Comment on "Particle Diffusion in a
Quasi-Two-Dimensional Bacterial Bath"

Guillaume Grégoire, Hugues Chaté, Yuhai Tu

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Guillaume Grégoire, Hugues Chaté, Yuhai Tu. Comment on "Particle Diffusion in a Quasi-Two-Dimensional Bacterial Bath". Physical Review Letters, 2001, 86 (3), pp.556. 10.1103/Phys-RevLett.86.556. hal-01053525

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Distributed under a Creative Commons Attribution 4.0 International License
Comment on “Particle Diffusion in a
Quasi-Two-Dimensional Bacterial Bath”

Recently, Wu and Libchaber (WL) reported on a fasci-
nating experiment in which bacteria move freely within a
fluid film seeded with polystyrene beads [1]. They studied
the dynamics of these beads as they are moved around by
the bacteria, and found superdiffusive motion ($\langle r^2 \rangle \sim t^\alpha$
with $\alpha \approx 1.5$) below some crossover scales $t_c$, $\ell_c$, beyond
which normal diffusion ($\alpha = 1$) is recovered. WL inter-
pret these scales as characteristic of the structures (swirls,
jets) that emerge from the collective motion of the bacte-
ria. A simple Langevin equation with a force term corre-
lated in time over the crossover scale $t_c$ was used to fit the
experimental data. Two problems arise from this descrip-
tion: First, the Langevin framework predicts ballistic be-
havior ($\alpha = 2$) at short scales, at odds with the nontrivial
exponents recorded in the experiment. Second, no attempt
is made to explain the origin of the collective motion and
how or why the crossover scales change with the bacteria
density $\rho$.

Here we show that a coherent and robust theoretical
framework for this experiment is that provided by the
“self-propelled XY spin” models studied recently [2] com-
plemented by a collection of passive beads. Let us describe
briefly the model used here (details will appear elsewhere,
together with evidence that our results are largely insensi-
tive to the particular modeling choices made [3]). Bacteria
move at discrete time steps with fixed-amplitude velocity
$v_0$ along a direction reflecting the action of two forces: a
noisy tendency to align with neighboring objects within ra-
dius $r_0$ [2], and a two-body repulsive force conferring them
a finite size $r_B$. A small number of passive beads of ra-
dius $r_B$ is added. They interact with bacteria via hard-core
repulsion plus some level of entrainment within range $r_0$
(i.e., they take part of the neighboring bacteria velocity).

Increasing ρ, ordered collective motion appears at a
value ρ* [2]. For ρ < ρ*, bacteria motion is characterized
by scales which diverge as ρ → ρ* (Fig. 1). Bead motion
is directly related to bacteria behavior, as evidenced by
their respective diffusive properties which both reveal su-
perdiffusion crossing over at the same time scale t_c to nor-
mal diffusion (Figs. 1b and 1c). The characteristic scales
of bead motion are thus given by the collective scales of
bacteria motion, as foreseen by WL, but the short-time be-
havior of the beads in our model is superdiffusive, which is
more consistent with the experimental data than the simple
Langevin ansatz.

The density dependence of the crossover scales is also
naturally explained by our model: as ρ increases, the sys-
tem is closer to the critical point ρ*, and the superdiffu-
sive behavior persists longer. The range of variation of
crossover scales recorded by WL is small (e.g., the maxi-
mum value of ρ_c is of the order of r_B). This explains
why a linear variation was found to be a good approxi-

<figure><img src="image_2.png" /><figcaption>FIG. 1. Minimal model for bacterial bath with passive beads with $v_0 = 0.3$, $r_0 = 1.0$, $r_B = 0.13$, $r_B = 0.38$ (for other details, see [3]). (a) Short-time (30 time steps) trajectories of bacteria (thin lines) and beads (thick lines) for $\rho < \rho^*$ in a system of size $32 \times 32$; (b) mean square displacement $\langle r^2 \rangle$ vs time for bacteria (solid lines) and beads (dashed lines) for $\rho = 2, 3, 3.5 < \rho^* \approx 4.2$; (c) $\rho$ variation of $t_c$ (□) and $\ell_c$ (Δ) and diffusion constant $D = \lim_{t \to \infty} d\langle r^2 \rangle/dt$ (○); (d) superdiffusion at $\rho = \rho^*$ with exponent $\alpha \approx 1.65(15)$.</figcaption></figure>

mation in [1], even though the scales are expected to di-
verge at threshold (Fig. 1c). We believe the framework
proposed here is more consistent than the Langevin ap-
proach of WL. A firmer statement will be available only
when experiments taking place closer to ρ* (i.e., at higher
densities) are conducted. They would allow for the su-
perdiffusive behavior to be observed over a wide range of
scales, and the expected universality of the associated ex-
ponent α (Fig. 1d) could be assessed.

Guillaume Grégoire and Hugues Chaté

CEA—Service de Physique de l'Etat Condensé

91191 Gif-sur-Yvette, France

Yuhai Tu
IBM T. J. Watson Research Center
Yorktown Heights, New York 10598

Received 5 May 2000

DOI: 10.1103/PhysRevLett.86.556

PACS numbers: 87.17.Jj, 45.70.Qj, 87.18.Hf

[1] X.-L. Wu and A. Libchaber, Phys. Rev. Lett. 84, 3017 (2000).

[2] A. Czirok, H. E. Stanley, and T. Vicsek, J. Phys. A 30, 1375 (1997); J. Toner and Y. Tu, Phys. Rev. E 58, 4828 (1998).

[3] G. Grégoire, H. Chaté, and Y. Tu (to be published).