2,276 research outputs found
Extreme Thouless effect in a minimal model of dynamic social networks
In common descriptions of phase transitions, first order transitions are
characterized by discontinuous jumps in the order parameter and normal
fluctuations, while second order transitions are associated with no jumps and
anomalous fluctuations. Outside this paradigm are systems exhibiting `mixed
order transitions' displaying a mixture of these characteristics. When the jump
is maximal and the fluctuations range over the entire range of allowed values,
the behavior has been coined an `extreme Thouless effect'. Here, we report
findings of such a phenomenon, in the context of dynamic, social networks.
Defined by minimal rules of evolution, it describes a population of extreme
introverts and extroverts, who prefer to have contacts with, respectively, no
one or everyone. From the dynamics, we derive an exact distribution of
microstates in the stationary state. With only two control parameters,
(the number of each subgroup), we study collective variables of
interest, e.g., , the total number of - links and the degree
distributions. Using simulations and mean-field theory, we provide evidence
that this system displays an extreme Thouless effect. Specifically, the
fraction jumps from to (in the
thermodynamic limit) when crosses , while all values appear with
equal probability at .Comment: arXiv admin note: substantial text overlap with arXiv:1408.542
Exact results for the extreme Thouless effect in a model of network dynamics
If a system undergoing phase transitions exhibits some characteristics of
both first and second order, it is said to be of 'mixed order' or to display
the Thouless effect. Such a transition is present in a simple model of a
dynamic social network, in which extreme introverts/extroverts always
cut/add random links. In particular, simulations showed that , the average fraction of cross-links between the two groups
(which serves as an 'order parameter' here), jumps dramatically when crosses the 'critical point' , as in typical
first order transitions. Yet, at criticality, there is no phase co-existence,
but the fluctuations of are much larger than in typical second order
transitions. Indeed, it was conjectured that, in the thermodynamic limit, both
the jump and the fluctuations become maximal, so that the system is said to
display an 'extreme Thouless effect.' While earlier theories are partially
successful, we provide a mean-field like approach that accounts for all known
simulation data and validates the conjecture. Moreover, for the critical system
, an analytic expression for the mesa-like stationary
distribution, , shows that it is essentially flat in a range
, with . Numerical evaluations of
provides excellent agreement with simulation data for .
For large , we find ,
though this behavior begins to set in only for . For accessible
values of , we provide a transcendental equation for an approximate
which is better than 1% down to . We conjecture how this approach
might be used to attack other systems displaying an extreme Thouless effect.Comment: 6 pages, 4 figure
Eigenvalue Separation in Some Random Matrix Models
The eigenvalue density for members of the Gaussian orthogonal and unitary
ensembles follows the Wigner semi-circle law. If the Gaussian entries are all
shifted by a constant amount c/Sqrt(2N), where N is the size of the matrix, in
the large N limit a single eigenvalue will separate from the support of the
Wigner semi-circle provided c > 1. In this study, using an asymptotic analysis
of the secular equation for the eigenvalue condition, we compare this effect to
analogous effects occurring in general variance Wishart matrices and matrices
from the shifted mean chiral ensemble. We undertake an analogous comparative
study of eigenvalue separation properties when the size of the matrices are
fixed and c goes to infinity, and higher rank analogues of this setting. This
is done using exact expressions for eigenvalue probability densities in terms
of generalized hypergeometric functions, and using the interpretation of the
latter as a Green function in the Dyson Brownian motion model. For the shifted
mean Gaussian unitary ensemble and its analogues an alternative approach is to
use exact expressions for the correlation functions in terms of classical
orthogonal polynomials and associated multiple generalizations. By using these
exact expressions to compute and plot the eigenvalue density, illustrations of
the various eigenvalue separation effects are obtained.Comment: 25 pages, 9 figures include
A complete devil's staircase in the Falicov-Kimball model
We consider the neutral, one-dimensional Falicov-Kimball model at zero
temperature in the limit of a large electron--ion attractive potential, U. By
calculating the general n-ion interaction terms to leading order in 1/U we
argue that the ground-state of the model exhibits the behavior of a complete
devil's staircase.Comment: 6 pages, RevTeX, 3 Postscript figure
Nonlinear evolution of surface morphology in InAs/AlAs superlattices via surface diffusion
Continuum simulations of self-organized lateral compositional modulation
growth in InAs/AlAs short-period superlattices on InP substrate are presented.
Results of the simulations correspond quantitatively to the results of
synchrotron x-ray diffraction experiments. The time evolution of the
compositional modulation during epitaxial growth can be explained only
including a nonlinear dependence of the elastic energy of the growing epitaxial
layer on its thickness. From the fit of the experimental data to the growth
simulations we have determined the parameters of this nonlinear dependence. It
was found that the modulation amplitude don't depend on the values of the
surface diffusion constants of particular elements.Comment: 4 pages, 3 figures, published in Phys. Rev. Lett.
http://link.aps.org/abstract/PRL/v96/e13610
Nonequilibrium critical dynamics of the relaxational models C and D
We investigate the critical dynamics of the -component relaxational models
C and D which incorporate the coupling of a nonconserved and conserved order
parameter S, respectively, to the conserved energy density rho, under
nonequilibrium conditions by means of the dynamical renormalization group.
Detailed balance violations can be implemented isotropically by allowing for
different effective temperatures for the heat baths coupling to the slow modes.
In the case of model D with conserved order parameter, the energy density
fluctuations can be integrated out. For model C with scalar order parameter, in
equilibrium governed by strong dynamic scaling (z_S = z_rho), we find no
genuine nonequilibrium fixed point. The nonequilibrium critical dynamics of
model C with n = 1 thus follows the behavior of other systems with nonconserved
order parameter wherein detailed balance becomes effectively restored at the
phase transition. For n >= 4, the energy density decouples from the order
parameter. However, for n = 2 and n = 3, in the weak dynamic scaling regime
(z_S <= z_rho) entire lines of genuine nonequilibrium model C fixed points
emerge to one-loop order, which are characterized by continuously varying
critical exponents. Similarly, the nonequilibrium model C with spatially
anisotropic noise and n < 4 allows for continuously varying exponents, yet with
strong dynamic scaling. Subjecting model D to anisotropic nonequilibrium
perturbations leads to genuinely different critical behavior with softening
only in subsectors of momentum space and correspondingly anisotropic scaling
exponents. Similar to the two-temperature model B the effective theory at
criticality can be cast into an equilibrium model D dynamics, albeit
incorporating long-range interactions of the uniaxial dipolar type.Comment: Revtex, 23 pages, 5 eps figures included (minor additions), to appear
in Phys. Rev.
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