Q2R Cellular Automata Research Articles

Equilibrium and nonequilibrium in the three-dimensional Q2R cellular automata

I study the equilibrium and nonequilibrium dynamics of a conservative and reversible Q2R cellular automata. This system exhibits a configuration space with 2^{2N} states, which grows with the size of the system. In this context, for small size, the phase space has fixed points and cycles. Through numerical studies and using a statistical approach, I can observe stable and unstable behaviors as well as a phase transition around a critical energy E_{c}. I introduce a coupling constant as a perturbation to the classic Q2R model and show through the phase diagram how this modified model exhibits three different phases.

The Approach Towards Equilibrium in a Reversible Ising Dynamics Model: An Information-Theoretic Analysis Based on an Exact Solution

We study the approach towards equilibrium in a dynamic Ising model, the Q2R cellular automaton, with microscopic reversibility and conserved energy for an infinite one-dimensional system. Starting from a low-entropy state with positive magnetisation, we investigate how the system approaches equilibrium characteristics given by statistical mechanics. We show that the magnetisation converges to zero exponentially. The reversibility of the dynamics implies that the entropy density of the microstates is conserved in the time evolution. Still, it appears as if equilibrium, with a higher entropy density is approached. In order to understand this process, we solve the dynamics by formally proving how the information-theoretic characteristics of the microstates develop over time. With this approach we can show that an estimate of the entropy density based on finite length statistics within microstates converges to the equilibrium entropy density. The process behind this apparent entropy increase is a dissipation of correlation information over increasing distances. It is shown that the average information-theoretic correlation length increases linearly in time, being equivalent to a corresponding increase in excess entropy.

Irreversible statistical mechanics from reversible motion: Q2R automata

Q2R cellular automata are a nice pedagogical example for addressing a century-old question discussed at the 20th IUPAP Statistical Physics Conference in Paris last July: How is thermodynamic irreversibility, as seen in entropy increase, compatible with microscopic reversibility, as in Newton's equations of motion? The Boltzmann–Zermelo argument says that the times for returning to the low-entropy initial state of a large system are far longer than the age of the universe. How can we test this assertion? Molecular dynamics in its usual form is inadequate to address this question, since arithmetic rounding errors as well as discretization of space and time preclude return to the exact initial configuration. Monte Carlo simulations of Ising models avoid such errors but require random numbers and are therefore not reversible in the usual sense. The Q2R update rules of cellular automata for microcanonical Ising models are reversible and the system returns exactly to the initial configuration after an exponentially long time. Computer simulation of Q2R cellular automata help us to understand some very old fundamental problems. This property will be reviewed, as well well as unexplained critical exponents obtained in large-scale simulations.

Ising Cellular Automata for Proton Diffusion on Protein Surfaces

Q2R cellular automata are combined with Kawasaki dynamics to give an Ising ferromagnet with conserved magnetization and conserved energy. In its lattice gas interpretation, it may serve to describe the diffusion of hydrogen ions on the surface of globular proteins. Our computer simulations test to what extent the phase transition at the Curie point is still visible in small 10×10 lattices and t~104.

Critical 2D and 3D Dynamics of Q2R Cellular Automata

Simulation of critical relaxation of the magnetization in a microcanonical Ising model agrees with the expected exponent z ≃2 after very long times on the square lattice, while on the simple cubic lattice z≃3.3 seems different.

Numerical method for analyzing surface fluctuations

A new method for measuring the width of an interface in the presence of overhangs and holes is presented, using a local Gibbs dividing surface. It is compared with the conventional method based on the order-parameter profile. Dynamics and size dependence of an interface are studied numerically in the two dimensional Q2R cellular automaton. Two types of interface fluctuations can be distinguished: the capillary waves dominate on length scales larger than the bulk correlation length, whereas order parameter fluctuations on small scales give rise to an intrinsic interface width. The scaling behavior of the capillary waves is shown to be the same as for the Ising model. The divergence of the intrinsic width at the critical point and its critical slowing down are investigated.

Q2R simulations of two-dimensional spin glasses

Q2R cellular automata are shown to attain thermal equilibration in the two-dimensional Edwards-Anderson model (1975) of a spin glass. Some problems associated with equilibration at low temperatures are identified as finite-size effects. The authors use a thermalization test due to Bhatt and Young (1988). Their results indicate the possibility of calculating Gibbs averages in this model by means of the Q2R algorithm. A rough efficiency comparison with the conventional (Metropolis) Monte Carlo algorithm is also made.

Spin dynamics confinement in Q2R

For low concentrations, the occupied sites in Q2R cellular automata on the square lattice are known to be confined to the smallest rectangles surrounding them. However, they do not necessarily fill out the whole rectangle allowed for them. The effective confinement transition is found numerically to agree well with that of bootstrap percolation.

Damage spreading in the Q2R Ising model

We find evidence of metastability in the spreading of damage in the Q2R cellular automaton approximation for a 2D Ising system which is first equilibrated with a standard Metropolis simulation. A subsequent study of both single-site and whole-line damage spreading in Metropolis/Q2R systems as well as in pure Q2R systems shows a logarithmic dependence of the damage-spreading threshold on time, and saturation effects in single-site damage suggest a transition in the damage-spreading threshold at low initial energy values. Examination of the magnetization as a function of energy in pure Q2R shows extremely long relaxation times for p < p c , which may bear some relation to the extremely slow convergence of the damage-spreading threshold.

On the evaluation of magnetisation fluctuations with Q2R cellular automata

The authors discuss the evaluation of fluctuations with the Q2R cellular automata. Explicit calculations are performed for the two-dimensional case. They argue that when Q2R results are compared properly with Monte Carlo simulations there are no inconsistencies at low temperatures as claimed recently.

Multitasking case study on the cray-2: The Q2R cellular automaton

The Q2R cellular automaton which may be used for microcanonical simulation of the Ising model is implemented in parallel on the four processors of a Cray-2 supercomputer. The simulation speed is 4.3 GHz as opposed to 670 MHz for the previously fastest reported implementation using one processor of a Cray-XMP. We also simulated the largest Ising system ever studied, 15, 130, 968, 192 spins. A number of subtleties in implementing and optimizing the parallel algorithm are discussed, as well as problems in defining performance measurements. Performance and results from the Q2R algorithm are compared to those from the standard Metropolis algorithm, and a number of dynamic exponents are measured for the first time.

Dynamics of interface width in the three-dimensional Q2R Ising model

The Q2R cellular automaton is used to simulate the growth of the liquid-vapour interface in the 3D Ising model. At the critical point the authors find that its width increases with time in a power law manner with an exponent of 0.345+or-0.015; below the critical temperature they study the size dependence of the equilibrium width.

Energy transport in a cellular automaton

Energy transport in the deterministic Q2R cellular automaton is studied. Two different types of transport processes are found: a diffusion type and a very efficient transport on “highways.” The dependence of both types on the energy is investigated.