5,850 research outputs found
Differential Equations Modeling Crowd Interactions
Nonlocal conservation laws are used to describe various realistic instances
of crowd behaviors. First, a basic analytic framework is established through an
"ad hoc" well posedness theorem for systems of nonlocal conservation laws in
several space dimensions interacting non locally with a system of ODEs.
Numerical integrations show possible applications to the interaction of
different groups of pedestrians, and also with other "agents".Comment: 26 pages, 5 figure
NonLocal Systems of Balance Laws in Several Space Dimensions with Applications to Laser Technolog
For a class of systems of nonlinear and nonlocal balance laws in several
space dimensions, we prove the local in time existence of solutions and their
continuous dependence on the initial datum. The choice of this class is
motivated by a new model devoted to the description of a metal plate being cut
by a laser beam. Using realistic parameters, solutions to this model obtained
through numerical integrations meet qualitative properties of real cuts.
Moreover, the class of equations considered comprises a model describing the
dynamics of solid particles along a conveyor belt
Self-organized criticality as an absorbing-state phase transition
We explore the connection between self-organized criticality and phase
transitions in models with absorbing states. Sandpile models are found to
exhibit criticality only when a pair of relevant parameters - dissipation
epsilon and driving field h - are set to their critical values. The critical
values of epsilon and h are both equal to zero. The first is due to the absence
of saturation (no bound on energy) in the sandpile model, while the second
result is common to other absorbing-state transitions. The original definition
of the sandpile model places it at the point (epsilon=0, h=0+): it is critical
by definition. We argue power-law avalanche distributions are a general feature
of models with infinitely many absorbing configurations, when they are subject
to slow driving at the critical point. Our assertions are supported by
simulations of the sandpile at epsilon=h=0 and fixed energy density (no drive,
periodic boundaries), and of the slowly-driven pair contact process. We
formulate a field theory for the sandpile model, in which the order parameter
is coupled to a conserved energy density, which plays the role of an effective
creation rate.Comment: 19 pages, 9 figure
Diagrammatic routes to nonlocal correlations beyond dynamical mean field theory
Strong electronic correlations pose one of the biggest challenges to solid
state theory. We review recently developed methods that address this problem by
starting with the local, eminently important correlations of dynamical mean
field theory (DMFT). On top of this, non-local correlations on all length
scales are generated through Feynman diagrams, with a local two-particle vertex
instead of the bare Coulomb interaction as a building block. With these
diagrammatic extensions of DMFT long-range charge-, magnetic-, and
superconducting fluctuations as well as (quantum) criticality can be addressed
in strongly correlated electron systems. We provide an overview of the
successes and results achieved---hitherto mainly for model Hamiltonians---and
outline future prospects for realistic material calculations.Comment: 60 pages, 42 figures, replaced by the version to be published in Rev.
Mod. Phys. 201
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