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Critical behavior in a model of random walkers in processes with conservation of parity, branching, multiple occupation and annihilation.

Fabiana Carvalho dos Anjos

Published Jul 27, 2016• 0 citations
OpenAlex

Abstract

In this dissertation we address the problem of second order phase transitions with absorbing states. We analyze the critical behavior of a one dimensional model simulated on one dimensional chains. The sites on the linear chain can be multiple occupied by particles that annhilates each other with a finite probability. It also presents parity-conserving branching with an even number of offsprings. The number of particles are constant modulo 2. We determine the threshold of the phase transition between the statistically stationary active state and the absorbing state. From steady-state simulations and a finite-size scaling analysis, we calculate the order-parameter, order-parameter fluctuations, and special correlation length critical exponents. We also followed the short-time critical relaxation to determine some dynamical critical exponents. The hyperscaling relation is checked for both sets of stationary and dynamical critical exponents which we found to be consistent with the BARW-PC universality class.

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In this dissertation we address the problem of second order phase transitions with absorbing states. We analyze the critical behavior of a one dimensional model simulated on one dimensional chains. The sites on the linear chain can be multiple occupied by particles that annhilates each other with a finite probability. It also presents parity-conserving branching with an even number of offsprings. The number of particles are constant modulo 2. We determine the threshold of the phase transition between the statistically stationary active state and the absorbing state. From steady-state simulations and a finite-size scaling analysis, we calculate the order-parameter, order-parameter fluctuations, and special correlation length critical exponents. We also followed the short-time critical relaxation to determine some dynamical critical exponents. The hyperscaling relation is checked for both sets of stationary and dynamical critical exponents which we found to be consistent with the BARW-PC universality class.
In this work, we investigate the critical behavior of a model describing the parity-conserving branching and annihilating process of random walkers. The model is simulated on a one dimensional lattice on which the sites can be occupied by multiple particles with a finite annhilation probability. We determine the threshold of the phase transition between the statistically stationary active state and the absorbing state. From steady-state simulations and a finite-size scaling analysis, we calculate the order-parameter, order-parameter fluctuations, and spacial correlation length critical exponents. Further, we follow the short-time critical relaxation to provide a set of relevant dynamical critical exponents. We check the validity of the hyperscaling relation for both sets of stationary and dynamical critical exponents. These are consistent with the BARW-PC universality class.
Recently, Takayasu and Tretyakov [Phys. Rev. Lett. 68, 3060 (1992)] studied branching annihilating random walks (BAWs) with n=1--5 offspring. These models exhibit a continuous phase transition to an absorbing state. For odd n the models belong to the universality class of directed percolation. For even n the particle number is conserved modulo 2 and the critical behavior is not compatible with directed percolation. In this article I study the BAW with n=4 using time-dependent simulations and finite-size scaling, obtaining precise estimates for various critical exponents. The results are consistent with the conjecture: \ensuremath{\beta}/${\ensuremath{\nu}}_{\mathrm{\ensuremath{\perp}}}$=1/2, ${\ensuremath{\nu}}_{\mathrm{\ensuremath{\parallel}}}$/${\ensuremath{\nu}}_{\mathrm{\ensuremath{\perp}}}$=7/4, \ensuremath{\gamma}=0, \ensuremath{\delta}=2/7, \ensuremath{\eta}=0, and ${\mathrm{\ensuremath{\delta}}}_{\mathit{h}}$=9/2. These critical exponents characterize, respectively, the dependence of the order parameter (\ensuremath{\beta}/${\ensuremath{\nu}}_{\mathrm{\ensuremath{\perp}}}$) and relaxation time (${\ensuremath{\nu}}_{\mathrm{\ensuremath{\parallel}}}$/${\ensuremath{\nu}}_{\mathrm{\ensuremath{\perp}}}$) on system size, the growth of fluctuations (\ensuremath{\gamma}) close to the critical point, the long-time behavior of the probability of survival (\ensuremath{\delta}) and average number of particles (\ensuremath{\eta}) when starting at time zero with just two particles, and finally the decay of the order parameter (${\mathrm{\ensuremath{\delta}}}_{\mathit{h}}$) at the critical point in the presence of an external source.
Recently, Takayasu and Tretyakov [Phys. Rev. Lett. 68, 3060 (1992)] studied branching annihilating random walks with n=1-5 offsprings. These models exhibit a continuous phase transition to an absorbing state. Steady-state simulations yielded an estimate for the order parameter critical exponent \ensuremath{\beta} different from that of directed percolation. This result is quite surprising, as the universality class of directed percolation is known to be very robust. I have studied the critical behavior of the one-dimensional model with n=1 and 3 using time-dependent Monte Carlo simulations, and determined three critical exponents, all of which are in agreement with directed percolation.
We study a one-dimensional, nonequilibrium Potts-like model that has q symmetric absorbing states. For q=2, as expected, the model belongs to the parity-conserving universality class. For q=3 the critical behavior depends on the dynamics of the model. Under a certain dynamics it remains generically in the active phase, which is also the feature of some other models with three absorbing states. However, a modified dynamics induces a parity-conserving phase transition. Relations with branching-annihilating random walk models are discussed in order to explain such a behavior.
Many non-equilibrium systems display dynamic phase transitions from active to absorbing states, where fluctuations cease entirely. Based on a field theory representation of the master equation, the critical behavior can be analyzed by means of the renormalization group. The resulting universality classes for single-species systems are reviewed here. Generically, the critical exponents are those of directed percolation (Reggeon field theory), with critical dimension d_c = 4. Yet local particle number parity conservation in even-offspring branching and annihilating random walks implies an inactive phase (emerging below d_c' = 4/3) that is characterized by the power laws of the pair annihilation reaction, and leads to different critical exponents at the transition. For local processes without memory, the pair contact process with diffusion represents the only other non-trivial universality class. The consistent treatment of restricted site occupations and quenched random reaction rates are important open issues.
Abstract We study the critical order parameter fluctuations of the absorbing-state phase-transition exhibited by branching and annihilating random walkers performing anomalous diffusion in a linear chain. The diffusion process is considered to follow a power-law distribution of jump lengths with a typical decay exponent α . We focus in the case of parity conserving dynamics for which deviations from the usual directed percolation universality class have been previously demonstrated even for the limiting cases of normal diffusion. Anomalous diffusion induces a continuous change of the critical exponents. By performing a finite-size scaling analysis of simulation data, we show that the critical order parameter moment ratio also varies continuously with α . We unveil that the critical order parameter distribution evolves from a nearly Gaussian to an exponential form as the range of the jump distribution is increased up to the limit α = 5 / 2 on which the active state predominates for any finite branching probability.