Probabilistic initial value problem for cellular automaton rule 172

We consider the problem of computing a response curve for binary cellular automata -- that is, the curve describing the dependence of the density of ones after many iterations of the rule on the initial density of ones. We demonstrate how this problem could be approached using rule 130 as an example. For this rule, preimage sets of finite strings exhibit recognizable patterns, and it is therefore possible to compute both cardinalities of preimages of certain finite strings and probabilities of occurrence of these strings in a configuration obtained by iterating a random initial configuration $n$ times. Response curves can be rigorously calculated in both one- and two-dimensional versions of CA rule 130. We also discuss a special case of totally disordered initial configurations, that is, random configurations where the density of ones and zeros are equal to 1/2.Comment: 13 pages, 3 figure

Exponential convergence to equilibrium in cellular automata asymptotically emulating identity

We consider the problem of finding the density of 1's in a configuration obtained by $n$ iterations of a given cellular automaton (CA) rule, starting from disordered initial condition. While this problems is intractable in full generality for a general CA rule, we argue that for some sufficiently simple classes of rules it is possible to express the density in terms of elementary functions. Rules asymptotically emulating identity are one example of such a class, and density formulae have been previously obtained for several of them. We show how to obtain formulae for density for two further rules in this class, 160 and 168, and postulate likely expression for density for eight other rules. Our results are valid for arbitrary initial density. Finally, we conjecture that the density of 1's for CA rules asymptotically emulating identity always approaches the equilibrium point exponentially fast.Comment: 20 pages, 4 figures, 2 table

Random Quantum Circuits

Quantum circuits -- built from local unitary gates and local measurements -- are a new playground for quantum many-body physics and a tractable setting to explore universal collective phenomena far-from-equilibrium. These models have shed light on longstanding questions about thermalization and chaos, and on the underlying universal dynamics of quantum information and entanglement. In addition, such models generate new sets of questions and give rise to phenomena with no traditional analog, such as new dynamical phases in quantum systems that are monitored by an external observer. Quantum circuit dynamics is also topical in view of experimental progress in building digital quantum simulators that allow control of precisely these ingredients. Randomness in the circuit elements allows a high level of theoretical control, with a key theme being mappings between real-time quantum dynamics and effective classical lattice models or dynamical processes. Many of the universal phenomena that can be identified in this tractable setting apply to much wider classes of more structured many-body dynamics.Comment: Review article for Annual Review of Condensed Matter Physics; comments welcom

An exploration and validation of computer modeling of evolution, natural selection, and evolutionary biology with cellular automata for secondary students.

The Evolutionary Tool Kit, a new software package, is the prototype of a concept simulator providing an environment for students to create microworlds of populations of artificial organisms. Its function is to model processes, concepts and arguments in natural selection and evolutionary biology, using either Mendelian asexual or sexual reproduction, or counterfactual systems such as \u27paint pot\u27 or blending inheritance. In this environment students can explore a conceptual What if? in evolutionary biology, test misconceptions and deepen understanding of inheritance and changes in populations. Populations can be defined either with typological, or with populational thinking, to inquire into the role and necessity of variation in natural selection. The approach is generative not tutorial. The interface is highly graphic with twenty traits set as icons that are moved onto the \u27phenotypes\u27. Activities include investigations of evolutionary theory of aging, reproductive advantage, sexual selection and mimicry. Design of the activities incorporates Howard Gardner\u27s Theory of Multiple Intelligences. Draft of a teacher and student manual are included