129,916 research outputs found
Opinion Dynamics in Heterogeneous Networks: Convergence Conjectures and Theorems
Recently, significant attention has been dedicated to the models of opinion
dynamics in which opinions are described by real numbers, and agents update
their opinions synchronously by averaging their neighbors' opinions. The
neighbors of each agent can be defined as either (1) those agents whose
opinions are in its "confidence range," or (2) those agents whose "influence
range" contain the agent's opinion. The former definition is employed in
Hegselmann and Krause's bounded confidence model, and the latter is novel here.
As the confidence and influence ranges are distinct for each agent, the
heterogeneous state-dependent interconnection topology leads to a
poorly-understood complex dynamic behavior. In both models, we classify the
agents via their interconnection topology and, accordingly, compute the
equilibria of the system. Then, we define a positive invariant set centered at
each equilibrium opinion vector. We show that if a trajectory enters one such
set, then it converges to a steady state with constant interconnection
topology. This result gives us a novel sufficient condition for both models to
establish convergence, and is consistent with our conjecture that all
trajectories of the bounded confidence and influence models eventually converge
to a steady state under fixed topology.Comment: 22 pages, Submitted to SIAM Journal on Control and Optimization
(SICON
Predictions of ultra-harmonic oscillations in coupled arrays of limit cycle oscillators
Coupled distinct arrays of nonlinear oscillators have been shown to have a
regime of high frequency, or ultra-harmonic, oscillations that are at multiples
of the natural frequency of individual oscillators. The coupled array
architectures generate an in-phase high-frequency state by coupling with an
array in an anti-phase state. The underlying mechanism for the creation and
stability of the ultra-harmonic oscillations is analyzed. A class of
inter-array coupling is shown to create a stable, in-phase oscillation having
frequency that increases linearly with the number of oscillators, but with an
amplitude that stays fairly constant. The analysis of the theory is illustrated
by numerical simulation of coupled arrays of Stuart-Landau limit cycle
oscillators.Comment: 24 pages, 9 figures, accepted to Phys. Rev. E, in pres
Gaussian approximation for finitely extensible bead-spring chains with hydrodynamic interaction
The Gaussian Approximation, proposed originally by Ottinger [J. Chem. Phys.,
90 (1) : 463-473, 1989] to account for the influence of fluctuations in
hydrodynamic interactions in Rouse chains, is adapted here to derive a new
mean-field approximation for the FENE spring force. This "FENE-PG" force law
approximately accounts for spring-force fluctuations, which are neglected in
the widely used FENE-P approximation. The Gaussian Approximation for
hydrodynamic interactions is combined with the FENE-P and FENE-PG spring force
approximations to obtain approximate models for finitely-extensible bead-spring
chains with hydrodynamic interactions. The closed set of ODE's governing the
evolution of the second-moments of the configurational probability distribution
in the approximate models are used to generate predictions of rheological
properties in steady and unsteady shear and uniaxial extensional flows, which
are found to be in good agreement with the exact results obtained with Brownian
dynamics simulations. In particular, predictions of coil-stretch hysteresis are
in quantitative agreement with simulations' results. Additional simplifying
diagonalization-of-normal-modes assumptions are found to lead to considerable
savings in computation time, without significant loss in accuracy.Comment: 26 pages, 17 figures, 2 tables, 75 numbered equations, 1 appendix
with 10 numbered equations Submitted to J. Chem. Phys. on 6 February 200
Synchronization Landscapes in Small-World-Connected Computer Networks
Motivated by a synchronization problem in distributed computing we studied a
simple growth model on regular and small-world networks, embedded in one and
two-dimensions. We find that the synchronization landscape (corresponding to
the progress of the individual processors) exhibits Kardar-Parisi-Zhang-like
kinetic roughening on regular networks with short-range communication links.
Although the processors, on average, progress at a nonzero rate, their spread
(the width of the synchronization landscape) diverges with the number of nodes
(desynchronized state) hindering efficient data management. When random
communication links are added on top of the one and two-dimensional regular
networks (resulting in a small-world network), large fluctuations in the
synchronization landscape are suppressed and the width approaches a finite
value in the large system-size limit (synchronized state). In the resulting
synchronization scheme, the processors make close-to-uniform progress with a
nonzero rate without global intervention. We obtain our results by ``simulating
the simulations", based on the exact algorithmic rules, supported by
coarse-grained arguments.Comment: 20 pages, 22 figure
Non-equilibrium Berezinskii-Kosterlitz-Thouless Transition in a Driven Open Quantum System
The Berezinskii-Kosterlitz-Thouless mechanism, in which a phase transition is
mediated by the proliferation of topological defects, governs the critical
behaviour of a wide range of equilibrium two-dimensional systems with a
continuous symmetry, ranging from superconducting thin films to two-dimensional
Bose fluids, such as liquid helium and ultracold atoms. We show here that this
phenomenon is not restricted to thermal equilibrium, rather it survives more
generally in a dissipative highly non-equilibrium system driven into a
steady-state. By considering a light-matter superfluid of polaritons, in the
so-called optical parametric oscillator regime, we demonstrate that it indeed
undergoes a vortex binding-unbinding phase transition. Yet, the exponent of the
power-law decay of the first order correlation function in the (algebraically)
ordered phase can exceed the equilibrium upper limit -- a surprising
occurrence, which has also been observed in a recent experiment. Thus we
demonstrate that the ordered phase is somehow more robust against the quantum
fluctuations of driven systems than thermal ones in equilibrium.Comment: 11 pages, 9 figure
Dipole-dipole interaction between orthogonal dipole moments in time-dependent geometries
In two nearby atoms, the dipole-dipole interaction can couple transitions
with orthogonal dipole moments. This orthogonal coupling accounts for a number
of interesting effects, but strongly depends on the geometry of the setup.
Here, we discuss several setups of interest where the geometry is not fixed,
such as particles in a trap or gases, by averaging over different sets of
geometries. Two averaging methods are compared. In the first method, it is
assumed that the internal electronic evolution is much faster than the change
of geometry, whereas in the second, it is vice versa. We find that the
orthogonal coupling typically survives even extensive averaging over different
geometries, albeit with qualitatively different results for the two averaging
methods. Typically, one- and two-dimensional averaging ranges modelling, e.g.,
low-dimensional gases, turn out to be the most promising model systems.Comment: 11 pages, 14 figure
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