79 research outputs found
Renormalization-group Studies of Three Model Systems Far from Equilibrium
This thesis describes the development of analytical and computational techniques for systems far from equilibrium and their application to three model systems. Each of the model systems reaches a non-equilibrium steady state and exhibits one or more phase transitions. We first introduce a new position-space renormalization-group approach and illustrate its application using the one-dimensional fully asymmetric exclusion process. We have constructed a recursion relation for the relevant dynamic parameters for this model and have reproduced all of the important critical features of the model, including the exact positions of the critical point and the first and second order phase boundaries. The method yields an approximate value for the critical exponent v which is very close to the known value. The second major part of this thesis combines information theoretic techniques for calculating the entropy and a Monte Carlo renormalization-group approach. We have used this method to study and compare infinitely driven lattice gases. This approach enables us to calculate the critical exponents associated with the correlation length v and the order parameter /3. These results are compared to the values predicted from different field theoretic treatments of the models. In the final set of calculations, we build position-space renormalization-group recursion relations from the master equations of small clusters. By obtaining the probability distributions for these clusters numerically, we develop a mapping connecting the parameters specifying the dynamics on different length scales. The resulting flow topology in some ways mimics equilibrium features, with sinks for each phase and fixed points associated with each phase boundary. In addition, though, there are added complexities in the flows, suggesting multiple regions within the ordered phase for some values of parameters, and the presence of an extra source fixed point within the ordered phase. Thus, this study illustrates the successful applicability of position-space renormalization- group and information theoretic approaches to driven lattice gases in one and two dimensions. These methods provide new insights into the critical properties and ordering in these systems, and set the stage for further development of these approaches and their application to additional, more realistic models
Spatial stochastic predator-prey models
We consider a broad class of stochastic lattice predator-prey models, whose
main features are overviewed. In particular, this article aims at drawing a
picture of the influence of spatial fluctuations, which are not accounted for
by the deterministic rate equations, on the properties of the stochastic
models. Here, we outline the robust scenario obeyed by most of the lattice
predator-prey models with an interaction "a' la Lotka-Volterra". We also show
how a drastically different behavior can emerge as the result of a subtle
interplay between long-range interactions and a nearest-neighbor exchange
process.Comment: 5 pages, 2 figures. Proceedings paper of the workshop "Stochastic
models in biological sciences" (May 29 - June 2, 2006 in Warsaw) for the
Banach Center Publication
A Position-Space Renormalization-Group Approach for Driven Diffusive Systems Applied to the Asymmetric Exclusion Model
This paper introduces a position-space renormalization-group approach for
nonequilibrium systems and applies the method to a driven stochastic
one-dimensional gas with open boundaries. The dynamics are characterized by
three parameters: the probability that a particle will flow into the
chain to the leftmost site, the probability that a particle will flow
out from the rightmost site, and the probability that a particle will jump
to the right if the site to the right is empty. The renormalization-group
procedure is conducted within the space of these transition probabilities,
which are relevant to the system's dynamics. The method yields a critical point
at ,in agreement with the exact values, and the critical
exponent , as compared with the exact value .Comment: 14 pages, 4 figure
Fluctuations and Correlations in Lattice Models for Predator-Prey Interaction
Including spatial structure and stochastic noise invalidates the classical
Lotka-Volterra picture of stable regular population cycles emerging in models
for predator-prey interactions. Growth-limiting terms for the prey induce a
continuous extinction threshold for the predator population whose critical
properties are in the directed percolation universality class. Here, we discuss
the robustness of this scenario by considering an ecologically inspired
stochastic lattice predator-prey model variant where the predation process
includes next-nearest-neighbor interactions. We find that the corresponding
stochastic model reproduces the above scenario in dimensions 1< d \leq 4, in
contrast with mean-field theory which predicts a first-order phase transition.
However, the mean-field features are recovered upon allowing for
nearest-neighbor particle exchange processes, provided these are sufficiently
fast.Comment: 5 pages, 4 figures, 2-column revtex4 format. Emphasis on the lattice
predator-prey model with next-nearest-neighbor interaction (Rapid
Communication in PRE
Voting and Catalytic Processes with Inhomogeneities
We consider the dynamics of the voter model and of the monomer-monomer
catalytic process in the presence of many ``competing'' inhomogeneities and
show, through exact calculations and numerical simulations, that their presence
results in a nontrivial fluctuating steady state whose properties are studied
and turn out to specifically depend on the dimensionality of the system, the
strength of the inhomogeneities and their separating distances. In fact, in
arbitrary dimensions, we obtain an exact (yet formal) expression of the order
parameters (magnetization and concentration of adsorbed particles) in the
presence of an arbitrary number of inhomogeneities (``zealots'' in the
voter language) and formal similarities with {\it suitable electrostatic
systems} are pointed out. In the nontrivial cases , we explicitly
compute the static and long-time properties of the order parameters and
therefore capture the generic features of the systems. When , the problems
are studied through numerical simulations. In one spatial dimension, we also
compute the expressions of the stationary order parameters in the completely
disordered case, where is arbitrary large. Particular attention is paid to
the spatial dependence of the stationary order parameters and formal
connections with electrostatics.Comment: 17 pages, 6 figures, revtex4 2-column format. Original title ("Are
Voting and Catalytic Processes Electrostatic Problems ?") changed upon
editorial request. Minor typos corrected. Published in Physical Review
Phase Transitions and Spatio-Temporal Fluctuations in Stochastic Lattice Lotka-Volterra Models
We study the general properties of stochastic two-species models for
predator-prey competition and coexistence with Lotka-Volterra type interactions
defined on a -dimensional lattice. Introducing spatial degrees of freedom
and allowing for stochastic fluctuations generically invalidates the classical,
deterministic mean-field picture. Already within mean-field theory, however,
spatial constraints, modeling locally limited resources, lead to the emergence
of a continuous active-to-absorbing state phase transition. Field-theoretic
arguments, supported by Monte Carlo simulation results, indicate that this
transition, which represents an extinction threshold for the predator
population, is governed by the directed percolation universality class. In the
active state, where predators and prey coexist, the classical center
singularities with associated population cycles are replaced by either nodes or
foci. In the vicinity of the stable nodes, the system is characterized by
essentially stationary localized clusters of predators in a sea of prey. Near
the stable foci, however, the stochastic lattice Lotka-Volterra system displays
complex, correlated spatio-temporal patterns of competing activity fronts.
Correspondingly, the population densities in our numerical simulations turn out
to oscillate irregularly in time, with amplitudes that tend to zero in the
thermodynamic limit. Yet in finite systems these oscillatory fluctuations are
quite persistent, and their features are determined by the intrinsic
interaction rates rather than the initial conditions. We emphasize the
robustness of this scenario with respect to various model perturbations.Comment: 19 pages, 11 figures, 2-column revtex4 format. Minor modifications.
Accepted in the Journal of Statistical Physics. Movies corresponding to
Figures 2 and 3 are available at
http://www.phys.vt.edu/~tauber/PredatorPrey/movies
Parafermion Hall states from coset projections of abelian conformal theories
The Z_k-parafermion Hall state is an incompressible fluid of k-electron
clusters generalizing the Pfaffian state of paired electrons. Extending our
earlier analysis of the Pfaffian, we introduce two ``parent'' abelian Hall
states which reduce to the parafermion state by projecting out some neutral
degrees of freedom. The first abelian state is a generalized (331) state which
describes clustering of k distinguishable electrons and reproduces the
parafermion state upon symmetrization over the electron coordinates. This
description yields simple expressions for the quasi-particle wave functions of
the parafermion state. The second abelian state is realized by a conformal
theory with a (2k-1)-dimensional chiral charge lattice and it reduces to the
Z_k-parafermion state via the coset construction su(k)_1+su(k)_1/su(k)_2. The
detailed study of this construction provides us a complete account of the
excitations of the parafermion Hall state, including the field identifications,
the Z_k symmetry and the partition function.Comment: Latex, 36 pages, 3 tables, 2 figure
SYMBA: An end-to-end VLBI synthetic data generation pipeline: Simulating Event Horizon Telescope observations of M 87
Context. Realistic synthetic observations of theoretical source models are essential for our understanding of real observational data. In using synthetic data, one can verify the extent to which source parameters can be recovered and evaluate how various data corruption effects can be calibrated. These studies are the most important when proposing observations of new sources, in the characterization of the capabilities of new or upgraded instruments, and when verifying model-based theoretical predictions in a direct comparison with observational data. Aims. We present the SYnthetic Measurement creator for long Baseline Arrays (SYMBA), a novel synthetic data generation pipeline for Very Long Baseline Interferometry (VLBI) observations. SYMBA takes into account several realistic atmospheric, instrumental, and calibration effects. Methods. We used SYMBA to create synthetic observations for the Event Horizon Telescope (EHT), a millimetre VLBI array, which has recently captured the first image of a black hole shadow. After testing SYMBA with simple source and corruption models, we study the importance of including all corruption and calibration effects, compared to the addition of thermal noise only. Using synthetic data based on two example general relativistic magnetohydrodynamics (GRMHD) model images of M 87, we performed case studies to assess the image quality that can be obtained with the current and future EHT array for different weather conditions. Results. Our synthetic observations show that the effects of atmospheric and instrumental corruptions on the measured visibilities are significant. Despite these effects, we demonstrate how the overall structure of our GRMHD source models can be recovered robustly with the EHT2017 array after performing calibration steps, which include fringe fitting, a priori amplitude and network calibration, and self-calibration. With the planned addition of new stations to the EHT array in the coming years, images could be reconstructed with higher angular resolution and dynamic range. In our case study, these improvements allowed for a distinction between a thermal and a non-thermal GRMHD model based on salient features in reconstructed images
THEMIS: A Parameter Estimation Framework for the Event Horizon Telescope
The Event Horizon Telescope (EHT) provides the unprecedented ability to directly resolve the structure and dynamics of black hole emission regions on scales smaller than their horizons. This has the potential to critically probe the mechanisms by which black holes accrete and launch outflows, and the structure of supermassive black hole spacetimes. However, accessing this information is a formidable analysis challenge for two reasons. First, the EHT natively produces a variety of data types that encode information about the image structure in nontrivial ways; these are subject to a variety of systematic effects associated with very long baseline interferometry and are supplemented by a wide variety of auxiliary data on the primary EHT targets from decades of other observations. Second, models of the emission regions and their interaction with the black hole are complex, highly uncertain, and computationally expensive to construct. As a result, the scientific utilization of EHT observations requires a flexible, extensible, and powerful analysis framework. We present such a framework, Themis, which defines a set of interfaces between models, data, and sampling algorithms that facilitates future development. We describe the design and currently existing components of Themis, how Themis has been validated thus far, and present additional analyses made possible by Themis that illustrate its capabilities. Importantly, we demonstrate that Themis is able to reproduce prior EHT analyses, extend these, and do so in a computationally efficient manner that can efficiently exploit modern high-performance computing facilities. Themis has already been used extensively in the scientific analysis and interpretation of the first EHT observations of M87
A Universal Power-law Prescription for Variability from Synthetic Images of Black Hole Accretion Flows
We present a framework for characterizing the spatiotemporal power spectrum of the variability expected from the horizon-scale emission structure around supermassive black holes, and we apply this framework to a library of general relativistic magnetohydrodynamic (GRMHD) simulations and associated general relativistic ray-traced images relevant for Event Horizon Telescope (EHT) observations of Sgr A*. We find that the variability power spectrum is generically a red-noise process in both the temporal and spatial dimensions, with the peak in power occurring on the longest timescales and largest spatial scales. When both the time-averaged source structure and the spatially integrated light-curve variability are removed, the residual power spectrum exhibits a universal broken power-law behavior. On small spatial frequencies, the residual power spectrum rises as the square of the spatial frequency and is proportional to the variance in the centroid of emission. Beyond some peak in variability power, the residual power spectrum falls as that of the time-averaged source structure, which is similar across simulations; this behavior can be naturally explained if the variability arises from a multiplicative random field that has a steeper high-frequency power-law index than that of the time-averaged source structure. We briefly explore the ability of power spectral variability studies to constrain physical parameters relevant for the GRMHD simulations, which can be scaled to provide predictions for black holes in a range of systems in the optically thin regime. We present specific expectations for the behavior of the M87* and Sgr A* accretion flows as observed by the EHT
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