164 research outputs found
Flocking with discrete symmetry: the 2d Active Ising Model
We study in detail the active Ising model, a stochastic lattice gas where
collective motion emerges from the spontaneous breaking of a discrete symmetry.
On a 2d lattice, active particles undergo a diffusion biased in one of two
possible directions (left and right) and align ferromagnetically their
direction of motion, hence yielding a minimal flocking model with discrete
rotational symmetry. We show that the transition to collective motion amounts
in this model to a bona fide liquid-gas phase transition in the canonical
ensemble. The phase diagram in the density/velocity parameter plane has a
critical point at zero velocity which belongs to the Ising universality class.
In the density/temperature "canonical" ensemble, the usual critical point of
the equilibrium liquid-gas transition is sent to infinite density because the
different symmetries between liquid and gas phases preclude a supercritical
region. We build a continuum theory which reproduces qualitatively the behavior
of the microscopic model. In particular we predict analytically the shapes of
the phase diagrams in the vicinity of the critical points, the binodal and
spinodal densities at coexistence, and the speeds and shapes of the
phase-separated profiles.Comment: 20 pages, 25 figure
Sedimentation, trapping, and rectification of dilute bacteria
The run-and-tumble dynamics of bacteria, as exhibited by \textit{E. coli},
offers a simple experimental realization of non-Brownian, yet diffusive,
particles. Here we present some analytic and numerical results for models of
the ideal (low-density) limit in which the particles have no hydrodynamic or
other interactions and hence undergo independent motions. We address three
cases: sedimentation under gravity; confinement by a harmonic external
potential; and rectification by a strip of `funnel gates' which we model by a
zone in which tumble rate depends on swim direction. We compare our results
with recent experimental and simulation literature and highlight similarities
and differences with the diffusive motion of colloidal particles
Active Brownian Particles and Run-and-Tumble Particles: a Comparative Study
Active Brownian particles (ABPs) and Run-and-Tumble particles (RTPs) both
self-propel at fixed speed along a body-axis that reorients
either through slow angular diffusion (ABPs) or sudden complete randomisation
(RTPs). We compare the physics of these two model systems both at microscopic
and macroscopic scales. Using exact results for their steady-state distribution
in the presence of external potentials, we show that they both admit the same
effective equilibrium regime perturbatively that breaks down for stronger
external potentials, in a model-dependent way. In the presence of collisional
repulsions such particles slow down at high density: their propulsive effort is
unchanged, but their average speed along becomes . A
fruitful avenue is then to construct a mean-field description in which
particles are ghost-like and have no collisions, but swim at a variable speed
that is an explicit function or functional of the density . We give
numerical evidence that the recently shown equivalence of the fluctuating
hydrodynamics of ABPs and RTPs in this case, which we detail here, extends to
microscopic models of ABPs and RTPs interacting with repulsive forces.Comment: 32 pages, 6 figure
Arrested phase separation in reproducing bacteria: a generic route to pattern formation?
We present a generic mechanism by which reproducing microorganisms, with a
diffusivity that depends on the local population density, can form stable
patterns. It is known that a decrease of swimming speed with density can
promote separation into bulk phases of two coexisting densities; this is
opposed by the logistic law for birth and death which allows only a single
uniform density to be stable. The result of this contest is an arrested
nonequilibrium phase separation in which dense droplets or rings become
separated by less dense regions, with a characteristic steady-state length
scale. Cell division mainly occurs in the dilute regions and cell death in the
dense ones, with a continuous flux between these sustained by the diffusivity
gradient. We formulate a mathematical model of this in a case involving
run-and-tumble bacteria, and make connections with a wider class of mechanisms
for density-dependent motility. No chemotaxis is assumed in the model, yet it
predicts the formation of patterns strikingly similar to those believed to
result from chemotactic behavior
Large deviations in boundary-driven systems: Numerical evaluation and effective large-scale behavior
We study rare events in systems of diffusive fields driven out of equilibrium
by the boundaries. We present a numerical technique and use it to calculate the
probabilities of rare events in one and two dimensions. Using this technique,
we show that the probability density of a slowly varying configuration can be
captured with a small number of long wave-length modes. For a configuration
which varies rapidly in space this description can be complemented by a local
equilibrium assumption
When are active Brownian particles and run-and-tumble particles equivalent? Consequences for motility-induced phase separation
Active Brownian particles (ABPs, such as self-phoretic colloids) swim at
fixed speed along a body-axis that rotates by slow angular
diffusion. Run-and-tumble particles (RTPs, such as motile bacteria) swim with
constant \u until a random tumble event suddenly decorrelates the
orientation. We show that when the motility parameters depend on density
but not on , the coarse-grained fluctuating hydrodynamics of
interacting ABPs and RTPs can be mapped onto each other and are thus strictly
equivalent. In both cases, a steeply enough decreasing causes phase
separation in dimensions , even when no attractive forces act between
the particles. This points to a generic role for motility-induced phase
separation in active matter. However, we show that the ABP/RTP equivalence does
not automatically extend to the more general case of \u-dependent motilities
A numerical approach to large deviations in continuous-time
We present an algorithm to evaluate the large deviation functions associated
to history-dependent observables. Instead of relying on a time discretisation
procedure to approximate the dynamics, we provide a direct continuous-time
algorithm, valuable for systems with multiple time scales, thus extending the
work of Giardin\`a, Kurchan and Peliti (PRL 96, 120603 (2006)).
The procedure is supplemented with a thermodynamic-integration scheme, which
improves its efficiency. We also show how the method can be used to probe large
deviation functions in systems with a dynamical phase transition -- revealed in
our context through the appearance of a non-analyticity in the large deviation
functions.Comment: Submitted to J. Stat. Mec
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