domain: info sdv

text:
Safa Alsaidi
email: safa.alsaidi@inria.fr

Miguel Couceiro
email: miguel.couceiro@loria.fr

Sophie Quennelle
email: sophie.quennelle@inria.fr

Anita Burgun
email: anita.burgun@aphp.fr

Nicolas Garcelon
email: nicolas.garcelon@institutimagine.org

Adrien Coulet
email: adrien.coulet@inria.fr

Exploring Analogical Inference in Healthcare

Keywords: analogy classification, electronic health records, patient representation learning IARML

Analogical proportions are statements of the form 𝐴 : 𝐵 :: 𝐶 : 𝐷 that are used to map similar relationships between two pairs of objects, 𝐴, 𝐵, and 𝐶, 𝐷. Analogies have long been a subject of research in the Natural Language Processing (NLP) community, where they have been applied to a variety of reasoning and classification tasks. Lately, machine and representation learning have shown to be useful for analogical reasoning. In this paper, we discuss the possibility of adapting the analogical framework to healthcare applications, in particular to medical decision support. We particularly hypothesize that as language representations help in analogical reasoning in NLP, patient representation learned from Electronic Health Records (EHRs) may help in healthcare. We define three different analogy based settings adapted to EHR data that we see as first steps to the development of analogical applications to this domain. We provide statistics on the first sets of analogies that we built from a publicly available dataset of EHRs, and report preliminary, but promising results to detect patient-stay analogies following our very first experimental setting.

Introduction and motivation

An analogical proportion, or an analogy, is a relation between four objects 𝐴, 𝐵, 𝐶, and 𝐷 that is expressed as "𝐴 is to 𝐵 as 𝐶 is to 𝐷" and formally denoted as 𝐴 : 𝐵 :: 𝐶 : 𝐷. There are two main tasks associated with analogical proportions: analogy detection and analogy solving. Analogy detection corresponds to the task of deciding whether a quadruple ⟨𝐴, 𝐵, 𝐶, 𝐷⟩ is a valid analogy. Analogy solving corresponds to finding a fourth element 𝑥 so that 𝐴 : 𝐵 :: 𝐶 : 𝑥 is a valid analogy. This can be done either by retrieving 𝑥 from a pool of candidates or by generating 𝑥. Analogies have been extensively studied and applied to various Natural Language Processing (NLP) tasks [START_REF] Murena | Solving analogies on words based on minimal complexity transformation[END_REF][START_REF] Lepage | De l'analogie rendant compte de la commutation en linguistique[END_REF][START_REF] Alsaidi | A neural approach for detecting morphological analogies[END_REF][START_REF] Hertzmann | Image analogies[END_REF]. Object representations called embeddings are low-dimensional representations of high-dimensional vectors, which have been used to improve deep learning methodologies. Some of these embeddings learn precise representations and are able to detect differences between objects. As a result they can discriminate between valid and invalid analogical proportions and solve analogical equations.

In this paper, we explore the possibility to leverage the analogy framework to solve tasks relevant to the healthcare domain. We particularly consider using Electronic Health Records (EHRs) to learn patient representations, i.e., patient embeddings. We initiate the construction of sets of patient-based analogies using relationships existing between patient hospital stays from a publicly available set of EHRs. These health records consist of clinical and administrative data collected during patient hospital stays. Generally they are composed of structured (e.g., diagnostic codes, lab tests) and unstructured data (e.g., clinical notes, nursing reports, discharge summaries), either static (e.g., patient demographics) or temporal (e.g., vital signs).

EHRs have been secondary used to conduct epidemiological and observational studies. They have also been used as real word data to train predictive models [START_REF] Jensen | Mining electronic health records: towards better research applications and clinical care[END_REF]. In particular, deep learning methods have become increasingly popular in medical informatics for general tasks such as predicting mortality, in-hospital readmission, diagnoses, etc. A key element for such tasks is to effectively convert patient data from the raw EHR format to embeddings that can be further processed [START_REF] Si | Deep representation learning of patient data from electronic health records (ehr): A systematic review[END_REF]. Representation learning thus consists of learning low-dimension feature representations from raw data. As EHR data are heterogeneous and complex, studies have shown that deep learning models are suited to encode complex EHR data to learn patient representations and that various architectures are suited to different biomedical tasks [START_REF] Li | Behrt: Transformer for electronic health records[END_REF][START_REF] Landi | Deep representation learning of electronic health records to unlock patient stratification at scale[END_REF][START_REF] Miotto | Deep patient: an unsupervised representation to predict the future of patients from the electronic health records[END_REF][START_REF] Huang | Patient representation from structured electronic medical records based on embedding technique: Development and validation study[END_REF][START_REF] Ruan | Representation learning for clinical time series prediction tasks in electronic health records[END_REF][START_REF] Zhang | Patient2vec: A personalized interpretable deep representation of the longitudinal electronic health record[END_REF]. For instance, Madhumita et al. [START_REF] Madhumita | Patient representation learning and interpretable evaluation using clinical notes[END_REF] used a stacked denoised autoencoder and a paragraph vector model to learn generalized patient representations directly from clinical notes. Si and Roberts [START_REF] Si | Patient representation transfer learning from clinical notes based on hierarchical attention network[END_REF] utilized a three-level hierarchical attention-based recurrent neural network (HAN) with greedy segmentation to learn patient representation from clinical notes. Zhang et al. [START_REF] Zhang | Combining structured and unstructured data for predictive models: a deep learning approach[END_REF] proposed 2 multi-modal neural network architectures to enhance patient representation learning by combining sequential unstructured notes with structured data.

Analogies have only been sporadically applied to healthcare. Nonetheless, analogical reasoning has been applied in clinical practice by physicians for diagnosis and prognosis, as a way of linking visible signs and symptoms to possible causes. Indeed, medical reasoning relies on observations of previous patients with similar signs and symptoms, who happened to have a certain disease. Several studies have investigated analogies in healthcare by applying various machine learning methods. For instance, Rather et al. [START_REF] Rather | Using deep learning towards biomedical knowledge discovery[END_REF] used analogical proportions to identify hidden or unknown biomedical knowledge from free text resources. In their work, they defined analogies of the form "acetaminophen is a type of drug as diabetes' is a type of disease. " Dynomant et al. [START_REF] Dynomant | Word embedding for the french natural language in health care: comparative study[END_REF] used analogical proportions to compare embedding methods trained on a corpus of French health-related documents. Each analogical proportion aimed to verify if (𝑇 𝑒𝑟𝑚1 ⃗ -𝑇 𝑒𝑟𝑚2 ⃗ ) + 𝑇 𝑒𝑟𝑚3 ⃗ ≈ 𝑇 𝑒𝑟𝑚4 ⃗ , allowing to check if the similarity between the first two terms is similar to the one between Term 3 and Term 4.

In this paper, we describe an ongoing work on analogical inference in healthcare. We introduce three analogy based settings, where each setting aims to investigate specific biomedical tasks, namely identity, predictive, and generative tasks. In comparison with previous studies [START_REF] Si | Patient representation transfer learning from clinical notes based on hierarchical attention network[END_REF][START_REF] Li | Behrt: Transformer for electronic health records[END_REF][START_REF] Landi | Deep representation learning of electronic health records to unlock patient stratification at scale[END_REF][START_REF] Miotto | Deep patient: an unsupervised representation to predict the future of patients from the electronic health records[END_REF][START_REF] Huang | Patient representation from structured electronic medical records based on embedding technique: Development and validation study[END_REF][START_REF] Ruan | Representation learning for clinical time series prediction tasks in electronic health records[END_REF][START_REF] Zhang | Patient2vec: A personalized interpretable deep representation of the longitudinal electronic health record[END_REF], we aim to build analogies based on patient-stay representations. One of the main contributions of our work is a framework to build sets of proportions for analogical inference in healthcare.

This paper is organized as follows. Section 2 provides a description of the MIMIC-III dataset. Section 3 defines our analogical settings and associated biomedical tasks, and justifies our task choices. Section 4 presents preliminary statistics of the analogical proportions built from MIMIC-III. Section 5 initiates a discussion addressing some analogical postulates that could be useful when generating our analogies. Section 6 illustrates the feasibility of our approach by providing preliminary results using one of our experimental settings. Section 7 discusses perspectives for future research.

Data description

We propose to use EHRs as a source of patient medical history data and aim to consider both its structured and unstructured data to define our analogies. In particular we experiment with a publicly available dataset of EHRs called MIMIC-III (Medical Information Mart for Intensive Care-III) [START_REF] Johnson | Mimic-iii, a freely accessible critical care database[END_REF]. MIMIC-III is a critical care database, developed by the Massachusetts Institute of Technology (MIT)'s Laboratory for Computational Physiology and distributed by PhysioNet [START_REF] Goldberger | Physiobank, physiotoolkit, and physionet: components of a new research resource for complex physiologic signals[END_REF]. It contains integrated, de-identifed health-related data in accordance with Health Insurance Portability and Accountability Act (HIPAA). It contains data associated with all patients admitted to the ICU (Intensive Care Unit) of Beth Israel Deaconess Medical Center between 2001 and 2012. It contains various data, such as patient demographics, vital signs, lab test results, medications, hospital length of stay, survival, clinical notes, imaging reports and more, structured into 26 tables. Each patient-stay is associated with diagnosis codes, motivating the stay and procedures performed during the stay. It encompasses data of more than 40,000 ICU patients and more than 60,000 ICU stays. Table 1 shows statistics for the subgroups of adult patients (aged 18 and above) with at least two stays, which is the subset that we consider in the rest of the article.

The database contains a combination of structured and unstructured data and is accessible to researchers under a data use agreement, where users are required to follow a HIPAA training course demanded by the National Institutes of Health (NIH).

Experimental settings and biomedical tasks

As we defined previously, an analogy is a 4-ary relation and is usually written as 𝐴 : 𝐵 :: 𝐶 : 𝐷.

In this paper, we define three analogy based settings and associated tasks that we are interested in investigating with EHR data. We name our three settings as follows: (i) Identity; (ii) Identity + Sequent; (iii) Identity + Directly Sequent. For these settings, we do not want to learn "full" patient representation, but patient-stay representations (i.e., learn a numeric vector representation of EHR data that belong to a single hospital stay) which we hope to be simpler.

Identity

In the first setting, we propose to build analogies of the form:

𝑠 𝑖 1 𝑡 1 : 𝑠 𝑖 1 𝑡 2 :: 𝑠 𝑖 2 𝑡 3 : 𝑠 𝑖 2 𝑡 4
where 𝑠 𝑖 𝑡 refers to the stay 𝑡 of patient 𝑖. Here, pairs of the analogy quadruples are made of two random stays belonging to the same patient. Since there is no constraint on the order of stays, 𝑠 𝑖 1 𝑡 1 can happen before 𝑠 𝑖 1 𝑡 2 or the inverse. Note that 𝑖 1 and 𝑖 2 could be the same patient, and that 𝑡 1 and 𝑡 2 , or 𝑡 3 and 𝑡 4 , could represent the same time stamp. Furthermore, 𝑡 1 and 𝑡 3 or 𝑡 2 and 𝑡 4 could be the same when 𝑖 1 = 𝑖 2 (but not when 𝑖 1 ̸ = 𝑖 2 ). In this setting we aim at investigating identity tasks, i.e., associating an unaffected sample of data to the patient it belongs. Note that this setting fits several data cleaning and data privacy related applications Identity + Sequent For this setting, we add a temporal constraint to analogies, as we force

𝑠 𝑖 1 𝑡 1 ≪ 𝑠 𝑖 1 𝑡 2 and 𝑠 𝑖 2 𝑡 3 ≪ 𝑠 𝑖 2 𝑡 4
where ≪ denotes temporal sequentiality between stays of a same patient, i.e., 𝑠 𝑖 1 𝑡 2 takes place after 𝑠 𝑖 1 𝑡 1 and 𝑠 𝑖 2 𝑡 4 takes place after 𝑠 𝑖 2 𝑡 3 but not necessarily directly right after. We consider cases where 𝑖 1 = 𝑖 2 . In this setting, we also define a relation named diagnosis, which forces 𝑠 𝑖 1 𝑡 1 and 𝑠 𝑖 2 𝑡 3 to have the same diagnosis. This relation provides more meaning to our analogies and gives us more medical insight into the relationship between our patients. For example, based on the different stays associated with a single patient we hope to see how a certain disease develops (similarly or differently) between two distinct patients.

Identity + Directly Sequent

In this third setting, we make the temporal constraint more strict as we force the two stays of the same patient to be directly sequent (no other stay can exist in between). We note this constraint

𝑠 𝑖 1 𝑡 1 ≺ 𝑠 𝑖 1 𝑡 2 and 𝑠 𝑖 2 𝑡 3 ≺ 𝑠 𝑖 2 𝑡 4
The diagnosis relation is kept between 𝑠 𝑖 1 𝑡 1 and 𝑠 𝑖 2 𝑡 3 , and cases where 𝑖 1 = 𝑖 2 are also considered. With these three settings, we aim at investigating the applicability of two tasks: analogy detection and analogy inference. For instance, given an analogy of the form 𝐴 : 𝐵 :: 𝐶 : 𝑥, we can either propose potential values for an unknown stay 𝑥 (i.e., predictive task) or generate stays which would enrich our dataset with synthetic stays (i.e., generative task).

For the last two settings, we define additional settings by considering three levels of relaxation of the diagnosis constraint. It is satisfied either if both stays are associated with the very same primary diagnostic code (level 4) or in more relax settings, i.e., if both codes belong to the same level-3 or level-2 branch of the hierarchy of the ICD-9-CM (International Classification of Diseases, Ninth Revision, Clinical Modification) [START_REF]Centers for Disease Control and Prevention, International Classification of Diseases, Ninth Revision, Clinical Modification (ICD-9-CM)[END_REF]. Statistical details on the influence of the constraints is discussed in the next section.

Preliminary statistics on MIMIC-III

We computed some preliminary statistics on MIMIC-III dataset to check how many analogical proportions can be formed for each of the three analogical settings and based on the three level diagnosis constraint as shown in Table 3. To form the analogies, we built tuples of each of the two stays that belong to a single patient 𝑖. We kept only adult patients (aged 18 and above) that have at least two hospital stays. For our first setting, we define a valid analogy as a quadruple of four stays

(𝑠 𝑖 1 𝑡 1 , 𝑠 𝑖 1 𝑡 2 , 𝑠 𝑖 2 𝑡 3 , 𝑠 𝑖 2 𝑡 4 )
, where each pair of two stays belong to a single patient 𝑖 𝑗 . Since we do not restrict the order of the stays for each of the pairs, our analogies were made of all the permutations of all the stays belonging to a patient.

For our second and third settings, we define a valid analogy to be a quadruple made of four stays

(𝑠 𝑖 1 𝑡 1 , 𝑠 𝑖 1 𝑡 2 , 𝑠 𝑖 2 𝑡 3 , 𝑠 𝑖 2 𝑡 4 )
, where each pair of two stays belong to a single patient 𝑖 𝑗 and 𝑠 𝑖 1 𝑡 1 and 𝑠 𝑖 2 𝑡 3 have the same diagnostic code. As an order constraint is introduced for these two settings, we had to make sure that 𝑠 𝑖 1 𝑡 1 takes place before 𝑠 𝑖 1 𝑡 2 and 𝑠 𝑖 2 𝑡 3 takes place before 𝑠 𝑖 2 𝑡 4 . For the second setting, the stays do not necessarily happen directly right after, where other stays can exist in between. As for the third setting, one stay immediately follows the other, i.e., there is no other stay in between them.

For the diagnosis constraint, we referred to the ICD-9-CM, which is the standard nomenclature for assigning diagnosis codes to each hospital stay. Indeed, each stay has a unique primary diagnosis code and a set of secondary codes. Diagnostic codes are organized hierarchically as follows: (1) chapter, (2) block, (3) 3-digit category, and (4) full code. As an example, the diagnosis code 767.4 and its hierarchy are presented Table 2.

Level Level name

Example of code range The MIMIC-III dataset associates each stay with an ICD-9 code (i.e. full code). For the second and third analogical settings, we preprocessed our diagnosis codes in the following manner. We kept only the primary diagnostic code associated with each patient-stay. We filtered ICD-9 codes that appeared only once. For the second level and third level diagnosis settings, we performed the same preprocessing except that we had to first convert the ICD-9 codes into their corresponding category and block formats.

Table 3 provides the number of valid analogies that can be formed with the defined settings. As shown, the number of analogies is the highest for the Identity setting, which could be explained as a result of the absence of constraint on both the order of stays and diagnosis. The more strict the order constraint is, the less the amount of valid analogies that could be formed. The diagnosis constraint also influences the number of analogies that could be generated. The number of analogies is the lowest for full code (level 4) settings, where less patients share the very same primary diagnosis code. In comparison, more patients can share a single diagnosis code in the category (level 3) and block (level 2) settings, where we observe the highest number of analogies in the block setting for both the second and third analogical settings.

Setting

Properties of analogies for data augmentation

As we are interested in exploring different deep learning models, we would need large amounts of data to train them. To enlarge the training datasets, one may use analogy properties to generate more analogies in a process called data augmentation. Training our model on different equivalent forms of the same analogy could help reduce overfitting. Previous works [START_REF] Miclet | Analogical dissimilarity: Definition, algorithms and two experiments in machine learning[END_REF][START_REF] Lepage | De l'analogie rendant compte de la commutation en linguistique[END_REF][START_REF] Antic | Analogical proportions[END_REF] have defined postulates that proportional analogy should obey; some of which include the following:

• However, not all these postulates hold for all our settings. Based on the current definitions of our analogical settings, we can apply reflexivity for all the three settings. Inner reflexivity can only be applied for the Identity setting. Adding this postulate for the second and third settings would require to loose our order constraint, which is inconsistent with the temporal aspect of predictive modeling. Determinism holds for all the settings. We include this postulate even if it produces trivial analogies. Central permutation can be applied on our analogies for the first setting only and in the very particular case when 𝑖 1 = 𝑖 2 . When 𝑖 1 ̸ = 𝑖 2 , central permutation cannot be applied to increase our dataset as it would enable to associate stays of distinct patients, which is inconsistent with the aim of the Identity setting. Concerning the second and third analogy settings, central permutation cannot be applied as it violates the order constraint in most cases. Note that central permutation can be applied for these two settings for cases when

𝑖 1 = 𝑖 2 , 𝑡 2 ≤ 𝑡 1 , 𝑡 3 ≤ 𝑡 4
, and the same diagnosis is associated to 𝑠 𝑖 1 𝑡 1 and 𝑠 𝑖 2 𝑡 3 . Inner symmetry can be applied for the Identity setting, but it violates the order constraints for the other two settings. For all the three analogical settings, by applying symmetry to one valid analogy, we can increase the number of valid analogies as it does not violate any of the three constraints. In addition to valid forms, we can also consider invalid forms (i.e., that contradict some of the setting constraints or that cannot be inferred from the base cases using the allowed postulates) for classification purposes.

Preliminary experiments: error analyses in the Identity setting

We set up a preliminary experiment on the analogy detection task, addressing our Identity setting. Inspired by [START_REF] Alsaidi | A neural approach for detecting morphological analogies[END_REF][START_REF] Lim | Solving word analogies: A machine learning perspective[END_REF], we consider a CNN classifier adapted to patient-stay, to determine whether a given (𝐴, 𝐵, 𝐶, 𝐷) constitutes a valid analogy. For the embedding model, we consider the Fusion CNN model developed by [START_REF] Zhang | Combining structured and unstructured data for predictive models: a deep learning approach[END_REF], which combines both structured and unstructured data to obtain patient-stay representations. For this very first experiment, we only consider structured data limited to demographics and admission-related information. For unstructured data, we group clinical notes associated with a hospital stay. We learn clinical note embeddings and concatenate them with static information following [START_REF] Zhang | Combining structured and unstructured data for predictive models: a deep learning approach[END_REF] to obtain our final patient-stay representations.

We considered hospital stays of 200 patients extracted randomly from MIMIC-III. We define a valid analogy as a quadruple of four stays

(𝑠 𝑖 1 𝑡 1 , 𝑠 𝑖 1 𝑡 2 , 𝑠 𝑖 2 𝑡 3 , 𝑠 𝑖 2 𝑡 4 )
, where each pair of two stays belong to a single patient 𝑖 𝑗 . We do not define any order constraint for our stays; therefore, 𝑠 𝑖 1 𝑡 1 can happen before 𝑠 𝑖 1 𝑡 2 . Quadruples where 𝑖 1 = 𝑖 2 are also included in the dataset. To generate other valid analogies, we make use of all postulates in Section 5, except for central permutation that is only applied in the case when 𝑖 1 = 𝑖 2 . As reflexivity forces 𝑖 1 = 𝑖 2 , it cannot be applied in the cases where 𝑖 1 ̸ = 𝑖 2 . Accordingly, given a valid analogy 𝐴 : 𝐵 :: 𝐶 : 𝐷, we generate 8 additional valid analogies, namely, 𝐶 : 𝐷 :: 𝐴 : 𝐵, 𝐷 : 𝐶 :: 𝐵 : 𝐴, 𝐵 : 𝐴 :: 𝐷 : 𝐶, 𝐴 : 𝐴 :: 𝐶 : 𝐶, 𝐵 : 𝐴 :: 𝐶 : 𝐷, 𝐴 : 𝐵 :: 𝐷 : 𝐶, 𝐶 : 𝐷 :: 𝐵 : 𝐴, 𝐷 : 𝐶 :: 𝐴 : 𝐵, and 2 invalid, namely, 𝐷 : 𝐴 :: 𝐵 : 𝐶 and 𝐴 : 𝐶 :: 𝐵 : 𝐷. When 𝑖 1 = 𝑖 2 , we generate one more valid analogy of the form 𝐴 : 𝐵 :: 𝐴 : 𝐵 and we consider invalid analogies as valid.

For training and evaluation, we split our dataset into 70% training set and 30% testing set, representing 939,638 analogies for training and 402,703 for testing. We randomly draw 50,000 analogies of each set (i.e., training and testing) when loading the data. Following the data augmentation procedure introduced before, given a valid analogy, we generate 9 valid analogies (i.e., positive examples) and 2 invalid analogies (i.e., negative examples) for cases when 𝑖 1 ̸ = 𝑖 2 . In contrast, we generate 12 valid analogies and no invalid analogies for cases when 𝑖 1 = 𝑖 2 . Based on this setting, we tend to generate more valid analogies than invalid ones. We trained our model on 10 epochs, with 3 random intializations to observe how the model behaves and how much it is able to learn. We only computed the accuracy and obtained 96.85 ± 1.75 for valid analogies and 70.31 ± 1.94 for invalid analogies. Our model performs the best for positive examples which can be explained as a result of the imbalance between valid and invalid examples in the training data. Nonetheless, these preliminary results seem to show that the model learns, to some extent, patient-stay identity relationships.

To gain a deeper insight on how our classification model works, we present four examples: one true positive, one false negative, one true negative, and one false positive. Patient ids in the examples below have been changed and dates have been shifted. We provide elements of interpretation to explain why our model correctly classifies some analogies and why in other cases it does not.

Analysis of a true positive example. We consider the stay 𝑠 𝑖 This example has been correctly classified as valid for all the 9 valid forms. As we do not introduce any order constraint for this setting, we can notice that for some forms like 𝐴 : 𝐵 :: 𝐶 : 𝐷 and 𝐴 : 𝐵 :: 𝐷 : 𝐶, 𝑠 𝑖 1 𝑡 2 would take place before 𝑠 𝑖 1 𝑡 1 in time. The model correctly classifies these analogy forms as valid.

Analysis of a false negative example. We consider the stays in this example to belong to the same patient 1109, who is a female. The stay 𝑠 𝑖 1 𝑡 1 of patient 1109, with 25yo, suffers from Malignant essential hypertension, admitted 7 times before, and with 1 Radiology report and 2 Nursing/Other reports documenting this stay. The second stay 𝑠 𝑖 1 𝑡 2 of patient 1109, but with 26yo, with Hypertensive chronic kidney disease, admitted 4 times before, and with 8 Physician reports and 4 Nursing reports. The third stay 𝑠 𝑖 2 𝑡 3 of patient 1109, but with 26yo, admitted once again for Hypertensive chronic kidney disease, admitted 3 times before, and with 3 Physician reports and 8 Nursing reports. The fourth stay 𝑠 𝑖 2 𝑡 4 of patient 1109, with 27yo, suffers from Vascular complications of medical care, admitted 5 times before, and with 3 Physician reports and 9 Nursing reports.

As we mentioned above, for cases where 𝑖 1 = 𝑖 2 , applying central permutation would also give us valid analogies. In this example, our model incorrectly classified the form of 𝐷 : 𝐴 :: 𝐵 : 𝐶 as invalid. As there were less analogies made of four stays that belong to the same patient included in our dataset, we noticed that our model is more likely to incorrectly classify these analogies, particularly for invalid forms.

Analysis of a true negative example. We consider the stay 𝑠 𝑖 1 𝑡 1 of patient 553, who is a male, with 23yo, suffers from Hypertensive chronic kidney disease, admitted 4 times before, and with 4 Physician reports and 8 Nursing reports documenting this stay. The stay 𝑠 𝑖 1 𝑡 2 belongs to the same patient 553, but with 24yo, admitted once again for Hypertensive chronic kidney disease, admitted 6 times before, and with 6 Physician reports and 6 Nursing reports. The third stay 𝑠 𝑖 2 𝑡 3 belongs to patient 2387, who is a male, with 52yo, with Unspecified disease of pericardium, admitted 7 times before, and with 2 Radiology reports and 2 Nursing/Other reports. The stay 𝑠 𝑖 2 𝑡 4 belongs to the same patient 2387, but with 53yo, with Unspecified pleural effusion, admitted 9 times before, and with 8 Radiology reports and 4 Nursing/Other reports.

This analogy has been correctly classified as invalid for both invalid forms, 𝐷 : 𝐴 :: 𝐵 : 𝐶 and 𝐴 : 𝐶 :: 𝐵 : 𝐷.

Analysis of a false positive example. We consider the stay 𝑠 𝑖 1 𝑡 1 of patient 2771, who is a male, with 71yo, suffers from Subendocardial infarction, admitted 16 times before, and with 9 Nursing/Other reports. The second stay 𝑠 𝑖 1 𝑡 2 belongs to the same patient 2771, but with 70yo, suffers from Other pulmonary embolism and infarction, admitted 5 before, and with 1 Radiology report and 3 Nursing/Other reports. The stay 𝑠 𝑖 2 𝑡 3 of patient 2222, who is a female, with 69yo, with Diverticulosis of colon with hemorrhage, admitted 9 times before, and with 2 Physician reports and 10 Nursing reports. The stay 𝑠 𝑖 2 𝑡 4 belongs to the same patient 2222, but with 67yo, with Arterial embolism and thrombosis of lower extremity, admitted 5 times before, and with 3 Nursing/Other reports.

This analogy has been incorrectly classified as valid for the invalid form, 𝐴 : 𝐶 :: 𝐵 : 𝐷. We noticed that when the category of the clinical notes is similar between two hospital stays and when our hospital stays do not include a lot of clinical notes, our model seems to struggle to distinguish between the two hospital stays. 𝑠 𝑖 1 𝑡 2 and 𝑠 𝑖 2 𝑡 4 have the same number of Nursing/Other reports. Thus these reports may not contain enough information to help our model differentiate between these two stays. As a result, the model incorrectly matches these stays to the same patient.

Conclusion

In this paper we discussed an exploratory approach to investigate analogical inference in healthcare. We started by briefly surveying some related work that address different applications of analogical reasoning in different domains. We defined three analogical settings for different healthcare tasks, and discussed the motivation behind our settings. We also presented preliminary statistics of the sets of analogical proportions that we built from MIMIC-III. The main contribution of our work is the formalization of settings that are meaningful in healthcare, and that guide the process of building sets of analogies in healthcare. We discuss the pertinence of certain widely used postulates in this healthcare context. Lastly, we also illustrated the Identity setting on which we addressed a preliminary experiment on the analogy detection task. These first results pave the way to conducting further experiments on the other two settings, and to an in depth analysis of the potential of coupling representation learning and analogical reasoning in healthcare.

  reflexivity: 𝐴 : 𝐵 :: 𝐴 : 𝐵 • inner reflexivity: 𝐴 : 𝐴 :: 𝐶 : 𝐶 • determinism: 𝐴 : 𝐴 :: 𝐴 : 𝐷 → 𝐷 = 𝐴 • symmetry: 𝐴 : 𝐵 :: 𝐶 : 𝐷 → 𝐶 : 𝐷 :: 𝐴 : 𝐵 • inner symmetry: 𝐴 : 𝐵 :: 𝐶 : 𝐷 → 𝐵 : 𝐴 :: 𝐷 : 𝐶 • central permutation: 𝐴 : 𝐵 :: 𝐶 : 𝐷 → 𝐴 : 𝐶 :: 𝐵 : 𝐷.

Table 1

 1 General statistics of the MIMIC-III EHR dataset, restricted to patients aged of 18 and above, with at least 2 stays.

	Statistics

Table 2

 2 Hierarchy of the ICD9-CM (International Classification of Disease, Ninth Revision, Clinical Modifications), with examples for the code 767.4 ("Injury to spine and spinal cord due to birth trauma").

	1	Chapter	760-779
	2	Block	764-779
	3	3-digit category code	767
	4	Full code	767.4

Table 3 Number

 3

		ICD Level	Analogies
	Identity	N/A	1, 100, 954, 350
		4	648, 169
	Identity + Sequent	3	1, 876, 445
		2	3, 243, 699
		4	545, 892
	Identity+Directly Sequent	3	1, 326, 518
		2	2, 780, 507

of valid analogies that can be generated from MIMIC-III, depending on the different settings and on the level of flexibility allowed on patient ICD diagnosis.

  1 𝑡 1 of patient 1249, who is a female, with 83yo, suffers from Measles keratitis, admitted twice before, and with 12 Radiology reports documenting this stay. The second stay 𝑠 𝑖 1 𝑡 2 of the same patient 1249, but with 81yo, suffers from Pancreat cyst/pseudocyst, only admitted once before, and with 5 Radiology reports and 7 Nursing/Other reports. The stay 𝑠 𝑖 2 𝑡 3 belongs to patient 4695, who is a female, 21yo, with Acute venous embolism and thrombosis of superficial veins of upper extremity, admitted 5 times before, and with 5 Radiology reports and 2 Nursing/Other reports. The stay 𝑠 𝑖 2 𝑡 4 of the same patient 4695, but with 22yo, suffers from Hypertensive chronic kidney disease, admitted 8 times before, and with 5 Physician reports and 7 Nursing reports documenting this stay.

Acknowledgments

We thank IARML reviewers for their constructive and positive feedback. Experiments presented in this paper were carried out using computational clusters equipped with GPU from the Grid'5000 testbed (see https://www.grid5000.fr).

The research work of the second named author is partially supported by TAILOR, a EU Horizon 2020 project (GA No 952215), and the Inria Project Lab "Hybrid Approaches for Interpretable AI" (HyAIAI).


Na Shao

Jean Rodriguez

Adrien Quintard
email: adrien.quintard@univ-grenoble-alpes.fr

Catalysis Driven Six-Step Synthesis of Apratoxin A Key Polyketide Fragment

Keywords: Optimization details, complete experimental procedures, products characterization, and spectra. (

Apratoxin A is a potent anticancer natural product whose key polyketide fragment constitutes a considerable challenge for organic synthesis, with five prior syntheses requiring 12 to 20 steps for its preparation. By combining di erent redoxeconomical catalytic stereoselective transformations, the key polyketide fragment could be rapidly prepared. Followed by a siteselective protection of the diol, this strategy enables the preparation of the apratoxin A fragment in only six steps, representing the shortest route to this polyketide.

P olyketides are among the most prominent natural products and given their variety of bioactivity profiles, represent an invaluable inspiration for drug discovery. 1 As a consequence, chemists have for a long time considered polyketide synthesis as a motivation to devise more e cient synthetic routes. However, most syntheses still too often involve lengthy stepwise sequences and activation by multiple stoichiometric reagents. The di culty at synthesizing polyketides stems from the challenge associated with the control of multiple remote acyclic stereogenic centers required for each distinct chiral reactant or catalyst. Therefore, most solutions consist of the stepwise elaboration of the di erent functionalities through iterative sequences involving, aside from the molecular skeleton formation, costly changes in the redox states of the carbon chain and protecting group manipulation.

Fulfilling the principles of redox economies, 2 di erent cascade transformations have been developed where the initial activation step consists of an alcohol dehydrogenation by a metal complex. 3 In situ generating reactive aldehydes, the catalytic cascades can trigger complex enantioselective couplings between initially unreactive alcohols, considerably limiting waste and steps. Meeting these requirements, the Krische group developed a broad array of coupling reactions involving alcohols and pro-nucleophilic carbon donors. [START_REF] For | Catalytic Carbonyl Addition through Transfer Hydrogenation: A Departure from Preformed Organometallic Reagents[END_REF] In a typical example, enantioselective crotylation of alcohols can be performed directly through the iridium catalyzed condensation of α-methyl allyl acetate and alcohols (Scheme 1a). 5,[START_REF] Kim | Enantioselective Iridium Catalyzed Carbonyl Allylation from the Alcohol or Aldehyde Oxidation Level Using Allyl Acetate as an Allyl Metal Surrogate[END_REF] Such strategies can considerably short-cut the construction of key natural product fragments which was notably applied in the context of polyketides synthesis. 7 Complementary to these couplings, our group developed the concept of enantioselective multicatalytic borrowing hydrogens (Scheme 1b). 8 Based on a combination between an iron borrowing hydrogen catalyst and a chiral pyrrolidine organocatalyst, such an approach enables the enantioselective condensation of di erent nucleophiles to allylic alcohols. Apratoxins are natural macrolides first isolated in 2001 from Lyngbya types cyanobacteria. 9 They feature strong cytotoxic activity at the nM level against various cancer lines. Among the di erent natural apratoxins and unnatural analogues, apratoxin A (Scheme 2) is the most potent one. The excellent bioactivity profile has pushed organic chemists to develop a wide array of synthetic routes toward its preparation.

Received

While the preparation of the peptidic part of the natural product is more classical and straightforward, the biggest challenge arises from the stereoselective construction of the complex C33-C39 advanced polyketide fragment. While Nature has engineered powerful enzymes for its construction, for now, chemists have relied on lengthy synthetic sequences for its preparation (Scheme 2a). All these e orts have been summarized in two reviews by Coltart. 10 Illustrating the di culty at finding a convenient route to assemble this polyketide sca old and control the di erent stereocenters, the first synthesis of this fragment by Forsyth required 13 steps and the use of two chiral auxiliaries for the construction of the carbon-carbon bonds. 11 Since then, the groups of Ma and Liu, Takahashi and Doi or Kazmaier have developed alternative strategies all still requiring between 12 and 20 steps and multiple chiral auxiliaries. 12 Synthesis of apratoxin analogues and other doubly protected diol fragments also required 13-14 steps such as in works by Cavelier, Xu, Gilles and Ye. 13 As a result, despite decades of e orts, the shortest sequence to date remains based on a minimum of 12 steps and multiple stoichiometric activating reagents.

Given the above-mentioned progress in the development of redox-economical catalytic transformations, we envisaged that a combination between a multicatalytic borrowing hydrogen and a redox-neutral coupling could provide a fast and highly enantioselective generation of this polyketide motif (Scheme 2b). This strategy would elaborate the desired C33-C39 chain in only three steps directly from widely available products. Development of a subsequent site-selective alcohol protection could unlock the final access to the desired functionalized key polyketide fragment. Herein, we present our success at developing such a strategy, culminating in the concise sixstep synthesis of the key polyetide fragment, a ording the most e cient to date formal synthesis of apratoxin A. This approach considerably shortens the access to this class of products and should find application in the preparation of other polyketide motifs.

The synthesis started with the condensation of commercial diketone 1 to crotyl alcohol 2 (Scheme 3). 8 By combining iron complex Fe-Cat and diphenyl prolinol silyl ether organocat, the cascade enantioselective borrowing hydrogen/Claisen type fragmentation [START_REF] Rodriguez | Acyl transfer strategies as transient activations for enantioselective synthesis[END_REF] directly a orded the desired adduct 3 in 56% yield. Following stereoselective CBS reduction of the ketone and simultaneous deprotection of the alcohol, diol 4 could be isolated in 70% yield. The excellent selectivity of the CBS reduction 15 enabled isolatation of diol 4 in >99:1 dr and improved 99% ee after separation upon silica gel chromatography of the minor diastereomer generated.

From this diol, key to the success for a short and e cient synthesis of apratoxin A was the implementation of a selective crotylation of the primary alcohol in the presence of the unprotected secondary alcohol. Even though allylation had been described on other types of sca olds containing

B

secondary alcohols notably on 1,3-diols, 16 crotylation proved challenging in the context of unprotected 1,5-diol 4 (Table 1). Among all the tested parameters (see Supporting Information (SI) for additional experiments), we found that the nature of the base, the temperature, the concentration, and the structure of the catalyst were crucial to obtain the desired product. First experiments applying standard conditions with Ir-1 and Cs 2 CO 3 as base a orded the expected adduct 6 in a low 26% yield (entry 1). Under those conditions, formation of 6 was accompanied by a large amount of undesired dehydrogenation of the secondary alcohol to the corresponding ketone. Turning to K 3 PO 4 improved the yield to 44% (entry 2). While the presence of water is often associated with beneficial e ects in the crotylation, herein, using dry THF improved both yield and selectivity, providing 6 in 53% yield and 4:1 dr (entry 3).

Similarly, crotylation is also mostly time conducted at high concentration (>1 M), which proved detrimental to the present reaction, a ording 6 in only 26% yield (entry 4). Using more diluted conditions (0.25 M) also did not prove satisfactory, generating only 9% of the desired adduct with lower diastereocontrol (2.8:1 dr) (entry 5). Finally, use of less electron-e cient Ir-2 totally shut down the reactivity in this transformation (entry 6). Altogether, applying conditions of entry 3 with Ir-1 as catalyst and K 3 PO 4 enabled the synthesis of diol 6 in 53% yield and 4:1 dr. It is important to highlight that by applying this approach, all the four stereogenic centers of the polyketide chain are constructed in only three steps. From diol 6, the site-selective protection of the supposedly less hindered C-35 homoallylic alcohol was then scrutinized (Table 2, see Supporting Information for additional experi-

C

ments). We focused on the protection by a silyl or a 2,2,2tricloroethoxycarbonyl (Troc) group, already applied in the synthesis of apratoxin A. 12 Initially, using TBSCl or TESCl did not provide any reactivity (entry 1). Using the more reactive TBSOTf, only the major protection of both alcohols 7-di was observed with poor conversion, indicating a total lack of site selectivity (entry 2). Turning to the use of TrocCl considerably improved the reactivity (entries 3-8). [START_REF] Kawabata | A Catalytic One-Step Process for the Chemoand Regioselective Acylation of Monosaccharides[END_REF][START_REF]For examples of site-selective acylation of diols[END_REF] Using pyridine as the base, DMAP did catalyze the formation of the expected monoprotected alcohol 7 together with still a large amount of bis-protected diol 7-di (entry 3). Using pyridine alone improved the selectivity in favor of the monoprotected alcohol at the expense of the conversion (entry 4). Using an excess of DMAP as base improved the selectivity a ording around 25% of the desired monoprotected alcohol together with a large amount of unprotected diol and some bis-protected diol (entry 5). Given these results, we hypothesized that use of a bulkier nucleophilic Lewis base catalyst could improve the siteselective conversion to the monoprotected alcohol. Indeed, using chiral nonracemic isothiourea Hyper-BTM catalysts 19 resulted in a good compromise between conversion and selective monoprotection (entries 6-8). Since both enantiomers of the chiral catalyst provide closely related results (compare entries 6 and 7), this indicates that the selectivity is not arising from a chiral recognition by the enantiopure catalyst but rather due to the nature of the catalyst, disfavoring the acylation at the more hindered C-39 alcohol. This is confirmed by using achiral DHPB, [START_REF] Kobayashi | Unexpected reactivity of annulated 3H-benzothiazol-2-ylideneamines as an acyl transfer catalyst[END_REF] providing a closely related selectivity but at the expense of the conversion (entry 9). Altogether, by combining 1 mol % of (2S,3R)-hyper-BTM and 2 equiv of DMAP, the desired monoprotected alcohol 7 could be isolated in 59% yield through this site-selective protection (entry 8). Of interest, 20% of the unreacted starting diol could be recovered and reused, providing a 79% yield for this process based on recovered starting material. With a convenient site-selective monoprotection giving access to 7, subsequent functionalization of the secondary alcohol with the protected proline derivative and oxidative cleavage of the double bond provided in two subsequent steps the key fragment 9, a known apratoxin A precursor (Scheme 3). 12 In conclusion, by appropriately combining redox-economical stereoselective construction of the functionalized carbon chain with site-selective protection of the resulting diol 6, a short formal synthesis of apratoxin A was developed. The key C33-C39 polyketide fragment, common to all apratoxin A synthesis, could be prepared in only 6 steps while the previous literature syntheses required between 12 and 20 steps. Given the e ciency of this approach and the similarity between apratoxin A and other natural sca olds, this strategy should also be applied in the future to the e cient synthesis of other polyketides such as tuelarin C or neopeltolide.

■ ASSOCIATED CONTENT

F

Scheme 2 .

 2 Scheme 2. (a) Apratoxin A and Known Syntheses of the Key Polyketide Fragment; (b) Proposed Strategy

:

  July 22, 2022

	pubs.acs.org/OrgLett	Letter
		Scheme 1. Examples of Redox-Economic Transformations

© XXXX American Chemical Society A https://doi.org/10.1021/acs.orglett.2c02482

Org. Lett. XXXX, XXX, XXX-XXX

Table 1 .

 1 Optimization of the Crotylation

	Entry	Base	Solvent	Cat.	Concn	Yield % (dr) a
	1	Cs 2 CO 3	THF (H 2 O, 5 equiv)	Ir-1	0.5 M	26% (3.6:1 dr)
	2	K 3 PO 4	THF (H 2 O, 5 equiv)	Ir-1	0.5 M	44% (3.1:1 dr)
	3	K 3 PO 4	THF	Ir-1	0.5 M	53% (4:1dr)
	4	K 3 PO 4	THF	Ir-1	1 M	26% (3.4:1 dr)
	5	K 3 PO 4	THF	Ir-1	0.25 M	9% (2.8:1 dr)
	6	K 3 PO 4	THF	Ir-2	0.5 M	trace

a dr of the isolated product.

Table 2 .

 2 Optimization of the Site-Selective ProtectionDetermined by 1 H NMR on the crude reaction mixture. b brsm = based on recovered starting material.

	Entry	Conditions	6/7-di/7 a
	1 2 3	TBSCl or TESCl, base (excess), solvent, 0 to 80 °C (see SI for full conditions) TBSOTf (1 equiv), imidazole (2 equiv), DMAP (0.5 equiv), DMF, 0 °C to rt TrocCl (1.2 equiv), pyridine (3 equiv), DMAP (5 mol %), DCM, 0 °C to rt	Conv <5% 93/6/1 3.3/4/1
	4	TrocCl (1.05 equiv), pyridine (2 equiv), DCM, rt	2.2/0.7/1
	5	TrocCl (1.05 equiv), DMAP (2 equiv), DCM, rt	0.9/0.2/1
	6	TrocCl (1.05 equiv), pyridine (2 equiv), (2S,3R)-hyper-BTM (1 mol %), DCM, rt	1/0.9/1
	7	TrocCl (1.05 equiv), pyridine (2 equiv), (2R,3S)-hyper-BTM (1 mol %), DCM, rt	1.3/0.8/1
	8	TrocCl (1.05 equiv), DMAP (2 equiv), (2S,3R)-hyper-BTM (1 mol %), DCM, rt	0.3/0.2/1 (59%, 7 yield, 79% brsm) b
	9	TrocCl (1.05 equiv), DMAP (2 equiv), DHPB (1 mol %), DCM, rt	2.4/0.1/1

a https://doi.org/10.1021/acs.orglett.2c02482 Org. Lett. XXXX, XXX, XXX-XXX

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Funding

The Agence National de la Recherche (ANR-19-CE07-0033), the Centre National de la Recherche Scientifique (CNRS) and Aix-Marseille Universitéare warmly acknowledged for financial support. N.S. thanks the China Scholarship Council for a PhD grant (No. 201908070004).

Notes

The authors declare no competing financial interest.

■ ACKNOWLEDGMENTS

The Agence National de la Recherche (ANR-19-CE07-0033), the Centre National de la Recherche Scientifique (CNRS), Université Grenoble-Alpes, and Aix-Marseille Université are warmly acknowledged for financial support. N.S. thanks the China Scholarship Council for PhD grants (No. 201908-070004). We warmly thank the personnel from Aix-Marseille University and Marseille Spectropole and Dr. Nicolas Vanthuyne from Aix-Marseille University for HPLC analysis. We also warmly thank Anestis Alexandridis from Université Grenoble-Alpes for the initial synthesis of some iridium complexes.


Pierre Rizkallah

Aurélien Grabsch

Pierre Illien

Olivier Bénichou

Duality relation in single-file diffusion

Single-file transport, which corresponds to the diffusion of particles that cannot overtake each other in narrow channels, is an important topic in out-of-equilibrium statistical physics. Various microscopic models of single-file systems have been considered, such as the simple exclusion process (SEP), which has reached the status of a paradigmatic model. Several different models of single-file diffusion have been shown to be related by a duality relation, which holds either microscopically or only in the hydrodynamic limit of large time and large distances. Here we show that, within the framework of fluctuating hydrodynamics, these relations are not specific to these models and that, in the hydrodynamic limit, every single-file system can be mapped onto a dual single-file system, which we characterise. This general duality relation allows to obtain new results for different models, by exploiting the solutions that are available for their dual.

I. INTRODUCTION

The study of the dynamical properties of interacting particle systems in out-of-equilibrium situations has been the focus of many works over the last decades [1,2]. The search for a general theory able to describe these systems relies in part on the study of simple microscopic models. This is especially useful for one dimensional systems, for which exact analytical results can be obtained [3][4][5][6][7][8][9][10][11][12]. These 1D models notably describe the important situation of single-file diffusion, which corresponds to the motion of particles in narrow channels with the constraint that they cannot bypass each other. This geometrical constraint has a strong effect on the motion of the particles, which exhibit a subdiffusive behaviour [4,13,14]. This theoretical prediction has been observed at different scales, from the diffusion of molecules in zeolites [15] to the motion of colloids in narrow trenches [16,17].

For diffusive systems (satisfying a Fourier law) of particles, the simple exclusion process (SEP) has reached the status of a paradigmatic model [START_REF] Mallick | International Summer School on Fundamental Problems in Statistical Physics[END_REF]. It consists of particles on a lattice, which can hop to neighbouring sites, with an exclusion rule that enforces that each site can host at most one particle (see Fig. 1, top). The SEP on the infinite line can be solved by Bethe ansatz, and has yielded a number of exact results on the current fluctuation or the displacement of a tagged particle [6,9,10,[START_REF] Derrida | [END_REF].

The SEP is well known to be related to another model of particles on a lattice: the zero range process (ZRP) [20] (see the reviews [21,22]). In the ZRP, the number of particles on a lattice site is not limited (see Fig. 1, bottom). The rate at which a particle hops from a site can depend on the site and on the number of particles on it, but not on the neighbouring sites. The interaction between the particles thus occur only on the same site (through the hopping rate), hence the name zero range. The SEP can be mapped onto the ZRP with constant hopping rates, by setting the occupations of the sites in the ZRP to be equal to the number of empty sites between the particles of the SEP (see Fig. 1). This relation between the two models has been used explicitly in several works, see for instance [23][24][25][26].

A similar relation has been made between two other well studied models of one dimensional diffusive systems: the random average process (RAP) and the Kipnis Marchioro Presutti (KMP) model. The RAP is a model of particles on the continuous line, on which particles can hop, in either direction, to a random fraction of the distance to the next particle [27]. This model can be mapped onto a mass transfer model [27][28][29][30], which, in the hydrodynamic limit of large time and large distances, becomes equivalent to the KMP model [31]. The KMP model has been introduced to study the heat conductivity [3]. It is again a lattice model, but now each site can host a continuous variable which represents a mass (or an energy), and can randomly exchange mass with its neighbor.

In this article, we show that these relations between different models, which hold either microscopically (for the SEP/ZRP) or in the hydrodynamic limit (for the RAP/KMP), are not specific to the examples mentioned above: by applying these transformations, any single-file system can, in the hydrodynamic limit, be mapped onto another single-file system, which we characterise. This allows to obtain new results on several models of single-file systems by exploiting the solutions available on their dual. For this, we rely on the framework of fluctuating hydrodynamics and macroscopic fluctuation theory (MFT), which is a powerful tool to study the large scale behavior of diffusive systems [32][33][34]. Its major feature is to allow a unified description of a wide range of systems in terms of only two transport coefficients: a diffusion coefficient D(ρ), and a mobility coefficient σ(ρ), which both depend on the local density ρ of the system. Macroscopic fluctuation theory has been applied with success to the computation of the large FIG. 1. Mapping of the simple exclusion process (SEP, top) onto the zero range process (ZRP, bottom) [20][21][22]. The particles of the SEP can hop on neighbouring sites, only if they are empty. The sites of the ZRP correspond to the particles of the SEP. The number of particles on a site in the ZRP is the number of empty sites on the right of the corresponding particle in the SEP. For the SEP, we have represented by two arrows the possible jumps of an individual particle. We have represented in the ZRP the corresponding dynamics, illustrated by the two arrows corresponding to the jumps of the SEP particle.

deviations of the steady-state current across a diffusive system maintained out-of-equilibrium by two reservoirs [35][36][37], and later to study the current on a infinite line with step initial conditions [START_REF] Derrida | [END_REF]38], or to tracer diffusion [8]. Recently, the MFT equations have been solved exactly in several cases [39][40][41].

Note that many microscopic models discussed here have asymmetric versions, such as the asymmetric simple exclusion process (ASEP) in which all the particles have different jump rates to the left and to the right. The mappings onto other models are still possible, but not the description in terms of fluctuating hydrodynamics. Here, we will only focus on the symmetric case, for which the density has a diffusive scaling and fluctuating hydrodynamics can be applied.

A. Summary of the main results

We show that a single-file system described at large scale by the two coefficients D(ρ) and σ(ρ), whose definitions are recalled in Section II, can be mapped onto a dual single-file system associated with the transport coefficients

D(ρ) = 1 ρ 2 D 1 ρ , σ(ρ) = ρ σ 1 ρ . ( 1
)
The corresponding transformation for the density field ρ(x, t) and the current j(x, t) are written in Section III C. While these transformations have already been introduced for the SEP/ZRP [21,22] and for the RAP/KMP [30,31], we stress that our main result is that (i) the transformed density still describes a single-file system, and (ii) this single-file system is explicitly characterised in terms of the original system through (1). We call such a mapping between two single-file systems a duality relation because, as we will argue in Section III, the two dual models actually correspond to two equivalent descriptions of the same system. This duality relation is different from the stochastic duality studied in the context of interacting particle systems, such as in [54].

We also consider in Section IV three other transformations (corresponding to translation or dilatation of the density and current), which, combined with the duality relation described above, allow to map several different single-file systems onto each other. We argue in Appendix A that the combination of these four elementary transformations gives all the possible mappings between two single-file systems.

As an application of these transformations, we show that we can obtain new results on several models of single-file diffusion by mapping them onto a model for which a solution has been previously derived. We thereby characterise the statistical properties of two important observables in single-file diffusion (position of a tracer and integrated current through the origin) for several models using the known results that are available on the SEP [6,9,10,12]. Indeed, we compute in Section V the cumulant generating function of these observables, for the different single-file systems listed in Table I, and represented in Fig. 4 below.

II. HYDRODYNAMIC DESCRIPTION

Consider a system of particles, labelled by an index k, located at positions x k (t). These positions can either be continuous variables (e.g. for Brownian particles) or discrete variables (e.g. for the SEP). The time evolution of these

D0 (1 -ρ) 2 2D0ρ(1 -2ρ) 1 -ρ TABLE I.
Transport coefficients D(ρ) and σ(ρ) for the different models studied in this paper (see Section V or the caption of Fig. 4 for the definitions). For the zero range process we will consider, σ(ρ) = 2D0ρ/(1 + ρ). For the double exclusion process, the transport coefficients are obtained by taking a limit of the more general Katz-Lebowitz-Spohn (KLS) model, whose transport coefficients are recalled in the Supplementary Material of Ref. [45].

positions is stochastic, and depends on the details of the microscopic model. The set of positions {x k (t)} can be described by a density

ρ(x, t) = k δ(x -x k (t)) . ( 2
)
At the macroscopic level (large time, large distances), this density can be replaced by a continuous field, which satisfy a continuity relation

∂ t ρ(x, t) + ∂ x j(x, t) = 0 , ( 3
)
where j(x, t) is the local current at position x and time t. The stochastic behaviour of the underlying microscopic model can be taken into account by assuming that the current j itself is random, and takes the form [START_REF] Spohn | Large scale dynamics of interacting particles[END_REF]

j(x, t) = -D(ρ)∂ x ρ(x, t) + σ(ρ) η(x, t) , ( 4
)
where η is a Gaussian white noise in space and time with unit variance, i.e.,

η(x, t)η(x , t ) = δ(x -x )δ(t -t ) . ( 5
)
This strength of this approach, named fluctuating hydrodynamics, is that all the microscopic details of the model (displacement, interaction, ...) are encoded into two transport coefficients:

• the diffusion coefficient D(ρ);

• the mobility σ(ρ), which characterises the amplitude of the fluctuations of the current j.

These coefficients can be computed by considering a finite system of length L, placed between two reservoirs at density ρ 1 and ρ 2 [START_REF] Derrida | [END_REF]. D(ρ) and σ(ρ) are related to the first two moments of the total current Q t that flows between the reservoirs:

D(ρ) = lim L→∞ lim ρ1→ρ2=ρ L ρ 1 -ρ 2 lim t→∞ Q t t , σ(ρ) = lim L→∞ lim ρ1→ρ2=ρ L lim t→∞ Q 2 t t . ( 6
)
In practice, these two coefficients can be difficult to compute analytically. Nevertheless, they have been obtained for different microscopic models (see for instance Table I).

FIG. 2. Duality relation between two equivalent descriptions of a particle system: through the positions of the particles (left) and the through the gaps between the particles (right). The particle system is described at the macroscopic level by a density field ρ(x, t) and a current j(x, t), while the dual system (gaps) is described by a density ρ(k, t) and a current j(k, t). These two descriptions are equivalent, and related by Eqs. (27,28). The inverse transformation is identical, and written explicitly in Eqs. (30,31). These two sets of fields (ρ, j) and (ρ, j) obey the equations of fluctuating hydrodynamics (3,4), with transport coefficients respectively given by (D, σ) and ( D, σ), which are related by the transformation (29).

III. PARTICLES/GAPS DUALITY RELATION

We now show that there is a duality relation between different systems, at the level of the fluctuating hydrodynamics. This duality relation is general, and can be applied to any single-file system (not necessarily systems of particles). Nevertheless, we first define the dual fields ρ and j, from the original fields ρ and j, by using heuristic arguments valid for systems of particles (section III A). These definitions coincide with the ones used in Refs. [21,22] for the SEP/ZRP, and in [30,31] for the RAP/KMP. We then proceed to show that, for any single-file system, with these definitions, the dual fields still obey the equations of fluctuating hydrodynamics (section III B), with new transport coefficients, which we determine.

A. Dual density and current fields

Let us consider a system of particles at positions {x k (t)}, described at the macroscopic level by the two continuous fields ρ(x, t) and j(x, t), and the two transport coefficients D(ρ) and σ(ρ). The idea behind the duality relation is that this system of particles can be equivalently described by considering the gaps {g k = x k+1 -x k } between the particles, as shown in Fig. 2. Note that we will introduce the new fields only at the macroscopic level, at which k can be considered as a continuous variable. We will however motivate the definitions of these fields by microscopic considerations (in which k represents the label of a particle and is in principle an integer). At large scale, the gaps are described by a density field

ρ(k, t) = g k ≡ x k+1 (t) -x k (t) (7)
and the associated current j(k, t). Microscopically, k is a discrete variable representing the label of the particle. Here, at the macroscopic level, it is a continuous variable which interpolates between these discrete values. The system of particles obeys the equations of the fluctuating hydrodynamics (3,4):

∂ t ρ(x, t) + ∂ x j(x, t) = 0 , ( 8
) j(x, t) + D(ρ)∂ x ρ(x, t) = σ(ρ) η(x, t) . ( 9
)
This is our starting point to obtain the equations satisfied by ρ and j. The first step is to express the label k(x, t) of the particle located at position x at time t in terms of the density ρ. Microscopically, this is a piecewise constant function, which gives the index of the closest particle to the left of x at time t. At the macroscopic level, this becomes a continuous function, which can be defined by writing the number of particles between 0 and x as

k(x, t) -k(0, t) = x 0 ρ(x , t)dx . ( 10
)
That requires the knowledge of k(0, t), which is the index of the particle located at x = 0. It can be determined by writing that the variation of k(x, t) during a short time dt corresponds to the number of particles that crossed x from right to left, i.e. -j(x, t)dt (see Fig. 3(a)). Therefore,

∂ t k(x, t) = -j(x, t) . ( 11
)
FIG. 3. (a) the flux j(x, t) of particles at position x at time t gives the variation of the index k(x, t) of the closest particle to the left of x, see Eq. ( 11). (b) The flux j(k, t) of gaps through particle k at time t gives the evolution of the position of this particle (13).

Note that this is coherent with (10) and the continuity relation (8). Taking the convention that at t = 0, the particle located at x = 0 has index k(0, 0) = 0, Eqs. (10,11) fully determine k(x, t) for all x and t.

We now relate the density fields ρ and ρ by defining the density of particles as the inverse of the gap between the particles located at position x at time t,

ρ(x, t) = 1 g k(x,t) = 1 ρ(k(x, t), t) , ( 12
)
from Eq. ( 7). The current j(k, t) corresponds to the "flux of gap" through the particle k, which is thus given by the motion of this particle (see Fig. 3

(b)), j(k, t) = -∂ t x k (t) . ( 13
)
We can express ∂ t x k in terms of ρ and j by writing that the number of particles between x p (t) and x is

x xp(t) ρ(x , t)dx = k(x, t) -k(x p (t), t) = k(x, t) -p , ( 14
)
since k(x p (t), t) = p. Taking the derivative of this relation with respect to t, and using Eqs. (8,11) we obtain

j(x p (t), t) -ρ(x p (t), t) ∂ t x p (t) = 0 . ( 15
)
Combining this relation with (13) yields

j(k(x, t), t) = - j(x, t) ρ(x, t) . ( 16
)
Equations (10,11) define the index k(x, t), while Eqs. (12,16) define the new fields ρ(k, t) and j(k, t) from the original fields ρ(x, t) and j(x, t). The same definitions can be found in Refs. [21,22] for the SEP/ZRP, and in [30,31] for the RAP/KMP. Although here we have introduced these definitions based on physical arguments for a system of particles, the dual fields can be defined from any fields ρ(x, t) and j(x, t) which obey the equations of fluctuating hydrodynamics (8,9). We will now show that these new fields ρ(k, t) and j(k, t) still obey these equations, with new transport coefficients.

B. Mapping the equations of fluctuating hydrodynamics

Let us now deduce the new equations satisfied by ρ and j from the fluctuating hydrodynamics equations (8,9). We first rewrite ( 16) using (12) as

j(x, t) = - j(k(x, t), t) ρ(k(x, t), t) . ( 17
)
Taking the time derivative of Eq. ( 12) we get

∂ t ρ(x, t) = ∂ t 1 ρ(k, t) + ∂ t k(x, t) ∂ k 1 ρ(k, t) = - 1 ρ(k, t) 2 (∂ t ρ(k, t) -j(x, t)∂ k ρ(x, t)) = - 1 ρ(k, t) 2 ∂ t ρ(k, t) + j(k, t) ρ(k, t) ∂ k ρ(x, t) (18)
where we have used (11) and (17). Similarly,

∂ x ρ(x, t) = ∂ x k(x, t) ∂ k 1 ρ(k, t) = - 1 ρ(k, t) 3 ∂ k ρ(k, t) , ( 19
)
∂ x j(x, t) = - 1 ρ(k, t) ∂ k j(k, t) ρ(k, t) = - 1 ρ(k, t) 2 ∂ k j(k, t) -j(k, t) ∂ k ρ(k, t) ρ(k, t) . ( 20
)
Plugging [START_REF] Mallick | International Summer School on Fundamental Problems in Statistical Physics[END_REF]20) into the continuity relation ( 3), we obtain a new continuity equation for the fields ρ and j:

∂ t ρ(k, t) + ∂ k j(k, t) = 0 . ( 21
)
Similarly, using [START_REF] Derrida | [END_REF] into the expression of the current (9), we obtain

j(k, t) + D 1 ρ 1 ρ2 (k, t) ∂ k ρ(k, t) = -ρ(k, t) σ 1 ρ η(x k (t), t) (22)
The transformation of the Gaussian white noise η(x, t) under the change of variables can be obtained from the distribution,

P [η] ∝ exp - T 0 dt ∞ -∞ dx η(x, t) 2 2 = exp - T 0 dt ∞ -∞ dk ∂x k ∂k η(x k (t), t) 2 2 . ( 23
)
Therefore, η(x k (t), t) is a Gaussian white noise with variance ∂k ∂x , which we denote

η(x k (t), t) (law) = ∂k ∂x η(k, t) = 1 ρ(k, t) η(k, t) , (24)
where

(law)
= indicates that the two noises have the same distribution, and η(k, t) is also a Gaussian white noise with unit variance. We thus get (using η(k, t)

(law) = -η(k, t)), j(k, t) + D(ρ)∂ k ρ(k, t) = σ(ρ) η(k, t) , ( 25
)
where we have denoted

D(ρ) = 1 ρ2 D 1 ρ , σ(ρ) = ρ σ 1 ρ . ( 26
)
The dual system, which describes the dynamics of the gaps, therefore obeys the equations of fluctuating hydrodynamics (21,25), with transport coefficients D and σ which are related to the original coefficients D and σ of the particle system by the transformations (26). The known duality relation between the RAP and the KMP model, used in Refs. [29][30][31] is a specific case of the duality relation described here, applied to the transport coefficients of these models, given in Table I.

C. Summary and discussion

We have started from a system of particles, and showed that the system can be equivalently described in terms of the gaps between the particles. The two systems are related by the transformation

ρ(x, t) = 1 ρ(k(x, t), t) , j(x, t) = - j(k(x, t), t) ρ(k(x, t), t) , ( 27
)
where

k(x, t) -k(0, t) = x 0 ρ(x , t)dx , ∂ t k(x, t) = -j(x, t) , ( 28
)
which maps the original equations (3,4) on ρ and j, onto the same equations of fluctuating hydrodynamics, but with the new transport coefficients

D(ρ) = 1 ρ2 D 1 ρ , σ(ρ) = ρ σ 1 ρ . ( 29
)
Although we have derived these relations by using physical arguments, from a mathematical point of view, they can be applied to any system which obeys the equations (3,4) of fluctuating hydrodynamics, not necessarily systems of particles. In particular, by applying this transformation to the dual system, we recover the original system. Explicitly, this means that the inverse transformation of the current and density is identical:

ρ(k, t) = 1 ρ(x k (t), t) , j(k, t) = - j(x k (t), t) ρ(x k (t), t) , ( 30
)
with

x k (t) -x 0 (t) = k 0 ρ(k , t)dk , ∂ t x k (t) = -j(k, t) . ( 31
)
Note that these relations, which are valid on the stochastic equations of fluctuating hydrodynamics, also imply a similar duality relation relation on the equations of Macroscopic Fluctuation Theory (MFT), which is a deterministic rewriting of the fluctuating hydrodynamics. We give more details on this relation in Appendix B.

The transformations presented in this article can be applied to both finite and infinite systems. In the case of a finite system, one should carefully study how the transformations act on the boundary conditions.

We will show in Section V that this duality relation can be used to obtain results on a single-file system by studying its dual. But first, we discuss other transformations of the density ρ and current j that will be useful in the following.

IV. TRANSLATION AND DILATATIONS

We introduce 3 other transformations, acting on the fields ρ and j, which, combined with the duality relation above, give all the possible mapping between two single-file systems. This is shown in Appendix A.

• Translation of density (T). The first transformation consists in shifting the density by a constant c. We define new density and current fields as

ρ(x, t) = ρ(x, t) + c , j(x, t) = j(x, t) . ( 32
)
These two fields still obey the equations of fluctuating hydrodynamics (3,4), upon changing the transport coefficients as

D(ρ) = D(ρ -c) , σ(ρ) = σ(ρ -c) . ( 33
)
• Dilatation of the fields (Di). This transformation multiplies the two fields by a constant c. We define ρ(x, t) = c ρ(x, t) , j(x, t) = c j(x, t) , (34) which obey the equations of fluctuating hydrodynamics (3,4), with the transport coefficients

D(ρ) = D ρ c , σ(ρ) = c 2 σ ρ c . ( 35
)
• Rescaling of time (Rt). The last transformation corresponds to changing the time scale by a constant τ . We introduce ρ(x, t) = ρ(x, τ t) , j(x, t) = τ j(x, τ t) . (36) Note that j needs to be rescaled by τ in order to satisfy the continuity equation (3). These new fields again obey the equations of fluctuating hydrodynamics (3,4), with the transport coefficients

D(ρ) = τ D(ρ) , σ(ρ) = τ σ(ρ) . ( 37
)
V. APPLICATIONS: INTEGRATED CURRENT AND TRACER DIFFUSION

We now focus on the case of an infinite system, and consider two different observables, which have been the focus of many studies in the context of single-file diffusion [6, 8-12, 19, 40, 48, 49]:

• The integrated current though the origin Q t , which counts the total number of particles that crossed the origin from left to right (minus the number from right to left) up to time t,

Q t = t 0 j(0, t )dt = ∞ 0 (ρ(x, t) -ρ(x, 0))dx . ( 38
)
• The position x 0 (t) of a tagged particle (a tracer), which was initially placed at the origin. This position can be deduced from the density ρ only, using that the number of particles located to the right of x 0 (t) is conserved [8]:

x0(t) 0 ρ(x, t)dx = ∞ 0 (ρ(x, t) -ρ(x, 0))dx . ( 39
)
We will consider the cumulant generating functions associated to these two observables, which in the long time limit take the form,

ln e λQt √ t ψ Q (λ) , ln e λx0(t) √ t ψ T (λ) . ( 40
)
The rescaled cumulant generating functions ψ Q and ψ T have been computed for different models of single file systems [6,9,10,[START_REF] Derrida | [END_REF]49]. We now show that, using the transformations we have identified above, the cumulants of these two observables for several other single-file systems can be deduced from these results. We first show how these transformations can be used to relate known results on the SEP and the KMP. Then, we apply them to obtain new results on the different single-file systems shown in Fig. 4.

A. Transformation of the observables

In order to use the different mappings identified in Section III and IV, we first study how they transform the two observables x 0 (t) and Q t . As in the previous sections, we consider a model described by the density ρ and current j, which is mapped to a new single-file system by one of the transformations defined above. The new system is described by a density ρ and a flux j. We also denote x0 (t) the position of a tracer in the new system, defined as in (39) with ρ replaced by ρ, and the flux through the origin in the new system is denoted Qt , defined as in (38).

• Duality relation (D). We can express the position of the tracer in terms of the integrated current of the dual system, using Eq. ( 31):

x 0 (t) = x 0 (0) - t 0 j(0, t )dt = -Qt , ( 41
)
since x 0 (0) = 0. The position of the tracer is exactly the opposite of the flux through the origin in the dual model. Applying this relation to the dual system, we also obtain . Each site hosts a continuous variable (a mass). At random times, the total mass of two neighbouring sites is randomly redistributed between them. (e) the double exclusion process. A particle can hop on a neighbouring site if the destination site and the next one is empty. It represents an exclusion process with particles that occupy the volume of two sites (shaded region). (f ) the gas of hard rods. Particles of length (rods) perform a Brownian motion, with the condition that the rods cannot overlap. The corresponding transport coefficients for these models are given in Table I.

x0 (t) = -Q t . (42)
• Translation (T). This transformation conserves the current j, therefore the integrated current is unchanged, Qt = Q t . For the position x 0 (t) of the tracer, Eq. ( 39) gives

x0(t) 0 (ρ(x, t) -c)dx = ∞ 0 (ρ(x, t) -ρ(x, 0))dx ≡ x0(t) 0 ρ(x, t)dx . ( 43
)
This yields the relation

x0(t) x0(t) ρ(x, t)dx = -c x 0 (t) . ( 44
)
• Dilatation (Di). This transformation multiplies both ρ and j by a constant c. Therefore, the integrated current is also multiplied by c, Qt = cQ t , while due to (39), the position of the tracer is unchanged: x0 (t) = x 0 (t).

• Rescaling of time (Rt). This transformation only changes the time scale by a factor τ (and the current accordingly), such that both the integrated current and the position of the tracer are only rescaled in time: Qt = Q tτ and x0 (t) = x 0 (tτ ).

These transformations are summarised in Table II. We can now apply these transformations to study the different models of single-file diffusion represented in Fig. 4.

B. Known results: the SEP and the KMP model

We summarise here the known results on the cumulant generating functions (40) for two models of single-file systems, the simple exclusion process and the Kipnis Marchioro Presutti model. In particular we show that the results on the KMP model can be deduced from those on the SEP.

The simple exclusion process

The simple exclusion process (SEP) is a model of particles on a lattice. Each site can host at most one particle. At random times (picked exponentially, with rate D 0 ), each particle can jump to a neighbouring site (with same probability), only if that site is empty (see Fig. 4(a)). For this model, the transport coefficients take the form [START_REF] Derrida | [END_REF]35]

D(ρ) = D 0 , σ(ρ) = 2D 0 ρ(1 -ρ) . ( 45
)
Transformation Original system Mapped system Transformation Original system Mapped system Duality relation (Du)

D(ρ) D(ρ) = 1 ρ 2 D 1 ρ Dilatation (Di) D(ρ) D(ρ) = D(ρ/c) σ(ρ) σ(ρ) = ρ σ 1 ρ σ(ρ) σ(ρ) = c 2 σ(ρ/c) ρ ρ = 1 ρ ρ ρ = c ρ x0(t) x0(t) = -Qt x0(t) x0(t) = x0(t) Qt Qt = -x0(t) Qt Qt = c Qt Translation (T) D(ρ) D(ρ) = D(ρ -c)
Rescaling time (Rt)

D(ρ) D(ρ) = τ D(ρ) σ(ρ) σ(ρ) = σ(ρ -c) σ(ρ) σ(ρ) = τ σ(ρ) ρ ρ = ρ + c ρ ρ = ρ Qt Qt = Qt x0(t) x0(t) = x0(tτ ) Qt Qt = Qtτ TABLE II.
Transformation of the transport coefficients D(ρ) and σ(ρ), of the mean density ρ, and of the observables (i) position of the tracer x0(t) and (ii) integrated current through the origin Qt, under the different transformations identified in Sections III and IV. The translation of density (T) has a complex effect on the dynamics of the tracer, see Eq. ( 44), so it is not written explicitly here.

The cumulant generating function of the integrated current in the SEP has been computed using Bethe ansatz [6], and reads [50]

ψ (SEP) Q (λ) = - D 0 π Li 3 2 (-ω) , ω = ρ(1 -ρ)(e λ + e -λ -2) , ( 46
)
where Li ν (z) = n≥1 z n /n ν is the polylogarithm function.

The cumulant generating function of the position of the tracer has been computed more recently [9], and is given as a double Legendre transform

ψ (SEP) T (s) = -D 0 min ξ [2sξ + φ(ξ)] , where φ(ξ) = µ(ξ, λ (ξ)) , (47)
where

µ(ξ, λ) = n≥1 (-ω) n n 3/2 A( √ n ξ) + ξ ln 1 + ρ(e λ -1) 1 + ρ(e -λ -1) , ∂µ ∂λ (ξ, λ (ξ)) = 0 , ( 48
)
with ω defined as in [START_REF] Spohn | Large scale dynamics of interacting particles[END_REF] and A(ξ) = e -ξ 2 / √ π + ξ erf(ξ). An alternative parametric representation is given in [12]. Expanding in powers of λ gives the first cumulants (the odd order cumulants vanish by symmetry),

x 0 (t) 2 = 2 1 -ρ ρ D 0 t π , ( 49
)
x 0 (t) 4 c ≡ x 0 (t) 4 -3 x 0 (t) 2 = 2 (1 -ρ) πρ 3 12(1 -ρ) 2 -π(3 -3(4 - √ 2)ρ + (8 -3 √ 2)ρ 2 ) D 0 t π . ( 50
)
Note that, in the low density limit ρ → 0, the SEP reduces to a model of Brownian particles with hard-core repulsion [9] (upon rescaling the distances by ρ and the time by ρ 2 ), which is described by the coefficients

D(ρ) = D 0 , σ(ρ) = 2D 0 ρ . ( 51
)
By taking the low density limit in the expressions [START_REF] Spohn | Large scale dynamics of interacting particles[END_REF][START_REF] Derrida | [END_REF], we obtain the cumulant generating functions for the Brownian particles. For instance, this yields [9,49]

ψ (B) T (λ) = -ρ D 0 min ξ [2λξ + φ(ξ)] , where φ(ξ) = Ξ(ξ) -Ξ(-ξ) 2 , Ξ(ξ) = ∞ ξ erfc(x)dx . ( 52
)

A direct consequence: general quadratic mobility

As a first illustration of the transformations identified in Sections III and IV, we show that we can relate any quadratic σ (with σ(0) = 0) to the SEP. This is similar to the extension discussed in Ref. [START_REF] Derrida | [END_REF]. We consider a single-file described by the transport coefficients:

D(ρ) = D 0 , σ(ρ) = 2D 0 ρ(a -bρ) . ( 53
)
Applying the dilatation (Di) both for this system and its dual, we get the following expressions for the transport coefficients of the transformed systems:

D 0 (Di) ----→ c=1/a D 0 (Du) ---→ D 0 ρ 2 (Di) ----→ c=1/b D 0 b 2 ρ 2 (Du) ---→ D 0 b 2 , 2D 0 ρ(a -bρ) (Di) ----→ c=1/a 2D 0 ρ(1 -bρ) (Du) ---→ 2D 0 1 - b ρ (Di) ----→ c=1/b 2D 0 b 2 1 - 1 ρ (Du) ---→ 2D 0 b 2 ρ(1 -ρ) . ( 54
)
Using finally a rescaling of time (Rt), we recover the transport coefficients of the SEP ( 45)

D 0 b 2 (Rt) ---→ τ =b 2 D 0 ≡ D(ρ) , 2D 0 b 2 ρ(1 -ρ) (Rt) ---→ τ =b 2 2D 0 ρ(1 -ρ) ≡ σ(ρ) , (55)
Under these transformations, the average density of the system becomes

ρ (Di) ----→ c=1/a ρ a (Du) ---→ a ρ (Di) ----→ c=1/b a bρ (Du) ---→ bρ a (Rt) ---→ τ =b 2 bρ a ≡ ρ SEP , ( 56
)
and the two observables become (on the first line, the integrated current of the transformed system is expressed in terms of the observables Q t and x 0 (t) of the original system; on the second line, same for the tracer's position):

Q t (Di) ----→ c=1/a Q t a (Du) -x 0 (t) (Di) ----→ c=1/b - x 0 (t) b (Du) Q t a (Rt) ---→ τ =b 2 Q tb 2 a ≡ Qt , x 0 (t) (Di) ----→ c=1/a x 0 (t) - Q t a (Di) ----→ c=1/b - Q t a x 0 (t) b (Rt) ---→ τ =b 2 x 0 (tb 2 ) b ≡ x0 (t) . ( 57
)
Therefore, we can relate the cumulant generating functions (40) by

ψ Q (λ) ≡ lim t→∞ 1 √ t ln e λQt = 1 b lim t→∞ 1 t/b 2 ln e λa Qt/b 2 ≡ 1 b ψ (SEP) Q (aλ) , ( 58
)
ψ T (λ) ≡ lim t→∞ 1 √ t ln e λx0(t) = 1 b lim t→∞ 1 t/b 2 ln e λbx0(t/b 2 ) ≡ 1 b ψ (SEP) T (bλ) , ( 59
)
where ψ (SEP) Q and ψ

(SEP) T are the cumulant generating functions of the SEP [START_REF] Spohn | Large scale dynamics of interacting particles[END_REF][START_REF] Derrida | [END_REF], evaluated at the density ρ SEP given by (56). We now use these two relations to recover known results on the KMP model.

The Kipnis Marchioro Presutti model

The Kipnis Marchioro Presutti (KMP) model is a stochastic model for mass or heat transfer [3]. It describes a lattice on which each site k hosts a continuous variable which represents a mass (or an energy). At random times picked from an exponential distribution with rate D 0 , a site can exchange mass with its right neighbour such that the total mass of the two neighbouring sites is randomly redistributed (with uniform distribution) between them (see Fig. 4(d)). This model has been applied to various situations, such as force fluctuations in packs of granular beads, or distribution of wealth in a society (see Ref. [51] and references therein).

The transport coefficients of this model are given by [42]

D(ρ) = D 0 , σ(ρ) = σ 0 ρ 2 . ( 60
)
This model can be deduced from the SEP by using the transformations described in (54,55) by setting ρ SEP = bρ/a, b = -σ 0 /(2D 0 ) and letting a → 0. This requires to assume that the results [START_REF] Spohn | Large scale dynamics of interacting particles[END_REF][START_REF] Derrida | [END_REF] obtained on the SEP are valid for ρ SEP < 0. Under this assumption, which is also used in [START_REF] Derrida | [END_REF], we can deduce the cumulant generating function of the integrated current in the KMP from (58),

ψ (KMP) Q (λ) = 2D 0 σ 0 D 0 π Li 3 2 σ 0 ρλ 2D 0 2 , ( 61
)
given in [6,[START_REF] Derrida | [END_REF].

Although the KMP is a mass transfer model and not a particle model, one can consider a tracer, at position x0 (t) defined as in (39). Its position can be interpreted as a fictitious "wall" that separates the systems into two regions in which the mass is conserved. Applying the same method as for the current, the corresponding cumulant generating function ψ (KMP) T can be deduced from the one of the SEP using (59), and reads,

ψ (KMP) T (λ) = - 2D 0 σ 0 lim a→0 ψ (SEP) T - σ 0 2D 0 λ ρ SEP=-σ 0 ρ/(2D 0 a) . ( 62
)
The limit is not easily taken on the expressions [START_REF] Derrida | [END_REF], but this can be done straightforwardly on the cumulants. For instance, this gives

x 0 (t) 2 = σ 0 t D 0 π , x 0 (t) 4 c = σ 3 0 12 + π(3 √ 2 -8) 4D 2 0 π t D 0 π . ( 63
)
An alternative parametric representation of ψ (KMP) T (λ) is given in [12].

C. The random average process

The random average process (RAP) [27,28,48] is a system of particles placed on an infinite line. At random times, picked from an exponential distribution with rate 1 2 , a particle can move, either to the left or to the right, to a random fraction of the distance to the next particle (see Fig. 4(c)). In the hydrodynamic limit, only the first two moments µ 1 and µ 2 of the distribution of this random fraction are relevant. The transport coefficients depend on these two parameters only [28,31]:

D(ρ) = µ 1 2ρ 2 , σ(ρ) = 1 ρ µ 1 µ 2 µ 1 -µ 2 . ( 64
)
Applying the duality relation (29), we obtain that the dual system is described by the coefficients

D(ρ) = D 0 = µ 1 2 , σ(ρ) = σ 0 ρ2 , σ 0 = µ 1 µ 2 µ 1 -µ 2 , ( 65
)
which correspond to the KMP model, with a specific choice of the factors D 0 and σ 0 in the coefficients (see Table I). Note that, under this duality transformation, the mean density ρ of the RAP becomes

ρ KMP = 1 ρ (66)
for the dual system.

Integrated current through the origin

The integrated current through the origin Q t is related to the position of the tracer in the dual system x0 (t) via (42). This implies that the cumulant generating function of

Q t reads ψ (RAP) Q (λ) ≡ lim t→∞ 1 √ t ln e λQt = lim t→∞ 1 √ t ln e -λx0(t) ≡ ψ (KMP) T (-λ) , (67)
where ψ

(KMP) T

is the cumulant generating function of a tracer in the KMP model (62), at density ρ KMP given by (66). In particular, the first cumulants read,

Q 2 t = σ 0 t D 0 π , Q 4 t c ≡ Q 4 t -3 Q 2 t = σ 3 0 12 + π(3 √ 2 -8) 4D 2 0 π t D 0 π . ( 68
)
Remarkably, these cumulants (and also the cumulant generating function) do not depend on the mean density ρ of the RAP. This is due to the fact that the dynamics of the RAP is invariant under a rescaling of the positions of all the particles (which does not change the integrated current Q t ).

Position of a tracer

The position x 0 (t) of the tracer in the RAP can be expressed in terms of the flux of the dual system (the KMP model) using (41). This relation was used in Refs. [29][30][31] to study the fluctuations of the position of a tracer, and correlations between different particles in the RAP. Here, we use it to express the generating function of the cumulants of x 0 (t) as

ψ (RAP) T (λ) ≡ lim t→∞ 1 √ t ln e λx0(t) = lim t→∞ 1 √ t ln e -λ Qt ≡ ψ (KMP) Q (-λ) , (69) hence, from (61)
,

ψ (RAP) T (λ) = (µ 1 -µ 2 ) µ 2 µ 1 2π Li 3 2 µ 2 λ ρ(µ 1 -µ 2 ) 2 . ( 70
)
Expanding this expression in powers of λ, we obtain the cumulants of the position of the tracer. For instance,

x 0 (t) 2 = µ 2 (µ 1 -µ 2 )ρ 2 2µ 1 t π , x 0 (t) 4 c ≡ x 0 (t) 4 -3 x 0 (t) 2 = 6µ 3 2 (µ 1 -µ 2 ) 3 ρ 4 µ 1 t π . ( 71
)
The variance coincides with the known result [48].

D. The zero range process

Zero range processes (ZRP) [22] correspond to lattice models, in which each site k can be occupied by any number n k of particles. At random times picked from an exponential distribution, each occupied site can lose a particle, which hops either to the left or to the right with equal probability (see Fig. 4(b)). The hopping rates from a site k can depend on the occupation number n k of the site, but not on the occupations of the other sites (unlike exclusion processes), hence the name "zero range". At the macroscopic scale, they correspond to a wide class of transport coefficients (depending on the hopping rates), such that they are related by D(ρ) = σ (ρ)/2 [35].

Here, we consider one specific ZRP, which corresponds to the simplest case in which the hoppings are constant, so they depend neither on the site nor on the number of particles on it (note that this is different from a system of independent particles, as in this case the hopping rate on site k would be proportional to n k ). At the macroscopic level, this system is described by the transport coefficients [23,35]

D(ρ) = D 0 (1 + ρ) 2 , σ(ρ) = 2D 0 ρ 1 + ρ . ( 72
)
This model is remarkable because it is an example of system whose transport coefficients are left unchanged by the duality transformation (Du). This property will be useful below. This model is well-known to be related to the SEP [21,22], and this relation was used explicitly in [23]. The mapping is the following: to each site k is associated a particle at position x k . The occupation of site k in the ZRP is given by the number of empty sites in the SEP between particles k + 1 and k (see Fig. 1). Inversely, the distance between particles k + 1 and k is given by n k + 1. Therefore, the SEP can be obtained from the ZRP by combining the translation of density (T) and the duality relation (Du):

D(ρ) = D 0 (1 + ρ) 2 (T) --→ c=1 D 0 ρ 2 (Du) ---→ D 0 = D(ρ) , σ(ρ) = 2D 0 ρ 1 + ρ (T) --→ c=1 2D 0 ρ (ρ -1) (Du) ---→ 2D 0 ρ(1 -ρ) = σ(ρ) . ( 73
)
Under theses transformations the observables become (again, the first line represents the current in the transformed system in terms of the observables of the original system; the second line is the expression of the position of the tracer):

Q t (T) --→ Q t (Du) -X t [x 0 , ρ] ≡ Qt , x 0 (t) (T) --→ X t [x 0 , ρ] -Q t ≡ x0 (t) , ( 74
)
where we denoted X t [x 0 , ρ] the result of the complex transformation of the position of the tracer ( 44) under (T). The mean density of the ZRP is transformed as

ρ (T) --→ c=1 ρ + 1 (Du) ---→ 1 ρ + 1 = ρ SEP . ( 75
)
Using (74), we can directly obtain the cumulant generating function for the current

ψ (ZRP) Q (λ) ≡ lim t→∞ 1 √ t ln e λQt = lim t→∞ 1 √ t ln e -λx0(t) ≡ ψ (SEP) T (-λ) , ( 76
)
where ψ

(SEP) T is given by ( 47), evaluated at the density ρ SEP (75). For instance,

Q 2 t = 2ρ D 0 t π , Q 4 t c = 2ρ 12ρ 2 + π(1 + 3(2 - √ 2)ρ -3ρ 2 ) π D 0 t π . ( 77
)
In order to obtain the cumulant generating function for the tracer in the ZRP, one should in principle use the complex transformation X t [x 0 , ρ], which requires the knowledge of the statistical properties of the density, as well as its correlations with the position of the tracer. These correlations are however more complex objects than the cumulant generating functions [11], and are not known for the ZRP. However, here, we can circumvent this problem since the ZRP we consider is self dual:

D(ρ) = D 0 (1 + ρ) 2 (Du) ---→ D 0 (1 + ρ) 2 = D(ρ) , σ(ρ) = 2D 0 ρ 1 + ρ (Du) ---→ 2D 0 ρ 1 + ρ = σ(ρ) . ( 78
)
The corresponding action on the observables is an exchange of integrated current and flux :

Q t (Du) -x 0 (t) ≡ Qt , x 0 (t) -Q t ≡ x0 (t) , ( 79
)
and the initial mean density is mapped in the following way :

ρ (Du) ---→ 1 ρ = ρ . ( 80
)
This mapping thus implies a direct relation of the cumulant generating functions for the tracer and the current in the ZRP:

ψ (ZRP) T (λ, ρ) = ψ (ZRP) Q (-λ, ρ -1 ) , ( 81
)
from which we can deduce for instance, using (77),

x 0 (t) 2 = 2 ρ D 0 t π , x 0 (t) 4 c = 2 12 + π(ρ 2 + 3(2 - √ 2)ρ -3) ρ 3 π D 0 t π . ( 82
)
E. The gas of hard rods

Consider a system of rods of length , which perform a Brownian motion, with the condition that two rods cannot overlap (see Fig. 4(f)). The main interest of this model compared to the point-like Brownian particles is that the system of hard rods has a maximal density 1/ . It is the one dimensional version of the hard sphere gas. Its transport coefficients are given by [17]

D(ρ) = D 0 (1 -ρ) 2 , σ(ρ) = 2D 0 ρ . ( 83
)
We can map the gas of hard rods onto the model of point-like Brownian particles by decreasing the size of the rods by their length . This is naturally done in the dual model (which describes the gaps between the rods) by removing a constant length to all the gaps. We can therefore construct this model by going to the dual of the hard rods model, translating the density by a constant -and going back to the particles using the duality relation again:

D(ρ) = D 0 (1 -ρ) 2 (Du) ---→ D 0 (ρ -) 2 (T) ---→ c=- D 0 ρ 2 (Du) ---→ D 0 = D(ρ) , σ(ρ) = 2D 0 ρ (Du) ---→ 2D 0 (T) ---→ c=- 2D 0 (Du) ---→ 2D 0 ρ = σ(ρ) .
(84) D(ρ) and σ(ρ) are the transport coefficients for the gas of hard rods [17], and this series of transformation indeed yields the coefficients D and σ of the gas of Brownian particles (see Table I). Under these transformations (see Table II), the position of the tracer is unchanged (in each column we give the observables of the current system in terms of the observables of the original system; the first line corresponds to the integrated current, the second line is the position of the tracer):

Q t (Du) -x 0 (t) (T) ---→ c=- -x 0 (t) (Du) -? ≡ Qt , x 0 (t) -Q t (T) ---→ c=- ? x 0 (t) ≡ x0 (t) , (85)
where we again do not write explicitly the complex transformation (44) of the position of the tracer under the transformation (T). The mean density of the system becomes

ρ (Du) ---→ 1 ρ (T) ---→ c=- 1 ρ - (Du) ---→ ρ 1 -ρ ≡ ρ B . ( 86
)
Since the transformation of the flux is rather complex, and requires the tracer/density correlations in the dual system (which are not known), we will only consider the cumulant generating function of the tracer, which reads

ψ (HR) T (λ) ≡ lim t→∞ 1 √ t ln e λx0(t) = lim t→∞ 1 √ t ln e λx0(t) ≡ ψ (B) T (λ) , ( 87
)
in terms of the cumulant generating function ψ (B) T of the tracer in the Brownian model ( 52), evaluated at the density ρ B (86). In particular, expanding in powers of λ, we get the first cumulants:

x 0 (t) 2 = 2(1 -ρ) ρ D 0 t π , x 0 (t) 4 c = 6(4 -π) π 1 -ρ ρ 3 D 0 t π . ( 88
)

F. The double exclusion process

We consider an exclusion process, with additional repulsion between neighbouring sites: a particle located at site i can, at exponential times, jump to site i + 1 only if sites i + 1 and i + 2 are empty, or with the same probability to site i -1 only if sites i -1 and i -2 are empty. This model is equivalent to having particles which occupy a volume of two lattice sites, with the constraint that they cannot overlap (see Fig. 4(e)). Exclusion processes with extended particles have for instance been considered for applications to biological systems [52]. The transport coefficients are [44,45]

D(ρ) = D 0 (1 -ρ) 2 , σ(ρ) = 2D 0 ρ(1 -2ρ) 1 -ρ . ( 89
)
This model can be directly mapped onto the SEP by reducing the size of the particles to one lattice site, similarly as we did for the gas of hard rods:

D(ρ) = D 0 (1 -ρ) 2 (Du) ---→ D 0 (ρ -1) 2 (T) ---→ c=-1 D 0 ρ 2 (Du) ---→ D 0 = D(ρ) , σ(ρ) = 2D 0 ρ(1 -2ρ) 1 -ρ (Du) ---→ 2D 0 1 - 1 ρ -1 (T) ---→ c=-1 2D 0 1 - 1 ρ (Du) ---→ 2D 0 ρ(1 -ρ) = σ(ρ) , (90)
which are indeed the transport coefficients of the SEP (see Table I). Note that we can straightforwardly extend this procedure to a system in which particles have a volume on n sites, by applying a translation of density (T) of n -1 for the dual. Under this series of transformations, the position of a tracer becomes (we again do not write the action of (T) on the position of the tracer of the dual system because of its complexity)

Q t (Du) -x 0 (t) (T) ---→ c=-1 -x 0 (t) (Du) -? ≡ Qt , x 0 (t) -Q t (T) ---→ c=-1 ? x 0 (t) ≡ x0 (t) , (91)
and the density

ρ (Du) ---→ 1 ρ (T) ---→ c=-1 1 ρ -1 (Du) ---→ ρ 1 -ρ ≡ ρ SEP . ( 92
)
The cumulant generating function of the tracer in the double exclusion process is thus

ψ (DEP) T (λ) ≡ lim t→∞ 1 √ t ln e λx0(t) = lim t→∞ 1 √ t ln e λx0(t) ≡ ψ (SEP) T (λ) , (93)
where ψ

(SEP) T is given by ( 47), evaluated at density ρ SEP given by (92). In particular, expanding in powers of λ, we get the first cumulants:

x 0 (t) 2 = 2(1 -2ρ) ρ D 0 t π , ( 94
) x 0 (t) 4 c = 2(1 -2ρ) πρ 3 12(1 -2ρ) 2 -π((23 -6 √ 2)ρ 2 -3(6 - √ 2)ρ + 3) D 0 t π . ( 95
)
VI. CONCLUSION

We have shown that, in the framework of fluctuating hydrodynamics, the duality relation that is well known between the SEP and the ZRP, and between the RAP and the KMP model, is not restricted to these models. Any single-file system can be mapped onto a dual one, which still obeys the equations of fluctuating hydrodynamics, with new transport coefficients. We have also introduced three other transformations, which, combined with the duality relation, yield all the possible relations between two single-file models.

We have characterized the action of these transformations on the two main observables studied in single-file diffusion: the integrated current through the origin and the position of a tagged particle. As an application of these mappings, we have shown that the cumulant generating functions of these observables can be obtained for various models of single-file systems from the one computed for the SEP.

Dualities have been essential in the resolution of microscopic models such as the SEP, the KMP model or the RAP [3,23,29,53,54]. The mappings presented here, valid for any single-file system, could play an important role in the application of the MFT to new models of single-file diffusion. deterministic rewriting of the fluctuating hydrodynamics [34,[55][56][57]. We briefly recall the main steps of this process, as presented in [START_REF] Derrida | [END_REF]. The idea is to start from the distribution of the white noise η(x, t), Rescaling the time t → t/T and space x → x/ √ T , one can see that the argument of the exponential is proportional to √ T . Therefore, for large T , the integral can be estimated by the saddle point method. Denoting (q, p) the values of (ρ, H) which dominate the integral, one finds that they obey the MFT equations: ∂ t q = ∂ x (D(q)∂ x q) -∂ x (σ(q)∂ x p) , (B6)

P [η] ∝ exp - T 0 dt ∞ -∞ dx η(x,
∂ t p = -D(q)∂ 2 x p - σ (q) 2 (∂ x p) 2 . (B7)
Note that the Gaussian integral over j in (B4) can also be computed by a saddle point method, and this gives the optimal current j = -D(q)∂ x q + σ(q)∂ x p . (B8)

With this expression, the first MFT equation (B6) can be interpreted as the conservation relation (3) for the optimal density q and optimal current j . We now study the consequences of the duality relation discussed in Section III on the MFT equations (B6,B7). Relations (27,28) hold for any realisation of the fluctuating fields ρ and j. They thus remain valid for their typical realisation q and j . This allows to define a dual optimal field q(k(x, t), t) = 1 q(x, t) , k(x, t) = x 0 q(x , t)dx + k(0, t) , ∂ t k(x, t) = -j (x, t) , (B9) and a dual optimal current, j (k, t) = -j (x, t) q(x, t) = -q(k, t)j (x, t) = -D(q)∂ k q -σ(q)∂ x p , (B10)

with D and σ given by (29). We define p(k, t) such that ∂ k p(k, t) = -∂ x p(x k (t), t) , (B11) so that the dual current reads j = -D(q)∂ k q + σ(q)∂ k p . (B12)

Using relations [START_REF] Mallick | International Summer School on Fundamental Problems in Statistical Physics[END_REF][START_REF] Derrida | [END_REF]20), applied to the optimal fields, we then deduce

∂ t q = ∂ k ( D(q)∂ k q) -∂ k (σ(q)∂ k p) , ( B13
)
which is the first MFT equation (B6) for the dual system. In order to get the second equation, we compute the time evolution of ∂ k p using (B11), which yields

∂ t ∂ k p = ∂ k -D(q)∂ 2 k p - σ (q) 2 (∂ k p) 2 . ( B14
)
Integrating this relation from -∞ to k, we get

∂ t p = -D(q)∂ 2 k p - σ (q) 2 (∂ k p) 2 , ( B15
)
which is the second equation (B7) for the dual system.

FIG. 4 .

 4 FIG.4. The different models of single-file systems considered in this paper. (a) the simple exclusion process (SEP). Each particle can hop to a neighbouring empty site. (b) the zero range process (ZRP). The particles can hop on a neighbouring site. (c) the random average process (RAP). Particles on a continuous line can hop to a random fraction of the distance to the next particle. (d) the Kipnis Marchioro Presutti model (KMP). Each site hosts a continuous variable (a mass). At random times, the total mass of two neighbouring sites is randomly redistributed between them. (e) the double exclusion process. A particle can hop on a neighbouring site if the destination site and the next one is empty. It represents an exclusion process with particles that occupy the volume of two sites (shaded region). (f ) the gas of hard rods. Particles of length (rods) perform a Brownian motion, with the condition that the rods cannot overlap. The corresponding transport coefficients for these models are given in TableI.

  t)2 2 , (B1) and insert the relation (4) in order to get the probability of observing ρ and j at all positions and times between 0 and T : -function imposes the continuity relation (3) at all points (x, t). This constraint can be handled by using the Martin-Siggia-Rose formalism, which consists in replacing the δ by the integration over a conjugate field H(x, t),δ (∂ t ρ + ∂ x j) = DH exp -Inserting this relation in (B2) and integrating over j, we obtain the probability of observing a realisation ρ(x, t), P [{ρ(x, t)}] ∝ DHDj exp -

	P [{ρ(x, t), j(x, t)}] ∝ exp -	0	T	dt	∞ -∞	dx	[j(x, t) + D(ρ(x, t))∂ x ρ(x, t)] 2 2σ(ρ(x, t)) 2	δ (∂ t ρ + ∂ x j) ,	(B2)
	where the δT	∞
										dt	dx H(∂ t ρ + ∂ x j) .	(B3)
										0	-∞
	0	T	dt	∞ -∞	dx H∂ t ρ + D(ρ)∂ x ρ∂ x H -	σ(ρ) 2	(∂ x H) 2
						0	T	dt	∞ -∞	dx H∂ t ρ + D(ρ)∂ x ρ∂ x H -	σ(ρ) 2	(∂ x H) 2	.	(B5)

+ [j + D(ρ)∂ x ρ -σ(ρ)∂ x H] 2 2σ(ρ) 2 , (B4)

after integration by parts. The integral over j is Gaussian and can thus be performed straightforwardly, P [{ρ(x, t)}] ∝ DH exp -

We have thus shown that the duality relation discussed in the main text at the level of the fluctuating hydrodynamics implies a duality of the MFT equations, with the transformation of q and p given by (B9,B11).

Appendix A: General mapping between two single-file systems

Here we investigate the possibility of existence of other mappings than the physical ones presented in the main text. We look for transformations of the following general form, linear with respect to the flux (the case of a non linear transformation of the flux generates difficulties concerning the random white noise):

The definitions (A3), (A4) are necessary so that the new density and flux verify the conservation equation

Then, from (4), we get the following stochastic Fourier law

where

This mapping is not coherent for general F and G because of the compatibility condition

The factors in front of j∂ x ρ and ∂ t ρ must vanish, so we get after simplifications,

The only solutions to these equations are parametrized by a, b, c, d ∈ R:

The In the main text, we have used the formalism of fluctuating hydrodynamics, in which the density field ρ and the current j are stochastic. In practice, it is convenient to use the Macroscopic Fluctuation Theory (MFT), which is a