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Maxwell-Boltzmann velocity distribution for noninteracting active matter

Pedro Herrera and Mario Sandoval
Phys. Rev. E 103, 012601 – Published 4 January 2021
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    Abstract

    We theoretically and computationally find a Maxwell-Boltzmann-like velocity distribution for noninteracting active matter (NAM). To achieve this, mass and moment of inertia are incorporated into the corresponding noninteracting active Fokker-Planck equation (NAFP), thus solving for the first time, the underdamped scenario of NAM following a Fokker-Planck formalism. This time, the distribution results in a bimodal symmetric expression that contains the effect of inertia on transport properties of NAM. The analytical distribution is further compared to experiments dealing with vibrobots. A generalization of the Brinkman hierarchy for NAFP is also provided and used for systematically solving the NAFP in position space. This work is an important step toward characterizing active matter using an equivalent nonequilibrium statistical mechanics.

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    • Received 1 August 2020
    • Revised 13 November 2020
    • Accepted 11 December 2020

    DOI:https://doi.org/10.1103/PhysRevE.103.012601

    ©2021 American Physical Society

    Physics Subject Headings (PhySH)

    1. Techniques
    Fokker–Planck equationTheories of collective dynamics & active matter
    Physics of Living Systems

    Authors & Affiliations

    Pedro Herrera and Mario Sandoval*

    • Department of Physics, Universidad Autonoma Metropolitana-Iztapalapa, Mexico City 09340, Mexico

    • *sem@xanum.uam.mx

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    Vol. 103, Iss. 1 — January 2021

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    • Figure 1
      Figure 1

      Progress of nonequilibrium statistical mechanics on the description of passive and active Brownian particles. This article solves the gold oval case.

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    • Figure 2
      Figure 2

      Velocity distribution and MSS, as a function of propulsion (Pe) and inertia (ST,SR) for noninteracting self-propelled solid disks. Theoretical, experimental, and numerical results are presented, respectively, as green solid and dashed lines, yellow circles, and blue bars. In (a)–(c), speed is increased, while ST=0.09 and SR=0.06, according to experiments [23]. (a) Pe=U2τR/DT=151 from experiments. (b) Pe=2778 (hypothetical speed). (c) Pe=4938 (hypothetical speed). (d) Numerical (blue circles) and theoretical [Eq. (11)] MSS for figures (a)–(c). In (e)–(g), inertia is increased. All data were taken from experiments [23]. (e) Pe=16, ST=0.48 and SR=0.48. (f) Pe=58, ST=0.19 and SR=0.03. (g) Pe=41, ST=0.39 and SR=0.52. (h) Numerical (blue circles) and theoretical [Eq. (11)] MSS for (e) and (g).

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    • Figure 3
      Figure 3

      Theoretical (green-dashed lines) and numerical (blue circles) results for the probability distribution function of position P(x). (a) Same parameters as in Fig. 2. (b) Same parameters as in Fig. 2.

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