5 research outputs found
Minimal stochastic field equations for one-dimensional flocking
We consider the collective behaviour of active particles that locally align
with their neighbours. Agent-based simulation models have previously shown that
in one dimension, these particles can form into a flock that maintains its
stability by stochastically alternating its direction. Until now, this
behaviour has been seen in models based on continuum field equations only by
appealing to long-range interactions that are not present in the simulation
model. Here, we derive a set of stochastic field equations with local
interactions that reproduces both qualitatively and quantitatively the
behaviour of the agent-based model, including the alternating flock phase. A
crucial component is a multiplicative noise term of the voter model type in the
dynamics of the local polarization whose magnitude is inversely proportional to
the local density. We show that there is an important subtlety in determining
the physically appropriate noise, in that it depends on a careful choice of the
field variables used to characterise the system. We further use the resulting
equations to show that a nonlinear alignment interaction of at least cubic
order is needed for flocking to arise
FĂ©iniĂșlacht agus inspreagadh i gcĂĄs cĂșntĂłirĂ teanga i scoileanna Gaeltachta [Identity and motivation among language assistants in Gaeltacht schools]
San alt seo, cuirtear i lĂĄthair taighde a rinneadh ar ghnĂ©ithe Ă©agsĂșla de chur i bhfeidhm ScĂ©im na gCĂșntĂłirĂ Teanga (an ScĂ©im feasta) atĂĄ ĂĄ reĂĄchtĂĄil ag an Roinn TurasĂłireachta, CultĂșir, EalaĂon, Gaeltachta, SpĂłirt agus na MeĂĄn. Cuireann an ScĂ©im, atĂĄ ar bun Ăł bhĂ 1999 ann, cĂșntĂłirĂ teanga ar fĂĄil do scoileanna Gaeltachta dâfhonn an Ghaeilge a lĂĄidriĂș mar theanga labhartha i measc daltaĂ scoile sna ceantair sin. Ba Ă© aidhm mhĂłr an tionscadail taighde nĂĄ iniĂșchadh a dhĂ©anamh ar staid reatha na ScĂ©ime agus ina dhiaidh sin moltaĂ chun feabhais a chur ar fĂĄil. ĂsĂĄideadh ceistneoirĂ, agallaimh agus grĂșpaĂ fĂłcais le sonraĂ a bhailiĂș Ăł phĂĄirtithe leasmhara na ScĂ©ime. San alt seo, dĂrĂtear ar shonraĂ a bhaineann le fĂ©iniĂșlacht agus inspreagadh na gcĂșntĂłirĂ teanga a thĂĄinig chun cinn sna grĂșpaĂ fĂłcais ar ghlac 50 cĂșntĂłir pĂĄirt iontu. I gcomhthĂ©acs na fĂ©iniĂșlachta, is iad na tĂ©amaĂ a dâeascair Ăł na comhrĂĄite nĂĄ fĂ©indearcadh na gcĂșntĂłirĂ orthu fĂ©in, a rĂłl i suĂomh na hoibre, agus rĂłl na hoibre a dhĂ©anann siad i dtacĂș leis an phobal lena mbaineann siad. I dtaca leis an inspreagadh, bhain na tĂ©amaĂ leis an teanga fĂ©in, na pĂĄistĂ a mbĂonn siad ag obair leo, tacĂș le cĂ©ad ghlĂșn eile chainteoirĂ na Gaeilge agus todhchaĂ na Gaeltachta. PlĂ©itear rĂĄitis na gcĂșntĂłirĂ i gcomhthĂ©acs na litrĂochta ar fhĂ©iniĂșlacht teanga agus inspreagadh mĂșinteoirĂ teanga, mar aon le cĂĄs na Gaeilge agus na Gaeltachta.
[In this article we present data from a research project which reviewed various aspects of the implementation of ScĂ©im na gCĂșntĂłirĂ Teanga (The Language Assistants Scheme), which is run by the Department of Tourism, Culture, Arts, Gaeltacht, Sport and Media. The Scheme, established in 1999, provides language assistants to Gaeltacht schools in order to strengthen spoken Irish among pupils in those regions. The main aim of the research was to evaluate the current state of the Scheme and to make recommendations for its enhancement. Questionnaires, interviews and focus groups were used to gather data from the Schemeâs stakeholders. The present article focuses on issues of identity and motivation which were discussed by the 50 language assistants who took part in the focus groups. In relation to identity, the themes that emerged were assistantsâ self-image, their role in the workplace, and the role of the work they do to support their own community. As for motivation, the themes were connected to the language itself, the children they work with, supporting the next generation of Irish speakers, and the future of the Gaeltacht. The assistantsâ statements are discussed in relation to the literatures on language identity and language teacher motivation, as well as the Irish language and the Gaeltacht.
Stochastic effects in systems of aligning self-propelled particles
Systems of self-propelled particles are often capable of exhibiting complex
behaviours on a macroscopic scale with only simple interactions between the
active microscopic agents. In systems where the particles interact by attempting
to align their directions of motion, the ordered steady state tends towards a
dense coherent grouping of particles (a "flock") travelling in the same direction.
Continuum theories where the particles are treated as an active
fluid allow for
a greater understanding of the macroscopic behaviour of these
flocks, although
these theories have typically focussed on understanding the behaviour of the
flock
in a steady state. In this thesis, we are interested in deriving continuum theories
of aligning self-propelled particle systems and in understanding the role that
stochasticity has in ensuring the dynamical behaviour of the underlying agentbased
model is maintained.
The focus of this thesis is a family of models of aligning self-propelled particles
on a lattice that interact by aligning with a subset of nearby particles. We
use the Kramers-Moyal approximation to derive stochastic Langevin equations
directly from the microscopic interactions. Our goal is to obtain equations for
the evolution of the system's density and polarisation such that their trajectories
match the dynamic behaviour of the underlying agent-based models and, in doing
so, to demonstrate that the form of the stochastic prefactor in the polarisation
equation can greatly affect the macroscopic behaviour of the system.
In Chapter 2, we study the ordered state of a system of aligning self-propelled
particles on a one-dimensional lattice. The aligning interaction between particles
allows for the formation of a
flock capable of alternating the direction it travels
through the lattice. We derive a set of stochastic differential equations for the
density and polarisation of the system and introduce a numerical integration
scheme to demonstrate that the order parameter of each of the agent-based and
continuum systems scales identically with increasing noise strength. We then use
the continuum equations to obtain a minimal set of interactions for a
flock to
exist in one dimension and demonstrate how alignment interactions with three
particles are necessary for a
flock to form on a one dimensional lattice.
This motivates the work in the remainder of this thesis, wherein we examine a
family of two-dimensional models to explore whether we can derive stochastic
differential equations whose trajectories demonstrate the same behaviour as in
the agent-based models. We introduce a family of four lattice-based agent-based
models in Chapter 3 and map out the behaviour of the ordered state in each of
these models. These models consist of all combinations of two interaction types
(exponential or linear in local polarisation) and two interaction neighbourhoods
(fi xed or varying with local density). One of these models shows the "banding"
present in traditional Vicsek models, while the other three show the occasional
macroscopic change in direction observed in
flocks of birds such as starlings.
In Chapter 4, we use the Kramers-Moyal approximation again to derive stochastic
differential equations for the density and polarisation of the four models above.
Using linear stability analysis, we explain why the ordered state in each model
consists of a
flock that will either be capable or incapable of turning. The
linear stability analysis shows why the choice of interaction neighbourhood does
not affect the ability of the
flocks with a linear interaction to macroscopically
alter direction and why that choice does affect the ability of a
flock to turn for
systems with an exponential interaction, although some calculation remains here
to demonstrate linear stability exactly matching that of the agent-based systems.
We also explore a numerical integration scheme for the two dimensional models,
laying out a procedure that may result in integrated trajectories matching the
behaviour of the agent-based model as in one dimension.
The work in this thesis explores the effects of different stochastic terms in
continuum equations describing systems of aligning self-propelled particles and
introduces a mechanism to derive these terms to ensure the behaviour matches
that of the underlying agent-based models. We demonstrate the power of this
mechanism by identifying a minimal model of
flocking in one dimension and
by exploring when
flocks can turn in two dimensions. These examples provide a
pathway for exploring the dynamic behaviour of other interacting particle models
on a macroscopic scale