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Standard Models on Collective Motion

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Order and Fluctuations in Collective Dynamics of Swimming Bacteria

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Abstract

We start by reviewing previous theoretical and numerical studies on collective motion of self-propelled elements. Especially, we comprehensively review standard models on collective motion: Vicsek-style models and corresponding hydrodynamic theories. Then existing experimental works are carefully examined based on the theoretical understandings. We stress that giant number fluctuations rooted in the spontaneous rotational symmetry breaking need to be discussed in the homogeneous long-range ordered phase without any clusters. We conclude this chapter by mentioning the gaps between those previous experiments and the theoretical predictions. We clarify what is required to observe experimentally to test the theoretical predictions on the Vicsek universality class.

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Notes

  1. 1.

    In the definition here, \(\theta _j^t\) is updated prior to the update of \(\varvec{r}_j^t\). It is known that the Vicsek model is robust against the order of updating these two variables.

  2. 2.

    Note that, if you do not wait until the system relaxes to steady states, there exist apparent clusters.

  3. 3.

    In the first Vicsek’s paper [2], the number of simulated particles was \(10^4\) at the largest. On the other hand, in [1], the number of particles was more than \(5\times 10^5\).

  4. 4.

    The period T should be long enough so that the N(t) and \(N(t+T)\) are decorrelated.

  5. 5.

    True and quasi-long-range order can be distinguished by correlation functions. If the correlation remains positive at the large system size limit, the system has true long-range order. On the other hand, the correlation decays algebraically in the case of quasi-long-range order.

  6. 6.

    In their original papers [4, 5, 7], there are ‘fluctuations of notation’, which we have corrected in this thesis.

  7. 7.

    Such topological phases are often observed in two-dimensional cultures of mammalian cells with high aspect ratio such as neural progenitor cells [37] and epithelial cells [38].

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Authors and Affiliations

  1. Department of Physics, Graduate School of Science, The University of Tokyo, Bunkyo-ku, Tokyo, Japan

    Dr. Daiki Nishiguchi

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Nishiguchi, D. (2020). Standard Models on Collective Motion. In: Order and Fluctuations in Collective Dynamics of Swimming Bacteria. Springer Theses. Springer, Singapore. https://doi.org/10.1007/978-981-32-9998-6_2

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  • DOI: https://doi.org/10.1007/978-981-32-9998-6_2

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