* Hasan and Farri [2019] Hasan, S.A., Farri, O., 2019\.  Clinical Natural Language Processing with Deep Learning. Springer International Publishing, Cham.  pp. 147–171.  URL: https://doi.org/10.1007/978-3-030-05249-2_5, doi:10.1007/978-3-030-05249-2_5.
  * Hastie et al. [2009] Hastie, T., Tibshirani, R., Friedman, J., 2009.  Overview of Supervised Learning. Springer New York, New York, NY.  pp. 9–41.  URL: https://doi.org/10.1007/978-0-387-84858-7_2, doi:10.1007/978-0-387-84858-7_2.
  * Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J., 1997\.  Long short-term memory.  Neural Computation 9, 1735–1780.  URL: https://doi.org/10.1162/neco.1997.9.8.1735, doi:10.1162/neco.1997.9.8.1735, arXiv:https://doi.org/10.1162/neco.1997.9.8.1735.
  * Hospedales et al. [2013] Hospedales, T.M., Gong, S., Xiang, T., 2013.  Finding rare classes: Active learning with generative and discriminative models.  IEEE Transactions on Knowledge and Data Engineering 25, 374--386.  doi:10.1109/TKDE.2011.231.
  * Hou et al. [2017] Hou, S., Feng, Y., Wang, Z., 2017.  Vegfru: A domain-specific dataset for fine-grained visual categorization, in: 2017 IEEE International Conference on Computer Vision (ICCV), pp. 541--549.  doi:10.1109/ICCV.2017.66.
  * Huang et al. [2019] Huang, J., Osorio, C., Sy, L., 2019.  An empirical evaluation of deep learning for ICD-9 code assignment using MIMIC-III clinical notes.  Computer Methods and Programs in Biomedicine 177\.  doi:10.1016/j.cmpb.2019.05.024.
  * Ji et al. [2020] Ji, S., Cambria, E., Marttinen, P., 2020.  Dilated convolutional attention network for medical code assignment from clinical text, in: Proceedings of the 3rd Clinical Natural Language Processing Workshop, Association for Computational Linguistics, Online. pp. 73--78.  URL: https://www.aclweb.org/anthology/2020.clinicalnlp-1.8, doi:10.18653/v1/2020.clinicalnlp-1.8.
  * Kaur [2019] Kaur, R., 2019.  Distributed knowledge based clinical auto-coding system, in: Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics: Student Research Workshop, Association for Computational Linguistics, Florence, Italy. pp. 1--9.  URL: https://www.aclweb.org/anthology/P19-2001, doi:10.18653/v1/P19-2001.
  * Kaur and Ginige [2018] Kaur, R., Ginige, J.A., 2018\.  Comparative Analysis of Algorithmic Approaches for Auto-Coding with ICD-10-AM and ACHI.  Studies in Health Technology and Informatics 252, 73--79.
  * Kaur and Ginige [2019] Kaur, R., Ginige, J.A., 2019\.  Analysing effectiveness of multi-label classification in clinical coding, in: Proceedings of the Australasian Computer Science Week Multiconference, Association for Computing Machinery, New York, NY, USA.  URL: https://doi.org/10.1145/3290688.3290728, doi:10.1145/3290688.3290728.
  * Kaur and University. [2018] Kaur, R., University., W.S., 2018\.  A comparative analysis of selected set of natural language processing (NLP) and machine learning (ML) algorithms for clinical coding using clinical classification standards.  URL: http://hdl.handle.net/1959.7/uws:49614. a thesis presented to Western Sydney University, School of Computing, Engineering and Mathematics, in fulfilment of the requirements for the degree of Master of Research.
  * Kavuluru et al. [2013] Kavuluru, R., Han, S., Harris, D., 2013.  Unsupervised extraction of diagnosis codes from emrs using knowledge-based and extractive text summarization techniques, in: Zaïane, O.R., Zilles, S. (Eds.), Advances in Artificial Intelligence, Springer Berlin Heidelberg, Berlin, Heidelberg. pp. 77--88.
  * Kavuluru et al. [2015] Kavuluru, R., Rios, A., Lu, Y., 2015.  An empirical evaluation of supervised learning approaches in assigning diagnosis codes to electronic medical records.  Artificial Intelligence in Medicine 65, 155 -- 166.  URL: http://www.sciencedirect.com/science/article/pii/S0933365715000482, doi:https://doi.org/10.1016/j.artmed.2015.04.007. intelligent healthcare informatics in big data era.
  * Khattak et al. [2019] Khattak, F.K., Jeblee, S., Pou-Prom, C., Abdalla, M., Meaney, C., Rudzicz, F., 2019\.  A survey of word embeddings for clinical text.  Journal of Biomedical Informatics: X 4, 100057.  URL: https://www.sciencedirect.com/science/article/pii/S2590177X19300563, doi:https://doi.org/10.1016/j.yjbinx.2019.100057.
  * Kim [2014] Kim, Y., 2014.  Convolutional neural networks for sentence classification, in: Proceedings of the 2014 Conference on Empirical Methods in Natural Language Processing (EMNLP), Association for Computational Linguistics, Doha, Qatar. pp. 1746--1751.  URL: https://www.aclweb.org/anthology/D14-1181, doi:10.3115/v1/D14-1181.
  * Ko and Seo [2000] Ko, Y., Seo, J., 2000.  Automatic text categorization by unsupervised learning, in: Proceedings of the 18th Conference on Computational Linguistics - Volume 1, Association for Computational Linguistics, Stroudsburg, PA, USA. pp. 453--459.  URL: https://doi.org/10.3115/990820.990886, doi:10.3115/990820.990886.
  * Kohavi [1995] Kohavi, R., 1995.  A study of cross-validation and bootstrap for accuracy estimation and model selection, in: Proceedings of the 14th International Joint Conference on Artificial Intelligence - Volume 2, Morgan Kaufmann Publishers Inc., San Francisco, CA, USA. pp. 1137--1143.  URL: http://dl.acm.org/citation.cfm?id=1643031.1643047.
  * Larkey and Croft [1996] Larkey, L.S., Croft, W.B., 1996\.  Combining classifiers in text categorization, in: Proceedings of the 19th Annual International ACM SIGIR Conference on Research and Development in Information Retrieval, Association for Computing Machinery, New York, NY, USA. p. 289–297.  URL: https://doi.org/10.1145/243199.243276, doi:10.1145/243199.243276.
  * LeCun et al. [2015] LeCun, Y., Bengio, Y., Hinton, G., 2015.  Deep learning.  Nature 521, 436\.  URL: https://doi.org/10.1038/nature14539.
  * Li and Yu [2020] Li, F., Yu, H., 2020.  ICD coding from clinical text using multi-filter residual convolutional neural network.  Proceedings of the AAAI Conference on Artificial Intelligence 34, 8180--8187.  URL: https://ojs.aaai.org/index.php/AAAI/article/view/6331, doi:10.1609/aaai.v34i05.6331.
  * Li et al. [2019] Li, M., Fei, Z., Zeng, M., Wu, F., Li, Y., Pan, Y., Wang, J., 2019.  Automated ICD-9 coding via a deep learning approach.  IEEE/ACM Transactions on Computational Biology and Bioinformatics PP, 1--1.  doi:10.1109/TCBB.2018.2817488.
  * Lin et al. [2017] Lin, C., Hsu, C.J., Lou, Y.S., Yeh, S.J., Lee, C.C., Su, S.L., Chen, H.C., 2017\.  Artificial intelligence learning semantics via external resources for classifying diagnosis codes in discharge notes.  J Med Internet Res 19, e380.  URL: http://www.jmir.org/2017/11/e380/, doi:10.2196/jmir.8344.
  * Lin et al. [2020] Lin, E., Chen, Q., Qi, X., 2020.  Deep reinforcement learning for imbalanced classification.  Applied Intelligence , 1--15.
  * Liu et al. [2019] Liu, Z., Miao, Z., Zhan, X., Wang, J., Gong, B., Yu, S., 2019.  Large-scale long-tailed recognition in an open world, pp. 2532--2541.  doi:10.1109/CVPR.2019.00264.
  * Marafino et al. [2014] Marafino, B.J., Davies, J.M., Bardach, N.S., Dean, M.L., Dudley, R.A., 2014.  N-gram support vector machines for scalable procedure and diagnosis classification, with applications to clinical free text data from the intensive care unit.  Journal of the American Medical Informatics Association 21, 871--875.  URL: https://doi.org/10.1136/amiajnl-2014-002694, doi:10.1136/amiajnl-2014-002694.
  * Mascio et al. [2020] Mascio, A., Kraljevic, Z., Bean, D., Dobson, R., Stewart, R., Bendayan, R., Roberts, A., 2020.  Comparative analysis of text classification approaches in electronic health records, in: Proceedings of the 19th SIGBioMed Workshop on Biomedical Language Processing, Association for Computational Linguistics, Online. pp. 86--94.  URL: https://www.aclweb.org/anthology/2020.bionlp-1.9, doi:10.18653/v1/2020.bionlp-1.9.
  * Moher et al. [2009] Moher, D., Liberati, A., Tetzlaff, J., Altman, D.G., Group, T.P., 2009.  Preferred reporting items for systematic reviews and meta-analyses: The prisma statement.  PLOS Medicine 6, 1--6.  URL: https://doi.org/10.1371/journal.pmed.1000097, doi:10.1371/journal.pmed.1000097.
  * Moons et al. [2020] Moons, E., Khanna, A., Akkasi, A., Moens, M.F., 2020\.  A comparison of deep learning methods for ICD coding of clinical records.  Applied Sciences 10.  URL: https://www.mdpi.com/2076-3417/10/15/5262, doi:10.3390/app10155262.
  * Moriyama et al. [2011] Moriyama, I.M., Loy, R.M., Robb-Smith, A.H.T., Rosenberg, H.M., Hoyert, D.L., 2011.  History of the statistical classification of diseases and causes of death URL: https://www.cdc.gov/nchs/data/misc/classification_diseases2011.pdf.
  * Mujtaba et al. [2019] Mujtaba, G., Shuib, L., Idris, N., Hoo, W.L., Raj, R.G., Khowaja, K., Shaikh, K., Nweke, H.F., 2019\.  Clinical text classification research trends: Systematic literature review and open issues.  Expert Systems with Applications 116, 494 -- 520.  URL: http://www.sciencedirect.com/science/article/pii/S0957417418306110.
  * Mullenbach et al. [2018] Mullenbach, J., Wiegreffe, S., Duke, J., Sun, J., Eisenstein, J., 2018.  Explainable prediction of medical codes from clinical text, in: Proceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long Papers), Association for Computational Linguistics, New Orleans, Louisiana. pp. 1101--1111.  URL: https://www.aclweb.org/anthology/N18-1100, doi:10.18653/v1/N18-1100.
  * Nguyen and Patrick [2016] Nguyen, H., Patrick, J., 2016\.  Text mining in clinical domain: Dealing with noise, in: Proceedings of the 22Nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, ACM, New York, NY, USA. pp. 549--558.  URL: http://doi.acm.org/10.1145/2939672.2939720, doi:10.1145/2939672.2939720.
  * Organisation [2016] Organisation, W.H., 2016.  ICD-11 Revision Conference Report URL: https://www.who.int/classifications/network/meeting2016/ICD-11RevisionConferenceReportTokyo.pdf?ua=1.
  * Pan and Yang [2010] Pan, S.J., Yang, Q., 2010.  A survey on transfer learning.  IEEE Transactions on Knowledge and Data Engineering 22, 1345--1359.  doi:10.1109/TKDE.2009.191.
  * Perotte et al. [2013] Perotte, A., Pivovarov, R., Natarajan, K., Weiskopf, N., Wood, F., Elhadad, N., 2013\.  Diagnosis code assignment: models and evaluation metrics.  Journal of the American Medical Informatics Association 21, 231--237.  URL: https://doi.org/10.1136/amiajnl-2013-002159, doi:10.1136/amiajnl-2013-002159.
  * Pestian et al. [2007] Pestian, J.P., Brew, C., Matykiewicz, P., Hovermale, D., Johnson, N., Cohen, K.B., Duch, W., 2007.  A shared task involving multi-label classification of clinical free text, in: Biological, translational, and clinical language processing, Association for Computational Linguistics, Prague, Czech Republic. pp. 97--104.  URL: https://www.aclweb.org/anthology/W07-1013.
  * Prakash et al. [2017] Prakash, A., Zhao, S., Hasan, S.A., Datla, V., Lee, K., Qadir, A., Liu, J., Farri, O., 2017.  Condensed memory networks for clinical diagnostic inferencing, in: Proceedings of the Thirty-First AAAI Conference on Artificial Intelligence, AAAI Press. p. 3274–3280.
  * Provost et al. [1998] Provost, F.J., Fawcett, T., Kohavi, R., 1998.  The case against accuracy estimation for comparing induction algorithms, in: Proceedings of the Fifteenth International Conference on Machine Learning, Morgan Kaufmann Publishers Inc., San Francisco, CA, USA. p. 445–453.
  * Reed et al. [2016] Reed, G.M., Drescher, J., Krueger, R.B., Atalla, E., Cochran, S.D., First, M.B., Cohen-Kettenis, P.T., Arango-de Montis, I., Parish, S.J., Cottler, S., Briken, P., Saxena, S., 2016\.  Disorders related to sexuality and gender identity in the ICD-11: revising the ICD-10 classification based on current scientific evidence, best clinical practices, and human rights considerations.  World Psychiatry 15, 205--221.  URL: https://onlinelibrary.wiley.com/doi/abs/10.1002/wps.20354, doi:10.1002/wps.20354, arXiv:https://onlinelibrary.wiley.com/doi/pdf/10.1002/wps.20354.
  * Rios and Kavuluru [2018a] Rios, A., Kavuluru, R., 2018a.  EMR coding with semi-parametric multi-head matching networks, in: Proceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long Papers), Association for Computational Linguistics, New Orleans, Louisiana. pp. 2081--2091.  URL: https://www.aclweb.org/anthology/N18-1189, doi:10.18653/v1/N18-1189.
  * Rios and Kavuluru [2018b] Rios, A., Kavuluru, R., 2018b.  Few-shot and zero-shot multi-label learning for structured label spaces, in: Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing, Association for Computational Linguistics, Brussels, Belgium. pp. 3132--3142.  URL: https://www.aclweb.org/anthology/D18-1352, doi:10.18653/v1/D18-1352.
  * Rios and Kavuluru [2019] Rios, A., Kavuluru, R., 2019\.  Neural transfer learning for assigning diagnosis codes to emrs.  Artificial Intelligence in Medicine 96, 116 -- 122.  URL: http://www.sciencedirect.com/science/article/pii/S0933365718304378, doi:https://doi.org/10.1016/j.artmed.2019.04.002.
  * Sadoughi et al. [2018] Sadoughi, N., Finley, G., Fone, J., Murali, V., Korenevski, M., Baryshnikov, S., Axtmann, N., Miller, M., Suendermann-Oeft, D., 2018.  Medical code prediction with multi-view convolution and description-regularized label-dependent attention.  arXiv preprint arXiv:1811.01468 .
  * Samonte et al. [2018] Samonte, M.J.C., Gerardo, B.D., Fajardo, A.C., Medina, R.P., 2018\.  ICD-9 tagging of clinical notes using topical word embedding, in: Proceedings of the 2018 International Conference on Internet and E-Business, Association for Computing Machinery, New York, NY, USA. p. 118–123.  URL: https://doi.org/10.1145/3230348.3230357, doi:10.1145/3230348.3230357.
  * Samonte et al. [2017] Samonte, M.J.C., Gerardo, B.D., Medina, R.P., 2017.  Towards enhanced hierarchical attention networks in ICD-9 tagging of clinical notes, in: Proceedings of the 3rd International Conference on Communication and Information Processing, Association for Computing Machinery, New York, NY, USA. p. 146–150.  URL: https://doi.org/10.1145/3162957.3163030, doi:10.1145/3162957.3163030.
  * Santos et al. [2008] Santos, S., Murphy, G., Baxter, K., Robinson, K.M., 2008\.  Organisational factors affecting the quality of hospital clinical coding.  Health Information Management Journal 37, 25--37.
  * Scheurwegs et al. [2017] Scheurwegs, E., Cule, B., Luyckx, K., Luyten, L., Daelemans, W., 2017.  Selecting relevant features from the electronic health record for clinical code prediction.  Journal of Biomedical Informatics 74, 92--103.  URL: https://www.sciencedirect.com/science/article/pii/S1532046417302010, doi:https://doi.org/10.1016/j.jbi.2017.09.004.
  * Schäfer and Friedrich [2019] Schäfer, H., Friedrich, C.M., 2019\.  UMLS mapping and word embeddings for ICD code assignment using the MIMIC-III intensive care database, in: 2019 41st Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC), pp. 6089--6092.
  * Searle et al. [2020] Searle, T., Ibrahim, Z., Dobson, R., 2020.  Experimental evaluation and development of a silver-standard for the MIMIC-III clinical coding dataset, in: Proceedings of the 19th SIGBioMed Workshop on Biomedical Language Processing, Association for Computational Linguistics, Online. pp. 76--85.  URL: https://www.aclweb.org/anthology/2020.bionlp-1.8, doi:10.18653/v1/2020.bionlp-1.8.
  * Shi et al. [2017] Shi, H., Xie, P., Hu, Z., Zhang, M., Xing, E.P., 2017\.  Towards automated ICD coding using deep learning.  CoRR abs/1711.04075.  URL: http://arxiv.org/abs/1711.04075, arXiv:1711.04075.
  * Socher et al. [2013] Socher, R., Perelygin, A., Wu, J., Chuang, J., Manning, C.D., Ng, A., Potts, C., 2013.  Recursive deep models for semantic compositionality over a sentiment treebank, in: Proceedings of the 2013 Conference on Empirical Methods in Natural Language Processing, Association for Computational Linguistics. pp. 1631--1642.  URL: http://aclweb.org/anthology/D13-1170.
  * Sokolova and Lapalme [2009] Sokolova, M., Lapalme, G., 2009\.  A systematic analysis of performance measures for classification tasks.  Information Processing & Management 45, 427 -- 437.  URL: http://www.sciencedirect.com/science/article/pii/S0306457309000259, doi:https://doi.org/10.1016/j.ipm.2009.03.002.
  * Sonabend W et al. [2020] Sonabend W, A., Cai, W., Ahuja, Y., Ananthakrishnan, A., Xia, Z., Yu, S., Hong, C., 2020.  Automated ICD coding via unsupervised knowledge integration (UNITE).  International Journal of Medical Informatics 139, 104135.  URL: http://www.sciencedirect.com/science/article/pii/S1386505619313024, doi:https://doi.org/10.1016/j.ijmedinf.2020.104135.
  * Song et al. [2020] Song, C., Zhang, S., Sadoughi, N., Xie, P., Xing, E., 2020.  Generalized zero-shot text classification for ICD coding, in: Bessiere, C. (Ed.), Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, IJCAI-20, International Joint Conferences on Artificial Intelligence Organization. pp. 4018--4024.  URL: https://doi.org/10.24963/ijcai.2020/556, doi:10.24963/ijcai.2020/556. main track.
  * Stanfill et al. [2010] Stanfill, M.H., Williams, M., Fenton, S.H., Jenders, R.A., Hersh, W.R., 2010.  A systematic literature review of automated clinical coding and classification systems.  Journal of the American Medical Informatics Association 17, 646--651.  URL: https://doi.org/10.1136/jamia.2009.001024, doi:10.1136/jamia.2009.001024.
  * Subotin and Davis [2014] Subotin, M., Davis, A., 2014\.  A system for predicting ICD-10-PCS codes from electronic health records, in: Proceedings of BioNLP 2014, Association for Computational Linguistics, Baltimore, Maryland. pp. 59--67.  URL: https://www.aclweb.org/anthology/W14-3409, doi:10.3115/v1/W14-3409.
  * Sun et al. [2018] Sun, W., Cai, Z., Li, Y., Liu, F., Fang, S., Wang, G., 2018.  Data processing and text mining technologies on electronic medical records: a review.  Journal of healthcare engineering 2018\.  URL: https://www.ncbi.nlm.nih.gov/pubmed/29849998.
  * Teng et al. [2020] Teng, F., Yang, W., Chen, L., Huang, L., Xu, Q., 2020\.  Explainable prediction of medical codes with knowledge graphs.  Frontiers in Bioengineering and Biotechnology 8, 867.  URL: https://www.frontiersin.org/article/10.3389/fbioe.2020.00867, doi:10.3389/fbioe.2020.00867.
  * Vani et al. [2017] Vani, A., Jernite, Y., Sontag, D., 2017.  Grounded recurrent neural networks.  arXiv:1705.08557.
  * Vu et al. [2020] Vu, T., Nguyen, D.Q., Nguyen, A., 2020.  A label attention model for ICD coding from clinical text, in: Bessiere, C. (Ed.), Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, IJCAI-20, International Joint Conferences on Artificial Intelligence Organization. pp. 3335--3341.  URL: https://doi.org/10.24963/ijcai.2020/461, doi:10.24963/ijcai.2020/461. main track.
  * Wang et al. [2018] Wang, G., Li, C., Wang, W., Zhang, Y., Shen, D., Zhang, X., Henao, R., Carin, L., 2018.  Joint embedding of words and labels for text classification, in: Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), Association for Computational Linguistics, Melbourne, Australia. pp. 2321--2331.  URL: https://www.aclweb.org/anthology/P18-1216, doi:10.18653/v1/P18-1216.
  * Xian et al. [2018] Xian, Y., Lorenz, T., Schiele, B., Akata, Z., 2018\.  Feature generating networks for zero-shot learning, pp. 5542--5551.  doi:10.1109/CVPR.2018.00581.
  * Xie and Xing [2018] Xie, P., Xing, E., 2018.  A neural architecture for automated ICD coding, in: Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), Association for Computational Linguistics, Melbourne, Australia. pp. 1066--1076.  URL: https://www.aclweb.org/anthology/P18-1098, doi:10.18653/v1/P18-1098.
  * Xie et al. [2019] Xie, X., Xiong, Y., Yu, P.S., Zhu, Y., 2019.  EHR coding with multi-scale feature attention and structured knowledge graph propagation, in: Proceedings of the 28th ACM International Conference on Information and Knowledge Management, Association for Computing Machinery, New York, NY, USA. p. 649–658.  URL: https://doi.org/10.1145/3357384.3357897, doi:10.1145/3357384.3357897.
  * Xu et al. [2019] Xu, K., Lam, M., Pang, J., Gao, X., Band, C., Mathur, P., Papay, F., Khanna, A.K., Cywinski, J.B., Maheshwari, K., Xie, P., Xing, E.P., 2019.  Multimodal machine learning for automated ICD coding, in: Doshi-Velez, F., Fackler, J., Jung, K., Kale, D., Ranganath, R., Wallace, B., Wiens, J. (Eds.), Proceedings of the 4th Machine Learning for Healthcare Conference, PMLR, Ann Arbor, Michigan. pp. 197--215.  URL: http://proceedings.mlr.press/v106/xu19a.html.
  * Yan et al. [2015] Yan, Z., Zhang, H., Piramuthu, R., Jagadeesh, V., DeCoste, D., Di, W., Yu, Y., 2015.  Hd-cnn: Hierarchical deep convolutional neural networks for large scale visual recognition, in: 2015 IEEE International Conference on Computer Vision (ICCV), pp. 2740--2748.  doi:10.1109/ICCV.2015.314.
  * Zeng et al. [2019] Zeng, M., Li, M., Fei, Z., Yu, Y., Pan, Y., Wang, J., 2019.  Automatic ICD-9 coding via deep transfer learning.  Neurocomputing 324, 43 -- 50.  URL: http://www.sciencedirect.com/science/article/pii/S0925231218306246, doi:https://doi.org/10.1016/j.neucom.2018.04.081. deep Learning for Biological/Clinical Data.
  * Zhang and Zong [2015] Zhang, J., Zong, C., 2015\.  Deep neural networks in machine translation: An overview.  IEEE Intelligent Systems 30, 16--25.  doi:10.1109/MIS.2015.69.
  * Zhang et al. [2020] Zhang, Z., Liu, J., Razavian, N., 2020.  BERT-XML: Large scale automated ICD coding using BERT pretraining, in: Proceedings of the 3rd Clinical Natural Language Processing Workshop, Association for Computational Linguistics, Online. pp. 24--34.  URL: https://aclanthology.org/2020.clinicalnlp-1.3, doi:10.18653/v1/2020.clinicalnlp-1.3.
  * Zhu and Bain [2017] Zhu, X., Bain, M., 2017.  B-CNN: Branch convolutional neural network for hierarchical classification.  arXiv:1709.09890.
  * Zhu and Goldberg [2009] Zhu, X., Goldberg, A.B., 2009\.  Introduction to semi-supervised learning.  Synthesis lectures on artificial intelligence and machine learning 3, 1--130.  URL: https://doi.org/10.2200/S00196ED1V01Y200906AIM006.


Longitudinal Momentum Spectra of pair created in a pulsed electric field at
finite times: Are Oscillations “Real” 111 Based on talk given on 12th
September 2023 in 5th Conference on Extremely High Intensity Laser Physics
(ExHILP 2023) 12-15 September 2023 at Center for Relativistic Laser Science
(CoReLS),Institute for Basic Science (IBS), Gwangju Korea

Deepak Sah ∗,1,2, Manoranjan P. Singh 1,2

1Homi Bhabha National Institute, Training School Complex, Anushakti Nagar,
Mumbai 400094, India 2Theory and Simulations Lab, Raja Ramanna Centre for
Advanced Technology, Indore-452013, India

<EMAIL_ADDRESS><EMAIL_ADDRESS>

## Abstract

We investigate the production of electron-positron pairs from the vacuum in a
time-varying, spatially uniform pulsed electric field given by
$E(t)=E_{0}sech^{2}(t/\tau)$, with height of $E_{0}$ and width of $\tau$. Such
background field has received extensive attention in the literature [1,2,3],
with a focus on its asymptotic behavior, particularly the probability of pair
production. However, the problem of pair production after a finite evolution
time has rarely been discussed [4,5]. This raises questions about the
instantaneous appearance of particles in pair production and their behavior at
intermediate times when using a formalism that involves solving an evolution
equation for a dynamical quantity. Is it possible to make general statements
about this behavior? To address these questions, we analytically compute the
probability of $(e^{+}e^{-})$ pair production in momentum space using the
exact solution of the one-particle time-dependent Dirac equation, and we
compare the result with quantum kinetic theory (QKT), which is rigorously
derived from QED by canonical quantization of the Dirac field and subsequent
Bogoliubov transformation to a quasi-particle representation [6]. Both
approaches allow us to study the particle momentum spectrum at any instant of
time and can potentially unveil valuable information regarding quantum non-
equilibrium physics. We analyze the Longitudinal Momentum Spectrum (LMS) of
the created particles at finite times in both approaches. We observe
oscillatory structure in the LMS at time $t>\tau$, and this oscillation
pattern continuously changes from $t>\tau$ up to $t<6\tau$. After that, this
oscillation slowly fades away, and we see smooth gaussian-shape spectra for
late times, reproducing Schwinger’s non-perturbative signature. This
oscillation behavior at finite time clearly illustrates the quantum
interference effects associated with particle production, as explained in
recent work [7,9]. It is worth noting that both approaches exhibit quantum
interference patterns at finite times, manifested as oscillations observed in
the LMS. This clearly reveals that these oscillations are not due to transient
excitations and basis-dependent signatures [8]. Additionally, this formulation
at finite time of particle numbers (or spectra) can enhance our understanding
of designing laser pulses that lower the critical electric field threshold and
provide insights into the time-dependent processes of pair recombination and
the back-reaction of pairs.

References:

[1] A. I. Nikishov, _Soviet Physics JETP_ 30 (1970), 660–662.

[2] A. B. Balantekin et al., _Int. J. Mod. Phys. A_ 6 (1991), 695–725.

[3]. S. P. Gavrilov and D. M. Gitman, _Phys. Rev. D_ 53 (1996), 7162–7175.

[4] J. Hallin and P. Liljenberg,_Phys. Rev. D_ 52 (1995), 1150–1164.

[5]A. Di Piazza, _Phys. Rev. D_ 70 (2004), 053013.

[6]Y. Kluger et al., _Phys. Rev. Lett._ 67, 2427 (1991).

[7]D. Sah and M. P. Singh, [arXiv:2301.06545 [hep-ph]].

[8] R. Dabrowski and G. V. Dunne, _Phys. Rev. D_ 90, 025021 (2014).

[9] D. Sah and M. P. Singh,[arXiv:2309.12079 [hep-ph]].


11institutetext:  Center for Combustion Energy, Key laboratory for Thermal
Science and Power Engineering of MoE, Department of Energy and Power
Engineering, Tsinghua University, Beijing 100084, China
Physics of Fluids Group, University of Twente, 7500 AE Enschede, The
Netherlands
Universit$\acute{e}$ de Lille, Unit$\acute{e}$ de M$\acute{e}$canique de Lille
- J. Boussinesq (UML) ULR 7512, F 59000 Lille, France

# Accelerations of large inertial particles in turbulence

Yaning Fan 11 Cheng Wang 11 Linfeng Jiang 1122 Chao
<EMAIL_ADDRESS>11 Enrico Calzavarini222enrico.calzavarini@univ-
lille.fr 33112233

###### Abstract

Understanding the dynamics of material objects advected by turbulent flows is
a long standing question in fluid dynamics. In this perspective article we
focus on the characterization of the statistical properties of non-interacting
finite-sized massive spherical particles advected by a vigorous turbulent
flow. We study the fluctuations and temporal correlations of particle
accelerations and explore their behaviours with respect to both the particle
size and the particle mass density by means of fully-resolved numerical
simulations. We observe that the measured trends can not be interpreted as the
simple multiplicative combination of the two dominant effects: the spatial
filtering of fluid accelerations and the added-mass-adjusted fluid-to-particle
density ratio. We argue that other hydrodynamical forces or effects (e.g.
preferential flow sampling) have still a significant role even at the largest
particle sizes, which are here of the order of the integral scale of
turbulence.

##  1 Introduction

Fluid dynamics turbulence is a lasting challenge in science. Rather than
representing a single fundamental question - the problem’s definition itself
changed over epochs and disciplines [1] \- it is a faceted topic with
ramifications into a plethora of open issues and applications. Among the many
problems connected to turbulent flows a long standing one concerns the
description of the transport by the flow of material objects[2]. Clearly, the
study of the forces exerted on a body immersed in a flow is a classic topic in
the broader field of fluid dynamics, and many researchers have devoted their
attention to it. When considering the case of spherical bodies, we can refer
to Stokes’s work on drag force in creeping flows, followed by subsequent
studies by Oseen, Boussinesq, and Basset on inertial and unsteady corrections
(history force) to drag. Tchen, Corrsin, and Lumley conducted studies on the
pressure gradient force, while Auton focused on added-mass and lift forces in
the context of rotational potential flows (see [3] for an historical
overview). The unified modern formulation on the dynamics of a spherical
particle in an unsteady and non uniform viscous flow is due to Maxey & Riley
and Gatignol (MRG equation) [4, 5]333The same equation, without Faxén
corrections [6], was derived by Shu-Tang Tsai in 1957 and published in Chinese
language, see [7] for its recent translation.. However, the complexity of the
problem increases considerably when the carrying flow is turbulent, due to its
non-smooth and erratic characteristics both in time and space. In this case,
any kind of quantitative investigation must embrace the statistical approach
[8, 9].

In the last decades, a large body of studies has been dedicated to the problem
of particles advected by turbulence. This is largely attributed to the
emerging methods of experimental fluid/particle tracking by digital cameras
[10, 11, 12] and by the ever increasing high-performance numerical simulations
[8, 13]. While the phenomenology is presently relatively well explored for
particles whose size is of the order of the dissipative scale and the
associated particle Reynolds number is small, there is only a limited number
of studies for the cases of less idealized objects of larger size [14, 15, 16,
17, 18] or with non regular shape [19, 20] or inhomogeneous mass density [21].
The question is important for applications, among the many we wish to mention
the highly topical issue of transport and dispersion in turbulent ocean of
plastic debris [22]. Such debris encompass ranges of scales from the
dissipative to the inertial ones, have a variety of shapes and buoyancy
properties [23] and can undergo turbulence induced fragmentation [24].
Advances in the comprehension of the problem are crucially linked to a better
insight into their translational dynamics.

In this article we aim at performing a step forward into the understanding of
this complex phenomenon. We aim at highlighting how the particle mass density
affects the acceleration properties for particles whose size is larger than
the dissipative scale of turbulence. Our approach is computational, and makes
uses of numerical simulations that are capable to resolve all the spatio-
temporal scales active in the problem. Due to the wide range of scales at
work, these simulations are possible at the price of considering only single
(or at best few non-interacting) particles and moderate turbulent flow
intensities. Despite this limitation, our study allows to identify trends that
are expected to be valid also at larger turbulent flow Reynolds numbers and it
provides new benchmark data for future studies.

##  2 Theoretical considerations

Recent experimental and numerical studies have shown that the translational
[25, 17], and in part the rotational [26], statistical properties of inertial-
scale-sized neutrally buoyant particles advected by a turbulent flow can be
explained in terms of a coarse-graining effect of the underlying turbulent
flow. In other words the particle acceleration behaves - statistically - the
same as the spatially-filtered fluid acceleration field unperturbed by the
particle. This mechanism corresponds to assuming that the particle feedback on
the carrying turbulent flow is negligible. A neutral particle has on average a
small slip velocity compared to the surrounding fluid [27], it is therefore
reasonable to assume that the dissipative (surface) force associated to
viscous drag is sub-leading with respect to the inertial (volumetric) one. In
summary it seems reasonable to state that the acceleration of a particle of
typical size $d$ goes as $\bm{a}_{d}\sim\langle D_{t}\bm{u}\rangle_{V}$ where
$D_{t}\bm{u}=\partial_{t}\bm{u}+\bm{u}\cdot\nabla\bm{u}$ is the fluid
acceleration field and $\langle\ldots\rangle_{V}$ denotes a spatial average
over a volume ($V$) equivalent to the one of the particle.

When the particle is non-neutrally buoyant this scenario is complicated by two
factors, on one hand the different inertia between the particle and the fluid
tends to suppress/enhance (respectively for heavy/light particles) the fluid
accelerations of the surrounding carrying flow. This effect is not plainly
proportional to fluid-to-particle density ratio, $\rho_{f}/\rho_{p}$, but is
instead proportional to the parameter $\beta=3\rho_{f}/(\rho_{f}+2\rho_{p})$
because of the added-mass force exerted by the fluid on the particle [28].
Note that $\beta$ varies in the interval $\left[0,3\right]$, and the limits
correspond to the cases of very massive particles (ballistic limit) and to
that of very light particles where the inertia is all in the displaced
surrounding fluid (such as for the case of air bubbles in water); $\beta=1$
identifies the neutrally buoyant case. The second factor is the occurrence of
Archimedes buoyancy, which leads to an extra acceleration term of the form
$(1-\beta)\bf{g}$. In this study for simplicity we neglect the effect of
gravity, and this is always possible for sufficiently intense turbulent
flows444However, note that at increasing the particle size in a fixed
intensity turbulent flow, the relative importance of the coarse grained fluid
acceleration decrease with respect to buoyancy.. Adding together the mentioned
volume and surface forces the following model as been put forward for the
motion of finite-sized particles [29, 30]:

$\displaystyle\ddot{\bm{x}}=\beta\left(\langle
D_{t}\bm{u}\rangle_{V}+\frac{12\ \nu\
c(Re_{d})}{d^{2}}(\langle\bm{u}\rangle_{S}-\dot{\bm{x}})\right)+(1-\beta)\bm{g}.$

(1)

This is an adaptation of the MRG equation, which retains the so called Faxén
terms [5], with the addition of the empirical Shiller-Naumann drag correction
$c(Re_{d})=1+0.15Re_{d}^{0.687}$ where
$Re_{d}=||\langle\bm{u}\rangle_{S}-\dot{\bm{x}}||d/\nu$ is the instantaneous
particle Reynolds number and the omission of the history force. The symbols
$\langle\ldots\rangle_{S}$ denotes, similarly to the volume average, a spatial
average over a surface (S) equivalent to the one of the particle. These
average terms account for the effect of the local flow non uniformity at the
scale of the particle. If the particle is small they account simply for the
effect of the curvature (i.e. the laplacian) of the local flow velocity and
acceleration fields, if the particle is large they include higher even-order
spatial-derivatives, see also [31]. This model, known as Faxén corrected (FC)
model [29, 30], predicts qualitatively a series of trends in the single- and
two-time acceleration statistics, that will be discussed later on in this
article.

We can now ask, what are the statistical features of the particle acceleration
in a turbulent flow for the non-neutral ($\beta\neq 1$) case? When particles
are below the dissipative scale, both the Faxén and the Shiller-Naumann
corrections can be neglected in eq. (1) and expanding in the small parameter
$d^{2}/(12\nu\beta)\ll 1$(the drag response time) one obtains:

$\ddot{\bm{x}}\simeq
D_{t}\bm{u}+\frac{d^{2}}{12\nu\beta}(\beta-1)\left(D_{t}\bm{u}\cdot\nabla\bm{u}+D_{t}^{2}\bm{u}\right).$
(2)

This tells that the particle acceleration variance begins to deviate from the
one of a fluid tracer quadratically with its size $d$ and only when $\beta\neq
1$. For finite-sized particles a similar perturbative estimate is not
possible. However, the response time associated to the drag becomes long and
one can assume that the associated force becomes negligible. Taken together,
all these considerations lead us to guess that the inertia term $\beta\langle
D_{t}\bm{u}\rangle_{V}$ is the most important in determining the statistics of
the acceleration of large particles. This study aims at testing this
hypothesis and discussing its implications.

##  3 Methods: Fully resolved numerical study

We address the above questions by solving the fluid-particle coupled problem
which comprises the incompressible Navier-Stokes equations for the fluid
dynamics and the Newton-Euler equation for the particle motion, with the
addition of no-slip boundary conditions at their interfaces. The numerical
methods, based on the coupled Lattice-Boltzmann and Immersed-Boundary
algorithms, have been already described elsewhere [32, 26]. For the present
study, we carried on new validations for the dynamics of non neutrally buoyant
particles. Besides resolution convergence checks we verified that in the
presence of gravity the particle trajectory in still fluid agrees with the
expected Stokes flow dynamics. The spatial domain where the turbulent flow
takes place is a tri-periodic cube. The Taylor-scale based Reynolds number is
kept constant at $Re_{\lambda}=32$, the values of the relevant numerical/
physical turbulent flow scales are reported in Table 1.

$N^{3}$ | $\eta/\Delta x$ | $\tau_{\eta}/\Delta t$ | $L/\eta$ | $T_{L}/\tau_{\eta}$ | $\lambda/\eta$ | $Re_{\lambda}$
---|---|---|---|---|---|---
$128^{3}(256^{3})$ | $2.1(4.2)$ | $153(607)$ | $24.1$ | $12.4$ | $11.3$ | $32$

Table 1: Parameter and relevant scales of the simulated turbulent flow.
$N^{3}$: number of spatial grid points (the larger resolution is used for
$\beta>1$ particles); $\eta=(\nu^{3}/\epsilon)^{1/4}$: Kolmogorov dissipation
length scale in grid space units $\Delta x$; $\tau_{\eta}$: Kolmogorov time
scale in time-step units $\Delta t$, $L=u^{\prime 3}/\epsilon$: integral
scale; $T_{L}=L/u^{\prime}$: large-eddy turnover time; $\lambda=(15\nu
u^{\prime}/\epsilon)^{1/2}$: Taylor micro-scale; $Re_{\lambda}$: Taylor-
Reynolds number.

The control particle parameters are varied in the range $d/\eta=[6.5,18.7]$,
note that these represents inertial scales of the turbulent flow, and
$\rho_{p}/\rho_{f}=[0.4,10]$ (corresponding to $\beta=[0.14,1.67]$). The
particle are evolved for $O(10^{2})$ large-eddy turnover times. Each
simulations contains just one particle, but multiple independent simulations
are performed for each particle case; the informations along their trajectory
are saved for the analysis.

##  4 Results on acceleration statistics

###  4.1 Acceleration variance

The first quantity we focus on is the single-component acceleration variance,
$\langle a_{d,i}^{2}\rangle$ where $\langle\ldots\rangle$ denotes here the
time and ensemble average over many particle trajectories (all the
measurements are also given in Table 2).

$\beta$ $d/\eta$ | 6.5419 | 9.3444 | 11.2119 | 13.0785 | 15.8774 | 18.6739
---|---|---|---|---|---|---
1.6667 | 0.4538 | 0.4121 | 0.2400 | 0.2499 | 0.2185 | 0.1821
1.3636 | 0.3761 | 0.3265 | 0.2517 | 0.2448 | 0.1505 | 0.1610
1.0000 | 0.2749 | 0.2118 | 0.1736 | 0.1475 | 0.1204 | 0.0974
0.7500 | 0.2241 | 0.1572 | 0.1311 | 0.1045 | 0.0905 | 0.0683
0.5000 | 0.1768 | 0.1326 | 0.1020 | 0.0871 | 0.0641 | 0.0530
0.2727 | 0.1547 | 0.0977 | 0.0788 | 0.0605 | 0.0453 | 0.0351
0.1429 | 0.1308 | 0.0740 | 0.0540 | 0.0449 | 0.0337 | 0.0248

Table 2: $\langle a_{i}^{2}\rangle/\langle D_{t}u_{i}^{2}\rangle$ single
cartesian component particle acceleration variance normalized by the fluid-
tracer acceleration variance for various $\beta$ and $d/\eta$ values at
$Re_{\lambda}=32$. Figure 1:  (a) Particle acceleration variance normalized by
fluid-tracer acceleration variance as a function of the particle diameter
($d$) in dissipative units $\eta$. Data for density ratios $\beta$ are
displayed. The corresponding results for the point-point particle (PP) model
are reported (solid line). (b) Same data with the normalization
$\beta^{2}\langle(D_{t}u_{i})^{2}\rangle$ for the acceleration variance, this
way the PP model tends asymptotically to the unit value. Note that the fully-
resolved data with different $\beta$ do not overlap.

Figure 1(a) displays the trends for such a quantity (normalized by the fluid
flow acceleration variance) with respect to the particle diameter (in
dissipative scale units) and for particle sets with different density ratios
($\beta$). For the neutral case, $\beta=1$, the acceleration intensity
progressively decreases from the unit value for growing particle diameters,
and this is due to the spatial filtering effect mentioned above. This feature
has been already highlighted in experiments [25, 17] and also reproduced in
fully resolved simulations [33, 26]. A one dimensional estimate based on the
Kolmogorov 1941 (K41) turbulence theory suggests that $\langle
a_{d,i}^{2}\rangle\sim\langle\langle
D_{t}u_{i}\rangle_{V}^{2}\rangle\sim(\delta_{d}u)^{2}/d\sim d^{-2/3}$, where
$\delta_{d}u$ stands for a typical increment of velocity over a scale $d$. It
has been shown both experimentally and numerically that $\langle
a_{d}^{2}\rangle\sim d^{\alpha}$ with the exponent $\alpha$ varying between
$-4/3$ at low Reynolds (i.e. $Re_{\lambda}\leq 100$ as here) [33, 26] and
$-2/3$ for developed turbulence [25, 17].
The measurements for $\beta\neq 1$, which represent the main novelty of the
present study, show that the particle acceleration grows with $\beta$. In
order to contrast the density-ratio and the particle-size effect we also trace
on Fig.1(a) the results for point-like particles (solid lines). The point-
particle (PP) model includes the added mass but does not account for the
spatial filtering. It can be obtained from (1) by replacing the volume and
surface averages by the point values and by setting $c(Re_{d})=1$ [29]. While
in the limit of small particles the PP model departs weakly from the fluid
acceleration value (as suggested by eq. (2)), in the large size limit it
converges to $\langle
a_{d}^{2}\rangle\simeq\beta^{2}\langle(D_{t}u_{i})^{2}\rangle$[29]. This is
better evidenced in Fig.1(b), where the acceleration variances are normalized
by $\beta^{2}\langle(D_{t}u_{i})^{2}\rangle$ and all the PP data converge to
the same plateau. Conversely, the fully resolved simulation results tend to
vanish in the asymptotic limit due to the progressively enhanced smoothing by
filtering of turbulent fluctuations.
The effect of fluid-to-particle density ratio on the acceleration variance
$\langle a_{d}^{2}\rangle$ can be examined by dividing it by the acceleration
variance of the neutral particle of the corresponding size $\langle
a_{d,\beta=1}^{2}\rangle$. This is shown in Fig. 2 (a), one can notice that
the data still depend on the values of $\beta$. At this point is worth testing
the hypothesis,

$\langle a^{2}_{d}\rangle\simeq\beta^{2}\langle a^{2}_{d,\beta=1}\rangle.$ (3)

This is done in Fig.2(b). Although the trend suggests that the curves might
overlap for very large-particles, beyond the integral scale of turbulence,
$L\simeq 24\eta$, the lack of collapse of the curves can be interpreted as an
evidence of the non-multiplicative effect of the added mass and spatial
filtering on inertial-scale particles. We note that this multiplicative effect
was instead true in the FC model studied in [29], see in particular their Fig.
1(a) where with a similar rescaling all data collapse for large particle
diameters. The origin of this dicrepancy in the FC model remains to be
understood. In particular it would be interesting to check if it is due to the
non-linear drag. Indeed [29] did not take into account the Shiller-Naumann
correction in their study. However, non-linear drag was considered in the FC
model studied in [30] for neutrally buoyant particles, where it was shown to
have a negligible effect on the acceleration properties.

Figure 2:  (a) Particle acceleration variance normalized by the acceleration
variance of the neutrally buoyant particle of the corresponding size $\langle
a_{d,\beta=1}^{2}\rangle$ as a function of the particle diameter ($d$) in
dissipative units $\eta$. Note that the curves as approximately horizontal for
same $\beta$ particle families. (b) Same data with the normalization $\langle
a_{d,\beta=1}^{2}\rangle$ for the acceleration variance. Although curves get
closer, they do not collapse on each other.  Figure 3:  Single component
correlation function acceleration variance for particles diameter (a)
$d/\eta=6.5$ and (b) $d/\eta=18.7$. (c) Integral correlation time of the
particle acceleration versus the particle size, both expressed in dissipative
units, for different $\beta$ families. (d) Integral correlation time divided
by $T_{I,\beta=1}/\sqrt{\beta}$ versus the particle size.

###  4.2 Acceleration temporal correlations

The study of the temporal correlation functions of the acceleration allows to
further reinforce the observations made for the instantaneous acceleration
variance. We focus on the integral time, $T_{I}$, here defined as the integral
of the autocorrelation function from time zero to the time it first reaches
the null value (i.e. first zero-crossing time $T_{0}$):

$T_{I}\equiv\int_{0}^{T_{0}}C(\tau)d\tau,~{}~{}C(\tau)\equiv\frac{\langle
a_{i}(t+\tau)a_{i}(t)\rangle}{\langle(a_{i}(t))^{2}\rangle}$ (4)

Figure 3(a,b) shows the $C(\tau)$ correlations functions for the two
small/large particle limiting case in this study, i.e. for (a) $d/\eta=6.5$
and (b) $d/\eta=18.7$.

$\beta$ $d/\eta$ | 6.5419 | 9.3444 | 11.2119 | 13.0785 | 15.8774 | 18.6739
---|---|---|---|---|---|---
1.6667 | 1.2126 | 1.5015 | 1.6117 | 1.6425 | 1.7564 | 2.0205
1.3636 | 1.4170 | 1.5184 | 1.6928 | 1.9592 | 1.9589 | 2.1629
1.0000 | 1.6263 | 1.7011 | 1.9216 | 2.0704 | 2.2102 | 2.4419
0.7500 | 1.7408 | 1.8971 | 2.0315 | 2.2539 | 2.5249 | 2.6728
0.5000 | 1.8086 | 2.0344 | 2.2258 | 2.4176 | 2.6733 | 3.0411
0.2727 | 2.1082 | 2.5053 | 2.6268 | 2.7740 | 3.2323 | 3.6138
0.1429 | 2.7127 | 3.1644 | 3.3539 | 3.5064 | 4.1050 | 4.5701

Table 3: $T_{I}/\eta$, integral correlation time of the particle acceleration
normalized by the Kolmogorov time scale for various $\beta$ and $d/\eta$
values at $Re_{\lambda}=32$.

The integral correlation times computed from these curves grow with the
particle size and decreases with $\beta$, see Fig. 3(c) (all $T_{I}$
measurements are also reported in Table 3). The first trend is usually
rationalized in term of the coarse-graining hypothesis [34]. A particle of
size $d$ is subjected to turbulence fluctuations of that scale, this
corresponds to an eddy turnover time $\tau_{d}=d/\delta_{d}u\sim d^{2/3}$.
This prediction is only approximately true for neutrally buoyant particles. In
fact, similarly to the trends observed for the acceleration variance also in
the case of the correlation time the measured scaling $\tau_{d}\sim
d^{\gamma}$ has a senstive Reynolds dependence, it is observed that
$\gamma\leq 2/3$ at small Taylor-Reynolds number [33, 26], while it is
$2/3\leq\gamma\leq 1$ at large Reynolds [17].

The above argument can not be straightforwardly extended to non-neutral
particles. A possible approximate adaptation is presented in the following. We
indicate with $X_{a}=\frac{1}{2}\langle D_{t}u^{2}\rangle^{1/2}\tau_{a}^{2}$
the length spanned by a fluid particle over the time ($\tau_{a}$) during which
the acceleration is correlated (and so approximately constant). Such a length
can be travelled by a finite sized inertial particle over a time
$\tau_{d}=\sqrt{2X_{a}/\langle a_{d}^{2}\rangle^{1/2}}$. Now using the
hypothesis (3) one obtains

$\tau_{d}=\frac{\tau_{d,\beta=1}}{\sqrt{\beta}}\sim\frac{d^{2/3}}{\sqrt{\beta}}.$
(5)

The above prediction is tested in Fig. 3(d), we observe that while the size
dependency of the correlation time for any particle seems to be properly
normalized by the neutral case (i.e. the coarse graining hypothesis holds
true), the density dependence is only approximately explained in terms of the
above scaling. The collapse is better for the case of light particles were the
above argument is more fitting.

###  4.3 Acceleration’s higher statistical moments

Last we consider statistical properties beyond the second order. This can be
done by examining the trends in the shape of the probability density functions
(PDF) of the acceleration normalized by its standard deviation. These
functions are reported in Figure 4; panel (a) shows the curves for the
smallest particles, while (b) for the largest particles explored in this
study. It is clear the trend towards a Gaussianization of the accelerations
when the particle size is increased, this feature was also predicted by the FC
model simulations [29]. A similar tendency is observed at increasing the
particle mass density: PDF of heavy particles have shorter tails than neutral
particles of the same size, on the contrary light particles tend to have more
extreme accelerations. This trend is evident for the smallest particles, and
agrees with former results from PP model simulations [35], while it is here
negligible for the largest particles.
More robust conclusions can be drawn from the flatness of the acceleration,
$F(a_{d})=\langle a_{d}^{4}\rangle/\langle a_{d}^{2}\rangle^{2}$. Theoretical
considerations suggests that $\langle a_{d}^{4}\rangle/\langle
a_{d}^{2}\rangle^{2}\sim\langle(\delta_{d}u)^{8}\rangle/\langle(\delta_{d}u)^{4}\rangle^{2}\sim
d^{\zeta_{8}-2\zeta_{4}<0}$, where $\zeta_{p}$ indicates the scaling exponent
of the velocity structure functions of order $p$, i.e.,
$\langle(\delta_{d}u)^{p}\rangle\sim d^{\zeta_{p}}$. The value of the exponent
$F(a_{d})\sim d^{\phi}$ can be estimated in various ways, it is
$\phi\simeq-0.44$ from Kolmogorov-Obukhov 1962 model [36] and
$\phi\simeq-0.56$ with She-Lévêque parametrization [37]. Measuring a scaling
behaviour for $F(a_{d})$ is delicate as it requires large datasets.
Furthermore, similar to the acceleration variance, large Reynolds numbers are
needed in order to have a large-scale separation. Although some experiments
and simulations did not observe any scaling trend ($\phi=0$) [34, 16, 25, 33],
there are recently increasing evidences towards a reduction of the flatness
with size, [17] measured $\phi\simeq-0.5\pm 0.1$ for inertial range neutrally
buoyant particles, [38] observed it for bubbles in turbulence. This trend is
also supported by the FC model. The reduction of flatness with particle size
increase is confirmed by the present fully resolved simulations (see Figure 4
panel (c)). Furthermore, we observe a weak but detectable flatness
amplification with $\beta$. This was previously observed only for dissipative
scale particles both in experiments and simulations [35]. Its physical
interpretation lies in the phenomenon of preferential sampling: light
particles explore preferentially vortex cores where the fluid centripetal
acceleration is large, while heavy particles sample the outside of vortices
which are calmer regions as far as acceleration is concerned, and this is
reflected on their normalized accelerations PDF [39, 40]. This points to the
fact that preferential sampling is a small but non vanishing effect for
inertial-scale particles.

Figure 4:  PDF of particle acceleration normalized by their standard deviation
for (a) $d/\eta=6.5$ and (b) $d/\eta=18.7$. PDF of fluid tracer acceleration
(bold solid line) and Gaussian (traced for comparison). (c) Flatness of the
acceleration of particles versus their size, for different $\beta$. The dashed
line reports the fluid tracer acceleration flatness in the same flow. Note
that here $F(D_{t}u_{i})=6$.

##  5 Conclusions and Perspectives

Non-interacting finite-sized massive spherical particles advected by a
vigorous turbulent flow have been studied by means of fully-resolved numerical
simulations. We examined the single- and two-time statistical properties of
particle accelerations and explored their behaviours with respect to the
particle size and to the particle mass density. In this conditions the
inertial forces dominate over the dissipative ones. This study confirms all
the tendencies predicted by the coarse graining picture, namely the fact that
both the acceleration variance and its flatness decrease with the particle
size and the opposite for the acceleration correlation time. Given the limited
extension of the inertial range at the present Reynolds number
($Re_{\lambda}=32$) it is difficult to estimate reliable scaling laws as a
function of the particle diameter $d$. However, it seems that the observed
behaviours deviate systematically from the expected scaling laws derived from
the values of the velocity structure function exponents $\zeta_{p}$. This
point is delicate and requires further attention. It might be that the scaling
laws derived from $\zeta_{p}$ only holds asymptotically in $Re_{\lambda}$, or
it could be that systematic subleading deviations are present due to the
influence of other hydrodynamics effects (such as the drag).
Looking at the particle density dependence, we have tried to assess the
hypothesis $a_{d}\sim\beta\langle D_{t}u_{i}\rangle_{V}$. Even if the particle
acceleration increases with $\beta$, the observed trends can not be
interpreted as the simple multiplicative combination of the two dominant
terms: the spatial filtering of fluid accelerations and the added-mass-
adjusted fluid-to-particle density ratio. A similar mismatch is observed in
the dependence of the correlation time with $\beta$, for which we provided a
simple model. The study of the acceleration flatness indicates that light
particles are more intermittent than heavy ones and this also when their size
is large. This features suggest a role of preferential sampling of the flow by
the particles. This interpretation shall be put under scrutiny in further
studies, in fact in the context of inertial-scale particles the evidences of
preferential sampling and the related preferential clustering are still not
univocal [41, 42].

In the future it will be interesting to perform simulations at larger Reynolds
number and in larger domain size in order reduce the impact of finite
inertial-range and finite-domain effects present in our simulations. It will
also be interesting to develop new analysis techniques for the detection of
preferential sampling by inertial-scale particles in turbulent flows.
Furthermore, the statistical relevance for large particles in turbulence of
forces such as history [43] or lift [44], remains to be clearly assessed.

These results may help in developing effective models for the dynamics of
large particles in different context where particles are large with respect to
the typical variation scale in the flow, such drifters and floaters in the
ocean [45] or rock crystals in magmatic chambers and primordial magma oceans
[46].

##  6 Acknowledgement

This work was supported by the National Natural Science Foundation of China
under grant no. 11988102, and the New Cornerstone Science Foundation through
the Xplorer Prize.

## References

  * [1] Eckert M. The Turbulence Problem (Springer Cham, Berlin) 2019.
  * [2] Brandt L. Coletti F. Ann. Rev. Fluid Mech.542022159.
  * [3] Michaelides E. E. J. Fluids Eng.1191997233.
  * [4] Maxey M. R. Riley J. J. Phys. Fluids261983883.
  * [5] Gatignol R. J. Mécanique théorique et appliquée11983.
  * [6] Faxén H. Arkive Mat. Astron. Oh. Fys.1819241.
  * [7] Tsai S. T. Theor. App. Mech. Lett.122022100392.
  * [8] Toschi F. Bodenschatz E. Ann. Rev. Fluid Mech.412009375.
  * [9] Bec J., Gustavsson K. Mehlig B. Ann. Rev. Fluid Mech.562024null.
  * [10] Adrian R. J. Ann. Rev. Fluid Mech.231991261.
  * [11] Porta A. L., Voth G. A., Crawford A. M., Alexander J. Bodenschatz E. Nature409200122.
  * [12] Voth G. A., Porta L. A., Crawford A. M., Alexander J. Bodenschatz E. J. Fluid Mech.4692002121.
  * [13] Lohse D. Phys. Rev. Fluids32018110504.
  * [14] Prosperetti A. iuz H. O. J. Comput. Phys.1672001196.
  * [15] Qureshi N. M., Bourgoin M., Baudet C., Cartellier A. Gagne Y. Phys. Rev. Lett.992007184502.
  * [16] Xu H. Bodenschatz E. Phys. D23720082095.
  * [17] Volk R., Calzavarini E., Leveque E. Pinton J. F. J. Fluid Mech.6682011223.
  * [18] Mathai V., Prakash V. N., Brons J., Sun C. Lohse D. Phys. Rev. Lett.1152015124501.
  * [19] Voth G. A. Soldati A. Ann. Rev. Fluid Mech.492017249.
  * [20] Verhille G. J. Fluid Mech.9332022A3.
  * [21] Will J. B. Krug D. J. Fluid Mech.9272021A7.
  * [22] Law K. L. Ann. Rev. Marine Sci.92017205.
  * [23] Sutherland B. R., DiBenedetto M., Kaminski A. van den Bremer T. Phys. Rev. Fluids82023070701.
  * [24] Brouzet C., Guiné R., Dalbe M. J., Favier B., Vandenberghe N., Villermaux E. Verhille G. Phys. Rev. Fluids62021024601.
  * [25] Brown R. D., Warhaft Z. Voth G. A. Phys. Rev. Lett.1032009194501.
  * [26] Jiang L., Wang C., Liu S., Sun C. Calzavarini E. J. Fluid Mech.9392022A22.
  * [27] Cisse M., Homann H. Bec J. J. Fluid Mech.7352013R1.
  * [28] Auton T. R., Hunt J. C. R. Prud’Homme M. J. Fluid Mech.1971988241–257.
  * [29] Calzavarini E., Volk R., Bourgoin M., Leveque E., Pinton J. F. Toschi F. J. Fluid Mech.6302009179.
  * [30] Calzavarini E., Volk R., Leveque E., Pinton J. F. Toschi F. Phys. D241D2012237.
  * [31] Rallabandi B. Phys. Rev. Fluids62021L012302.
  * [32] Calzavarini E. Software Impacts12019100002.
  * [33] Homann H. Bec J. J. Fluid Mech.651201081.
  * [34] Qureshi N. M., Bourgoin M., Baudet C., Cartellier A. Gagne Y. Phys. Rev. Lett.992007184502.
  * [35] Volk R., Calzavarini E., Verhille G., Lohse D., Mordant N., Pinton J. F. Toschi F. Phys. D23720082084.
  * [36] Kolmogorov A. N. J. Fluid Mech.13196282–85.
  * [37] She Z. S. Leveque E. Phys. Rev. Lett.721994336.
  * [38] Prakash V. N., Tagawa Y., Calzavarini E., Mercado J. M., Toschi F., Lohse D. Sun C. New J. Phys.142012105017.
  * [39] Bec J., Biferale L., Boffetta G., Celani A., Cencini M., Lanotte A., Musacchi S. Toschi F. J. Fluid Mech.5502006349–358.
  * [40] Calzavarini E., Kerscher M., Lohse D. Toschi F. J. Fluid Mech.607200813.
  * [41] Fiabane L., Zimmermann R., Volk R., Pinton J. F. Bourgoin M. Phys. Rev. E862012035301.
  * [42] Uhlmann M. Chouippe A. J. Fluid Mech.8122017991–1023.
  * [43] Jaganathan D., Prasath S. G., Govindarajan R. Vasan V. Front. Phys.112023.
  * [44] Mazzitelli I. M. Lohse D. New J. Phys.62004203.
  * [45] Beron-Vera F. J. Nonlinear Dyn.10320211.
  * [46] Patočka V., Tosi N. Calzavarini E. Earth Planet. Sci. Lett.5912022117622.