM acknowledges
financial support from the PNRR-MUR project and PE0000023-NQSTI project, co-
funded by the European Union – NextGeneration EU, and from Provincia Autonoma
di Trento (PAT). SVM acknowleges funding from the Knut and Alice Wallenberg
Foundation (Project No. 2016-0089). This project is co-funded by the European
Union (Quantum Flagship project ASPECTS, Grant Agreement No. 101080167). Views
and opinions expressed are however those of the authors only and do not
necessarily reflect those of the European Union, Research Executive Agency or
UKRI. Neither the European Union nor UKRI can be held responsible for them.

##  Appendix A Linear response theory for stationary state properties

We consider the stationary electron and energy currents for a small,
symmetrically applied bias voltage $\Delta V_{b}$ across the infinite
reservoirs, such that $\bar{\mu}_{L}=-\bar{\mu}_{R}~{}=~{}\Delta V_{b}/2$.
There is no temperature gradient across the system,
$\bar{T}_{L}=~{}\bar{T}_{R}~{}=~{}T$ and we assume equal tunneling rate
$\Gamma_{L}=\Gamma_{R}=\Gamma$. The chemical potential and temperature of the
finite reservoir are

$\displaystyle\mu_{L}$ $\displaystyle=\delta\mu_{L}^{s}$ (19) $\displaystyle
T_{L}$ $\displaystyle=T+\delta T_{L}^{s}.$

In the strict linear response regime, the quantities $\delta\mu_{L}^{s}$ and
$\delta T_{L}^{s}$ are proportional to the applied bias $\Delta V_{b}$. First,
the charge and energy currents between the left infinite and finite reservoirs
are given by

$\displaystyle I_{L}$
$\displaystyle=-\sigma\left(\delta\mu_{L}^{s}-\frac{\Delta V_{b}}{2}\right)$
(20) $\displaystyle J_{L}$ $\displaystyle=-\kappa_{\text{el}}\delta
T_{L}^{s}.$

Second, the particle and energy currents between the QD and the right infinite
reservoir are

$\displaystyle I_{R}^{\textrm{QD}}$
$\displaystyle=p_{1}^{s}\Gamma^{\text{out}}_{R}-p_{0}^{s}\Gamma^{\text{in}}_{R},$
(21) $\displaystyle J_{R}^{\textrm{QD}}$ $\displaystyle=\epsilon
I_{R}^{\textrm{QD}}.$

In the stationary state, $I_{L}=I_{R}^{\textrm{QD}}$ and
$J_{L}=J_{R}^{\textrm{QD}}$ are required. Solving for $\delta\mu_{L}^{s}$ and
$\delta T_{L}^{s}$ leads to

$\displaystyle\delta\mu_{L}^{s}$ $\displaystyle=\frac{\Delta
V_{b}}{2}\frac{\frac{\sigma}{G}+3\left(\frac{\epsilon}{\pi
T}\right)^{2}-1}{\frac{\sigma}{G}+3\left(\frac{\epsilon}{\pi
T}\right)^{2}+1},$ (22) $\displaystyle\delta T_{L}^{s}$ $\displaystyle=-\Delta
V_{b}\frac{3\frac{\epsilon}{\pi^{2}T}}{\frac{\sigma}{G}+3\left(\frac{\epsilon}{\pi
T}\right)^{2}+1},$

where we used the Wiedemann-Franz law $\kappa_{\text{el}}=\pi^{2}\sigma T/3$
and introduced the conductance

$\displaystyle
G=\frac{\Gamma}{T}\frac{1}{3\left[1+\cosh{\left(\epsilon/T\right)}\right]-\sinh{\left(\epsilon/T\right)}}.$
(23)

Here, we can identify two independent, dimensionless parameters, $\epsilon/T$
and $T\sigma/\Gamma$. The position of the QD level $\epsilon$ is crucial as
the electron current is exponentially suppressed for
$\lvert\epsilon-\bar{\mu}_{\alpha}\rvert>T$. Additionally, $\sigma/\Gamma$
gives a measure of the competing couplings $\sigma$ (between the infinite and
the finite reservoir) and $\Gamma$ (between finite reservoir and QD).

##  Appendix B Large coupling to the infinite reservoir

In the limit of large coupling to the infinite reservoir, $T\sigma/\Gamma\gg
1$, the finite reservoir is barely depleted. We can assume that
$\delta\mu_{L}$ and $\delta T_{L}$ in Eq. 15 are small, i.e.,
$\delta\mu_{L}\ll\bar{\mu}_{L}$ and $\delta T_{L}\ll T$, and we can expand the
Fermi functions in the equations of motion (see Eqs. 1 and 13). The expansion
goes beyond a linear response description as it is not limited to small
applied bias. To first order in $\delta\mu_{L},\,\delta T_{L}$ we find closed
form expressions in the stationary state, which are given by

$\displaystyle\delta\mu_{L}^{s}$
$\displaystyle=-\frac{\Gamma}{\sigma}\frac{2\bar{f}_{-}\bar{T}_{L}\left[6\frac{\Gamma}{\sigma}\bar{f}_{-}\left(\epsilon-\bar{\mu}_{L}\right)-\pi^{2}\left(2+\bar{f}_{+}\right)\bar{T}_{L}^{2}\right]}{\left(2+\bar{f}_{+}\right)\left[6\frac{\Gamma}{\sigma}\bar{f}_{-}\left(\epsilon-\bar{\mu}_{L}\right)-\pi^{2}\left(2+\bar{f}_{+}\right)\bar{T}_{L}^{2}\right]\bar{T}_{L}-4\frac{\Gamma}{\sigma}\left(1+\bar{f}_{R}\right)F_{L}\left[\pi^{2}\bar{T}_{L}^{2}+3\left(\epsilon-\bar{\mu}_{L}\right)^{2}\right]},$
(24) $\displaystyle\delta T_{L}^{s}$
$\displaystyle=\frac{\Gamma}{\sigma}\frac{6\bar{f}_{-}\bar{T}_{L}^{2}\left[2\frac{\Gamma}{\sigma}\bar{f}_{-}+\left(2+\bar{f}_{+}\right)\left(\epsilon-\bar{\mu}_{L}\right)\right]}{\left(2+\bar{f}_{+}\right)\left[6\frac{\Gamma}{\sigma}\bar{f}_{-}\left(\epsilon-\bar{\mu}_{L}\right)-\pi^{2}\left(2+\bar{f}_{+}\right)\bar{T}_{L}^{2}\right]\bar{T}_{L}-4\frac{\Gamma}{\sigma}\left(1+\bar{f}_{R}\right)F_{L}\left[\pi^{2}\bar{T}_{L}^{2}+3\left(\epsilon-\bar{\mu}_{L}\right)^{2}\right]}.$

Here, we defined

$\displaystyle\bar{f}_{-}$ $\displaystyle=\bar{f}_{L}-\bar{f}_{R},$ (25)
$\displaystyle\bar{f}_{+}$ $\displaystyle=\bar{f}_{L}+\bar{f}_{R},$ (26)
$\displaystyle F_{L}$ $\displaystyle=\bar{f}_{L}\left[1-\bar{f}_{L}\right],$
(27)

with $\bar{f}_{\alpha}=f(\epsilon,\bar{\mu}_{\alpha},\bar{T}_{\alpha})$. The
stationary solution for the dot occupation probability according to Eq. 14
within the small-deviations expansion is given by

$\displaystyle
p_{1}^{s}=\bar{p}_{1}^{s}+\frac{4F_{L}}{\left(2+\bar{f}_{+}\right)^{2}}\left[\frac{\delta\mu_{L}^{s}}{\bar{T}_{L}}+\frac{\epsilon-\bar{\mu}_{L}}{\bar{T}_{L}^{2}}\delta
T_{L}^{s}\right],$ (28)

with $\bar{p}_{1}^{s}=2\bar{f}_{+}/(2+\bar{f}_{+})$.

The stationary state solution for the small deviations from the infinite
reservoir quantities in Eq. 24 depends explicitly on the ratio
$\Gamma/\sigma$, and in the limit of large $\sigma$, $\delta\mu_{L}^{s},\delta
T_{L}^{s}\to 0$. Expanding Eq. 24 in the ratio $\Gamma/\sigma$ up to second
order leads to

$\displaystyle\delta\mu_{L}^{s}\approx$
$\displaystyle-\frac{2\bar{f}_{-}}{2+\bar{f}_{+}}\frac{\Gamma}{\sigma}+\frac{8\bar{f}_{-}\left(1+\bar{f}_{R}\right)F_{L}\left[\pi^{2}\bar{T}_{L}^{2}+3\left(\epsilon-\bar{\mu}_{L}\right)^{2}\right]}{\left(2+\bar{f}_{+}\right)^{3}\pi^{2}\bar{T}_{L}^{3}}\left(\frac{\Gamma}{\sigma}\right)^{2},$
(29) $\displaystyle\delta T_{L}^{s}\approx$
$\displaystyle-\frac{6\bar{f}_{-}\left(\epsilon-\bar{\mu}_{L}\right)}{\left(2+\bar{f}_{+}\right)\pi^{2}\bar{T}_{L}}\frac{\Gamma}{\sigma}-\frac{12\bar{f}_{-}\left[\bar{f}_{-}\left(2+\bar{f}_{+}\right)\bar{T}_{L}-2\left(1+\bar{f}_{R}\right)F_{L}\left(\epsilon-\bar{\mu}_{L}\right)\right]\left[\pi^{2}\bar{T}_{L}^{2}+3\left(\epsilon-\bar{\mu}_{L}\right)^{2}\right]}{\left(2+\bar{f}_{+}\right)^{3}\pi^{4}\bar{T}_{L}^{4}}\left(\frac{\Gamma}{\sigma}\right)^{2}.$

These solutions are used to calculate the analytical stationary charge and
heat current in Fig. 2(b) [dotted lines].

## References

  * Gehring  _et al._ [2019] P. Gehring, J. M. Thijssen, and H. S. J. van der Zant, Single-molecule quantum-transport phenomena in break junctions, Nat. Rev. Phys. 1, 381 (2019).
  * Laird  _et al._ [2015] E. A. Laird, F. Kuemmeth, G. A. Steele, K. Grove-Rasmussen, J. Nygård, K. Flensberg, and L. P. Kouwenhoven, Quantum transport in carbon nanotubes, Rev. Mod. Phys. 87, 703 (2015).
  * Bischoff  _et al._ [2015] D. Bischoff, A. Varlet, P. Simonet, M. Eich, H. C. Overweg, T. Ihn, and K. Ensslin, Localized charge carriers in graphene nanodevices, Applied Physics Reviews 2, 031301 (2015).
  * van der Wiel  _et al._ [2002] W. G. van der Wiel, S. De Franceschi, J. M. Elzerman, T. Fujisawa, S. Tarucha, and L. P. Kouwenhoven, Electron transport through double quantum dots, Rev. Mod. Phys. 75, 1 (2002).
  * Hanson  _et al._ [2007] R. Hanson, L. P. Kouwenhoven, J. R. Petta, S. Tarucha, and L. M. K. Vandersypen, Spins in few-electron quantum dots, Rev. Mod. Phys. 79, 1217 (2007).
  * Díez-Pérez  _et al._ [2009] I. Díez-Pérez, J. Hihath, Y. Lee, L. Yu, L. Adamska, M. A. Kozhushner, I. I. Oleynik, and N. Tao, Rectification and stability of a single molecular diode with controlled orientation, Nat. Chem. 1, 635 (2009).
  * Perrin  _et al._ [2015] M. L. Perrin, E. Galan, R. Eelkema, F. Grozema, J. M. Thijssen, and H. S. J. van der Zant, Single-Molecule Resonant Tunneling Diode, J. Phys. Chem. C 119, 5697 (2015).
  * Benenti  _et al._ [2017] G. Benenti, G. Casati, K. Saito, and R. S. Whitney, Fundamental aspects of steady-state conversion of heat to work at the nanoscale, Physics Reports 694, 1 (2017).
  * Jaliel  _et al._ [2019] G. Jaliel, R. K. Puddy, R. Sánchez, A. N. Jordan, B. Sothmann, I. Farrer, J. P. Griffiths, D. A. Ritchie, and C. G. Smith, Experimental realization of a quantum dot energy harvester, Phys. Rev. Lett. 123, 117701 (2019).
  * Malik and Fobelets [2022] F. K. Malik and K. Fobelets, A review of thermal rectification in solid-state devices, Journal of Semiconductors 43, 103101 (2022).
  * Tesser  _et al._ [2022] L. Tesser, B. Bhandari, P. A. Erdman, E. Paladino, R. Fazio, and F. Taddei, Heat rectification through single and coupled quantum dots, New Journal of Physics 24, 035001 (2022).
  * Josefsson  _et al._ [2018] M. Josefsson, A. Svilans, A. M. Burke, E. A. Hoffmann, S. Fahlvik, C. Thelander, M. Leijnse, and H. Linke, A quantum-dot heat engine operating close to the thermodynamic efficiency limits, Nat. Nanotechnol. 13, 920 (2018).
  * Landi  _et al._ [2022] G. T. Landi, D. Poletti, and G. Schaller, Nonequilibrium boundary-driven quantum systems: Models, methods, and properties, Rev. Mod. Phys. 94, 045006 (2022).
  * Reimann [2008] P. Reimann, Foundation of statistical mechanics under experimentally realistic conditions, Phys. Rev. Lett. 101, 190403 (2008).
  * Houck  _et al._ [2012] A. A. Houck, H. E. Türeci, and J. Koch, On-chip quantum simulation with superconducting circuits, Nat. Phys. 8, 292 (2012).
  * Schmidt and Koch [2013] S. Schmidt and J. Koch, Circuit QED lattices: Towards quantum simulation with superconducting circuits, Ann. Phys. 525, 395 (2013).
  * Brantut  _et al._ [2012] J.-P. Brantut, J. Meineke, D. Stadler, S. Krinner, and T. Esslinger, Conduction of ultracold fermions through a mesoscopic channel, Science 337, 1069 (2012).
  * Krinner  _et al._ [2017] S. Krinner, T. Esslinger, and J.-P. Brantut, Two-terminal transport measurements with cold atoms, J. Phys.: Condens. Matter 29, 343003 (2017).
  * Lebrat  _et al._ [2018] M. Lebrat, P. Grišins, D. Husmann, S. Häusler, L. Corman, T. Giamarchi, J.-P. Brantut, and T. Esslinger, Band and correlated insulators of cold fermions in a mesoscopic lattice, Phys. Rev. X 8, 011053 (2018).
  * Zwolak [2020] M. Zwolak, Analytic expressions for the steady-state current with finite extended reservoirs, The Journal of Chemical Physics 153, 224107 (2020).
  * Trushechkin  _et al._ [2022] A. S. Trushechkin, M. Merkli, J. D. Cresser, and J. Anders, Open quantum system dynamics and the mean force Gibbs state, AVS Quantum Science 4, 012301 (2022).
  * Schaller  _et al._ [2014] G. Schaller, C. Nietner, and T. Brandes, Relaxation dynamics of meso-reservoirs, New Journal of Physics 16, 125011 (2014).
  * Gallego-Marcos  _et al._ [2014] F. Gallego-Marcos, G. Platero, C. Nietner, G. Schaller, and T. Brandes, Nonequilibrium relaxation transport of ultracold atoms, Phys. Rev. A 90, 033614 (2014).
  * Amato  _et al._ [2020] G. Amato, H.-P. Breuer, S. Wimberger, A. Rodríguez, and A. Buchleitner, Noninteracting many-particle quantum transport between finite reservoirs, Phys. Rev. A 102, 022207 (2020).
  * Ajisaka and Barra [2013] S. Ajisaka and F. Barra, Nonequilibrium mesoscopic Fermi-reservoir distribution and particle current through a coherent quantum system, Phys. Rev. B 87, 195114 (2013).
  * Riera-Campeny  _et al._ [2021] A. Riera-Campeny, A. Sanpera, and P. Strasberg, Quantum systems correlated with a finite bath: Nonequilibrium dynamics and thermodynamics, PRX Quantum 2, 010340 (2021).
  * van den Berg  _et al._ [2015] T. L. van den Berg, F. Brange, and P. Samuelsson, Energy and temperature fluctuations in the single electron box, New Journal of Physics 17, 075012 (2015).
  * Moreira  _et al._ [2023] S. V. Moreira, P. Samuelsson, and P. P. Potts, Stochastic thermodynamics of a quantum dot coupled to a finite-size reservoir, Phys. Rev. Lett. 131, 220405 (2023).
  * Wang [2016] Y. Wang, Optimizing work output for finite-sized heat reservoirs: Beyond linear response, Phys. Rev. E 93, 012120 (2016).
  * Pozas-Kerstjens  _et al._ [2018] A. Pozas-Kerstjens, E. G. Brown, and K. V. Hovhannisyan, A quantum otto engine with finite heat baths: energy, correlations, and degradation, New Journal of Physics 20, 043034 (2018).
  * Strasberg  _et al._ [2021] P. Strasberg, M. G. Díaz, and A. Riera-Campeny, Clausius inequality for finite baths reveals universal efficiency improvements, Phys. Rev. E 104, L022103 (2021).
  * Yuan  _et al._ [2022] H. Yuan, Y.-H. Ma, and C. P. Sun, Optimizing thermodynamic cycles with two finite-sized reservoirs, Phys. Rev. E 105, L022101 (2022).
  * Brandner  _et al._ [2015] K. Brandner, K. Saito, and U. Seifert, Thermodynamics of micro- and nano-systems driven by periodic temperature variations, Phys. Rev. X 5, 031019 (2015).
  * Mahan and Sofo [1996] G. D. Mahan and J. O. Sofo, The best thermoelectric., Proceedings of the National Academy of Sciences 93, 7436 (1996).
  * Humphrey and Linke [2005] T. E. Humphrey and H. Linke, Reversible thermoelectric nanomaterials, Phys. Rev. Lett. 94, 096601 (2005).
  * Giazotto  _et al._ [2006] F. Giazotto, T. T. Heikkilä, A. Luukanen, A. M. Savin, and J. P. Pekola, Opportunities for mesoscopics in thermometry and refrigeration: Physics and applications, Rev. Mod. Phys. 78, 217 (2006).
  * Dubi and Di Ventra [2011] Y. Dubi and M. Di Ventra, Colloquium: Heat flow and thermoelectricity in atomic and molecular junctions, Rev. Mod. Phys. 83, 131 (2011).
  * Staring  _et al._ [1993] A. A. M. Staring, L. W. Molenkamp, B. W. Alphenaar, H. van Houten, O. J. A. Buyk, M. A. A. Mabesoone, C. W. J. Beenakker, and C. T. Foxon, Coulomb-blockade oscillations in the thermopower of a quantum dot, Europhysics Letters 22, 57 (1993).
  * Scheibner  _et al._ [2007] R. Scheibner, E. G. Novik, T. Borzenko, M. König, D. Reuter, A. D. Wieck, H. Buhmann, and L. W. Molenkamp, Sequential and cotunneling behavior in the temperature-dependent thermopower of few-electron quantum dots, Phys. Rev. B 75, 041301 (2007).
  * Svensson  _et al._ [2012] S. F. Svensson, A. I. Persson, E. A. Hoffmann, N. Nakpathomkun, H. A. Nilsson, H. Q. Xu, L. Samuelson, and H. Linke, Lineshape of the thermopower of quantum dots, New Journal of Physics 14, 033041 (2012).
  * Svilans  _et al._ [2016] A. Svilans, M. Leijnse, and H. Linke, Experiments on the thermoelectric properties of quantum dots, Comptes Rendus Physique 17, 1096 (2016).
  * Majidi  _et al._ [2022] D. Majidi, M. Josefsson, M. Kumar, M. Leijnse, L. Samuelson, H. Courtois, C. B. Winkelmann, and V. F. Maisi, Quantum confinement suppressing electronic heat flow below the Wiedemann–Franz law, Nano Letters 22, 630 (2022).
  * Prance  _et al._ [2009] J. R. Prance, C. G. Smith, J. P. Griffiths, S. J. Chorley, D. Anderson, G. A. C. Jones, I. Farrer, and D. A. Ritchie, Electronic refrigeration of a two-dimensional electron gas, Phys. Rev. Lett. 102, 146602 (2009).
  * Muhonen  _et al._ [2009] J. T. Muhonen, A. O. Niskanen, M. Meschke, Y. A. Pashkin, J. S. Tsai, L. Sainiemi, S. Franssila, and J. P. Pekola, Electronic cooling of a submicron-sized metallic beam, Applied Physics Letters 94, 073101 (2009).
  * Nguyen  _et al._ [2013] H. Q. Nguyen, T. Aref, V. J. Kauppila, M. Meschke, C. B. Winkelmann, H. Courtois, and J. P. Pekola, Trapping hot quasi-particles in a high-power superconducting electronic cooler, New Journal of Physics 15, 085013 (2013).
  * Brantut  _et al._ [2013] J.-P. Brantut, C. Grenier, J. Meineke, D. Stadler, S. Krinner, C. Kollath, T. Esslinger, and A. Georges, A thermoelectric heat engine with ultracold atoms, Science 342, 713 (2013).
  * Grenier  _et al._ [2016] C. Grenier, C. Kollath, and A. Georges, Thermoelectric transport and Peltier cooling of cold atomic gases, C. R. Phys. 17, 1161 (2016).
  * Ingold and Nazarov [1992] G.-L. Ingold and Y. V. Nazarov, Charge tunneling rates in ultrasmall junctions, in  _Single Charge Tunneling_, edited by H. Grabert and M. H. Devoret (Springer New York, 1992).
  * Attard [2012] P. Attard,  _Non-Equilibrium Thermodynamics and Statistical Mechanics_ (Oxford University Press, 2012).
  * Kittel [2005] C. Kittel,  _Introduction to Solid State Physics_ , 8th ed. (John Wiley & Sons, 2005).
  * Cahill  _et al._ [2014] D. G. Cahill, P. V. Braun, G. Chen, D. R. Clarke, S. Fan, K. E. Goodson, P. Keblinski, W. P. King, G. D. Mahan, A. Majumdar, H. J. Maris, S. R. Phillpot, E. Pop, and L. Shi, Nanoscale thermal transport. II. 2003–2012, Applied Physics Reviews 1, 011305 (2014).
  * Qian  _et al._ [2021] X. Qian, J. Zhou, and G. Chen, Phonon-engineered extreme thermal conductivity materials, Nat. Mater. 20, 1188 (2021).
  * Pekola and Karimi [2018] J. P. Pekola and B. Karimi, Quantum Noise of Electron–Phonon Heat Current, J. Low Temp. Phys. 191, 373 (2018).
  * Brange  _et al._ [2018] F. Brange, P. Samuelsson, B. Karimi, and J. P. Pekola, Nanoscale quantum calorimetry with electronic temperature fluctuations, Phys. Rev. B 98, 205414 (2018).
  * Turek and Matveev [2002] M. Turek and K. A. Matveev, Cotunneling thermopower of single electron transistors, Phys. Rev. B 65, 115332 (2002).