Sensitivity to noise and ergodicity of an assembly line of cellular automata that classifies density

We investigate the sensitivity of the composite cellular automaton of H. Fuk\'{s} [Phys. Rev. E 55, R2081 (1997)] to noise and assess the density classification performance of the resulting probabilistic cellular automaton (PCA) numerically. We conclude that the composite PCA performs the density classification task reliably only up to very small levels of noise. In particular, it cannot outperform the noisy Gacs-Kurdyumov-Levin automaton, an imperfect classifier, for any level of noise. While the original composite CA is nonergodic, analyses of relaxation times indicate that its noisy version is an ergodic automaton, with the relaxation times decaying algebraically over an extended range of parameters with an exponent very close (possibly equal) to the mean-field value.Comment: Typeset in REVTeX 4.1, 5 pages, 5 figures, 2 tables, 1 appendix. Version v2 corresponds to the published version of the manuscrip

Simply modified GKL density classifiers that reach consensus faster

The two-state Gacs-Kurdyumov-Levin (GKL) cellular automaton has been a staple model in the study of complex systems due to its ability to classify binary arrays of symbols according to their initial density. We show that a class of modified GKL models over extended neighborhoods, but still involving only three cells at a time, achieves comparable density classification performance but in some cases reach consensus more than twice as fast. Our results suggest the time to consensus (relative to the length of the CA) as a complementary measure of density classification performance.Comment: Short note, 3 pages, 1 table, 2 composite figures, 18 reference

Electromagnetic surface wave propagation in a metallic wire and the Lambert WW function

We revisit the solution due to Sommerfeld of a problem in classical electrodynamics, namely, that of the propagation of an electromagnetic axially symmetric surface wave (a low-attenuation single TM$_{01}$ mode) in a cylindrical metallic wire, and his iterative method to solve the transcendental equation that appears in the determination of the propagation wave number from the boundary conditions. We present an elementary analysis of the convergence of Sommerfeld's iterative solution of the approximate problem and compare it with both the numerical solution of the exact transcendental equation and the solution of the approximate problem by means of the Lambert $W$ function.Comment: REVTeX double column, 9 pages, 3 figures, minor differences between v3 and published version; "Editor's Pick" for June 2019 edition of AJ

Efficient generation of random derangements with the expected distribution of cycle lengths

We show how to generate random derangements efficiently by two different techniques: random restricted transpositions and sequential importance sampling. The algorithm employing restricted transpositions can also be used to generate random fixed-point-free involutions only, a.k.a. random perfect matchings on the complete graph. Our data indicate that the algorithms generate random samples with the expected distribution of cycle lengths, which we derive, and for relatively small samples, which can actually be very large in absolute numbers, we argue that they generate samples indistinguishable from the uniform distribution. Both algorithms are simple to understand and implement and possess a performance comparable to or better than those of currently known methods. Simulations suggest that the mixing time of the algorithm based on random restricted transpositions (in the total variance distance with respect to the distribution of cycle lengths) is $O(n^{a}\log{n}^{2})$ with $a \simeq \frac{1}{2}$ and $n$ the length of the derangement. We prove that the sequential importance sampling algorithm generates random derangements in $O(n)$ time with probability $O(1/n)$ of failing.Comment: This version corrected and updated; 14 pages, 2 algorithms, 2 tables, 4 figure

The influence of temporal coherence on the dynamical Casimir effect

We study the dynamical Casimir effect in the presence of a finite coherence time, which is associated with a finite quality factor of the optical cavity. We use the time refraction model, where a fixed cavity with a modulated optical medium, replaces the empty cavity with a vibrating mirror. Temporal coherence is described with the help of cavity quasi-mode operators. Asymptotic expressions for the number of photon pairs generated from vacuum are derived.Comment: 8 pages, 1 figur

Vacuum effects in a vibrating cavity: time refraction, dynamical Casimir effect, and effective Unruh acceleration

Two different quantum processes are considered in a perturbed vacuum cavity: time refraction and dynamical Casimir effect. They are shown to be physically equivalent, and are predicted to be unstable, leading to an exponential growth in the number of photons created in the cavity. The concept of an effective Unruh acceleration for these processes is also introduced, in order to make a comparison in terms of radiation efficiency, with the Unruh radiation associated with an accelerated frame in unbounded vacuum.Comment: 5 pages, version to appear in Physics Letters