scholarly journals CRITICAL BEHAVIOR OF NONEQUILIBRIUM MODELS WITH INFINITELY MANY ABSORBING STATES

1994 ◽  
Vol 08 (23) ◽  
pp. 3299-3311 ◽  
Author(s):  
IWAN JENSEN
Keyword(s):  
Heterogeneous Catalysis ◽  
Critical Behavior ◽  
Critical Exponents ◽  
Lattice Model ◽  
Universality Class ◽  
Directed Percolation ◽  
Two Dimensional ◽  
Absorbing States ◽  
Nonequilibrium Models

I study the critical behavior of a two-dimensional dimer-trimer lattice model, introduced by Köhler and ben-Avraham,17a for heterogeneous catalysis of the reaction ½A2 + ⅓B3 → AB. The model possesses infinitely many absorbing states in which the lattice is saturated by adsorbed particles and reactions cease because only isolated vacancies are left. Results for various critical exponents show that the model exhibits the same critical behavior as directed percolation, contrary to earlier findings by Köhler and ben-Avraham. Together with several other studies, reviewed briefly in this article, this confirms that directed percolation is the generic universality class for models with infinitely many absorbing states.

scholarly journals Critical behavior of density-driven and shear-driven reversible–irreversible transitions in cyclically sheared vortices

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
S. Maegochi ◽  
K. Ienaga ◽  
S. Okuma
Keyword(s):  
Power Law ◽  
Critical Behavior ◽  
Particle Density ◽  
Universality Class ◽  
Time Dependent ◽  
Directed Percolation ◽  
Two Dimensional ◽  
Significant Drop ◽  
Cyclic Shear ◽  
Vortex Density

AbstractRandom assemblies of particles subjected to cyclic shear undergo a reversible–irreversible transition (RIT) with increasing a shear amplitude d or particle density n, while the latter type of RIT has not been verified experimentally. Here, we measure the time-dependent velocity of cyclically sheared vortices and observe the critical behavior of RIT driven by vortex density B as well as d. At the critical point of each RIT, $$B_{\mathrm {c}}$$ B c and $$d_{\mathrm {c}}$$ d c , the relaxation time $$\tau $$ τ to reach the steady state shows a power-law divergence. The critical exponent for B-driven RIT is in agreement with that for d-driven RIT and both types of RIT fall into the same universality class as the absorbing transition in the two-dimensional directed-percolation universality class. As d is decreased to the average intervortex spacing in the reversible regime, $$\tau (d)$$ τ ( d ) shows a significant drop, indicating a transition or crossover from a loop-reversible state with vortex-vortex collisions to a collisionless point-reversible state. In either regime, $$\tau (d)$$ τ ( d ) exhibits a power-law divergence at the same $$d_{\mathrm {c}}$$ d c with nearly the same exponent.

scholarly journals DIRECTED PERCOLATION AND OTHER SYSTEMS WITH ABSORBING STATES: IMPACT OF BOUNDARIES

2001 ◽  
Vol 15 (12) ◽  
pp. 1761-1797 ◽  
Author(s):  
PER FRÖJDH ◽  
MARTIN HOWARD ◽  
KENT BÆKGAARD LAURITSEN
Keyword(s):  
Critical Behavior ◽  
Mean Field ◽  
Critical Dimension ◽  
Scaling Theory ◽  
Directed Percolation ◽  
Upper Critical Dimension ◽  
Duality Symmetry ◽  
Universality Classes ◽  
Absorbing States ◽  
The Impact

We review the critical behavior of nonequilibrium systems, such as directed percolation (DP) and branching-annihilating random walks (BARW), which possess phase transitions into absorbing states. After reviewing the bulk scaling behavior of these models, we devote the main part of this review to analyzing the impact of walls on their critical behavior. We discuss the possible boundary universality classes for the DP and BARW models, which can be described by a general scaling theory which allows for two independent surface exponents in addition to the bulk critical exponents. Above the upper critical dimension d c , we review the use of mean field theories, whereas in the regime d<d c , where fluctuations are important, we examine the application of field theoretic methods. Of particular interest is the situation in d=1, which has been extensively investigated using numerical simulations and series expansions. Although DP and BARW fit into the same scaling theory, they can still show very different surface behavior: for DP some exponents are degenerate, a property not shared with the BARW model. Moreover, a "hidden" duality symmetry of BARW in d=1 is broken by the boundary and this relates exponents and boundary conditions in an intricate way.

CRITICAL BEHAVIOR AND THRESHOLD OF COEXISTENCE OF A PREDATOR–PREY STOCHASTIC MODEL IN A 2D LATTICE

2010 ◽  
Vol 20 (02) ◽  
pp. 309-314 ◽  
Author(s):  
C. ARGOLO ◽  
H. OTAVIANO ◽  
IRAM GLERIA ◽  
EVERALDO ARASHIRO ◽  
TÂNIA TOMÉ
Keyword(s):  
Critical Behavior ◽  
Critical Exponents ◽  
Order Parameter ◽  
Species Coexistence ◽  
Finite Size ◽  
Universality Class ◽  
Square Lattice ◽  
Scaling Analysis ◽  
Predator Prey ◽  
Prey System

We investigate the critical behavior of a stochastic lattice model describing a predator–prey system. By means of Monte Carlo procedure we simulate the model defined on a regular square lattice and determine the threshold of species coexistence, that is, the critical phase boundaries related to the transition between an active state, where both species coexist and an absorbing state where one of the species is extinct. A finite size scaling analysis is employed to determine the order parameter, order parameter fluctuations, correlation length and the critical exponents. Our numerical results for the critical exponents agree with those of the directed percolation universality class. We also check the validity of the hyperscaling relation and present the data collapse curves.

CRITICAL BEHAVIOR OF A FOUR-STATE IRREVERSIBLE MODEL WITH POTTS SYMMETRY

1999 ◽  
Vol 13 (14) ◽  
pp. 471-477 ◽  
Author(s):  
A. BRUNSTEIN ◽  
T. TOMÉ
Keyword(s):  
Monte Carlo ◽  
Phase Transitions ◽  
Critical Behavior ◽  
Potts Model ◽  
Present Model ◽  
Universality Class ◽  
Two Dimensional ◽  
Dynamical Phase ◽  
Dynamical Phase Transitions ◽  
Symmetry Operations

We analyze the critical behavior of a two-dimensional irreversible cellular automaton whose dynamic rules are invariant under the same symmetry operations as those of the three-state Potts model. We study the dynamical phase transitions that take place in the model and obtain the static and dynamical critical exponents through Monte Carlo simulations. Our results indicate that the present model is in the same universality class as the three-state Potts model.

A Test of Scaling Theory on the Critical Isotherm

1972 ◽  
Vol 50 (24) ◽  
pp. 3117-3122 ◽  
Author(s):  
D. D. Betts ◽  
L. Filipow
Keyword(s):  
Free Energy ◽  
Ising Model ◽  
Excellent Agreement ◽  
Critical Behavior ◽  
Critical Exponents ◽  
High Field ◽  
Two Dimensional ◽  
Amplitude Ratios ◽  
Critical Amplitude ◽  
Critical Isotherm

Using recently extended data of Sykes et al. on the high field expansion of the free energy of the two-dimensional spin-1/2 Ising model the critical behavior of the magnetization and its first six temperature derivatives are examined on the critical isotherm. The estimates of the critical exponents and the critical amplitude ratios are found to be in reasonable to excellent agreement with scaling theories.

scholarly journals Critical behavior near the reversible-irreversible transition in periodically driven vortices under random local shear

2019 ◽  
Vol 9 (1) ◽  
Author(s):  
S. Maegochi ◽  
K. Ienaga ◽  
S. Kaneko ◽  
S. Okuma
Keyword(s):  
Power Law ◽  
Critical Behavior ◽  
Amorphous Film ◽  
Universality Class ◽  
Initial Position ◽  
Directed Percolation ◽  
General Phenomenon ◽  
Periodically Driven ◽  
Local Shear ◽  
Absorbing Phase Transition

Abstract When many-particle (vortex) assemblies with disordered distribution are subjected to a periodic shear with a small amplitude $${\boldsymbol{d}}$$ d , the particles gradually self-organize to avoid next collisions and transform into an organized configuration. We can detect it from the time-dependent voltage $${\boldsymbol{V}}{\boldsymbol{(}}{\boldsymbol{t}}{\boldsymbol{)}}$$ V ( t ) (average velocity) that increases towards a steady-state value. For small $${\boldsymbol{d}}$$ d , the particles settle into a reversible state where all the particles return to their initial position after each shear cycle, while they reach an irreversible state for $${\boldsymbol{d}}$$ d above a threshold $${{\boldsymbol{d}}}_{{\boldsymbol{c}}}$$ d c . Here, we investigate the general phenomenon of a reversible-irreversible transition (RIT) using periodically driven vortices in a strip-shaped amorphous film with random pinning that causes local shear, as a function of $${\boldsymbol{d}}$$ d . By measuring $${\boldsymbol{V}}{\boldsymbol{(}}{\boldsymbol{t}}{\boldsymbol{)}}$$ V ( t ) , we observe a critical behavior of RIT, not only on the irreversible side, but also on the reversible side of the transition, which is the first under random local shear. The relaxation time $${\boldsymbol{\tau }}{\boldsymbol{(}}{\boldsymbol{d}}{\boldsymbol{)}}$$ τ ( d ) to reach either the reversible or irreversible state shows a power-law divergence at $${{\boldsymbol{d}}}_{{\boldsymbol{c}}}$$ d c . The critical exponent is determined with higher accuracy and is, within errors, in agreement with the value expected for an absorbing phase transition in the two-dimensional directed-percolation universality class. As $${\boldsymbol{d}}$$ d is decreased down to the intervortex spacing in the reversible regime, $${\boldsymbol{\tau }}{\boldsymbol{(}}{\boldsymbol{d}}{\boldsymbol{)}}$$ τ ( d ) deviates downward from the power-law relation, reflecting the suppression of intervortex collisions. We also suggest the possibility of a narrow smectic-flow regime, which is predicted to intervene between fully reversible and irreversible flow.

scholarly journals CRITICAL BEHAVIOR OF AN EPIDEMIC MODEL OF DRUG RESISTANT DISEASES

2004 ◽  
Vol 15 (09) ◽  
pp. 1279-1290 ◽  
Author(s):  
C. R. DA SILVA ◽  
U. L. FULCO ◽  
M. L. LYRA ◽  
G. M. VISWANATHAN
Keyword(s):  
Critical Behavior ◽  
Finite Size ◽  
Universality Class ◽  
Active Phase ◽  
Propagation Model ◽  
Mutation Rates ◽  
Scaling Analysis ◽  
Directed Percolation ◽  
Drug Resistant ◽  
Finite Size Scaling

In this work, we study the critical behavior of an epidemic propagation model that considers individuals that can develop drug resistance. In our lattice model, each site can be found in one of the four states: empty, healthy, normally infected (not drug resistant) and strain infected (drug resistant) states. The most relevant parameters in our model are related to the mortality, cure and mutation rates. This model presents two distinct stationary active phases: a phase with co-existing normal and drug resistant infected individuals, and an intermediate active phase with only drug resistant individuals. We employed a finite-size scaling analysis to compute the critical points and the critical exponents, β/ν and 1/ν, governing the phase transitions between these active states and the absorbing inactive state. Our results are consistent with the hypothesis that these transitions belong to the directed percolation universality class.

scholarly journals Criticality between cortical states

2018 ◽  
Author(s):  
Antonio J. Fontenele ◽  
Nivaldo A. P. de Vasconcelos ◽  
Thaís Feliciano ◽  
Leandro A. A. Aguiar ◽  
Carina Soares-Cunha ◽  
...  
Keyword(s):  
Phase Transition ◽  
Cerebral Cortex ◽  
Critical Exponents ◽  
Slow Wave Sleep ◽  
Mean Field ◽  
Universality Class ◽  
Directed Percolation ◽  
Intermediate Value ◽  
Neuronal Avalanches ◽  
The Brain

Since the first measurements of neuronal avalanches [1], the critical brain hypothesis has gained traction [2]. However, if the brain is critical, what is the phase transition? For several decades it has been known that the cerebral cortex operates in a diversity of regimes [3], ranging from highly synchronous states (e.g. slow wave sleep [4], with higher spiking variability) to desynchronized states (e.g. alert waking [5], with lower spiking variability). Here, using independent signatures of criticality, we show that a phase transition occurs in an intermediate value of spiking variability. The critical exponents point to a universality class different from mean-field directed percolation (MF-DP). Importantly, as the cortex hovers around this critical point [6], it follows a linear relation between the avalanche exponents that encompasses previous experimental results from different setups [7, 8] and is reproduced by a model.

Equilibrium and nonequilibrium models on Solomon networks

2016 ◽  
Vol 27 (11) ◽  
pp. 1650134 ◽  
Author(s):  
F. W. S. Lima
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Method ◽  
Critical Exponents ◽  
Majority Vote ◽  
Universality Class ◽  
Nonequilibrium Systems ◽  
Two Dimensional ◽  
Critical Properties ◽  
The Monte Carlo Method ◽  
Regular Lattices

We investigate the critical properties of the equilibrium and nonequilibrium systems on Solomon networks. The equilibrium and nonequilibrium systems studied here are the Ising and Majority-vote models, respectively. These systems are simulated by applying the Monte Carlo method. We calculate the critical points, as well as the critical exponents ratio [Formula: see text], [Formula: see text] and [Formula: see text]. We find that both systems present identical exponents on Solomon networks and are of different universality class as the regular two-dimensional ferromagnetic model. Our results are in agreement with the Grinstein criterion for models with up and down symmetry on regular lattices.

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