Effect of a Mode of Update on Universality Class for Coupled Logistic Maps: Directed Ising to Ising Class

Ising model at zero temperature leads to a ferromagnetic state asymptotically. There are two such possible states linked by symmetry, and Glauber–Ising dynamics are employed to reach them. In some stochastic or deterministic dynamical systems, the same absorbing state with [Formula: see text] symmetry is reached. This transition often belongs to the directed Ising (DI) class where dynamic exponents and persistence exponent are different. In asymmetrically coupled sequentially updated logistic maps, the transition belongs to the DI class. We study changes in the nature of transition with an update scheme. Even with the synchronous update, the transition still belongs to the DI class. We also study a synchronous probabilistic update scheme in which each site is updated with the probability [Formula: see text]. The order parameter decays with an exponent [Formula: see text] in this scheme. Nevertheless, the dynamic exponent [Formula: see text] is less than [Formula: see text] even for small values of [Formula: see text] indicating a very slow crossover to the Ising class. However, with a random asynchronous update, we recover [Formula: see text]. In the presence of feedback, synchronous update leads to a transition in the DI universality class which changes to Ising class for synchronous probabilistic update.

From random walk to critical dynamics

Chapter 22 studies stochastic dynamical equations, consistent with the detailed balance condition, which are generalized Langevin equations which describe a wide range of phenomena from Brownian motion to critical dynamics in continuous phase transitions. In the latter case, a dynamic action can be associated to the Langevin equation, which can be renormalized with the help of BRST symmetry. Dynamic renormalization group equations, describing critical dynamics, are then derived. Dynamic scaling follows, with a correlation time that exhibits critical slowing down governed by a dynamic exponent. In addition, Jarzinsky’s relation is derived in the case of a time–dependent driving force.

ELECTRODYNAMICS à la HOŘAVA

We study an electrodynamics consistent with anisotropic transformations of spacetime with an arbitrary dynamic exponent z. The equations of motion and conserved quantities are explicitly obtained. We show that the propagator of this theory can be regarded as a quantum correction to the usual propagator. Moreover, we obtain that both the momentum and angular momentum are not modified, but their conservation laws do change. We also show that in this theory the speed of light and the electric charge are modified with z. The magnetic monopole in this electrodynamics and its duality transformations are also investigated. For that we found that there exists a dual electrodynamics, with higher derivatives in the electric field, invariant under the same anisotropic transformations.

IS IT POSSIBLE TO RELATE MOND WITH HOŘAVA GRAVITY?

In this work we present a scalar field theory invariant under space-time anisotropic transformations with a dynamic exponent z. It is shown that this theory possesses symmetries similar to Hořava gravity and that in the limit z = 0 the equations of motion of the non-relativistic MOND theory are obtained. This result allow us to conjecture the existence of a Hořava type gravity that in the limit z = 0 is consistent with MOND.

DYNAMICAL BOUNDS FOR STURMIAN SCHRÖDINGER OPERATORS

The Fibonacci Hamiltonian, that is a Schrödinger operator associated to a quasiperiodical Sturmian potential with respect to the golden mean has been investigated intensively in recent years. Damanik and Tcheremchantsev developed a method in [10] and used it to exhibit a non trivial dynamical upper bound for this model. In this paper, we use this method to generalize to a large family of Sturmian operators dynamical upper bounds and show at sufficently large coupling anomalous transport for operators associated to irrational number with a generic diophantine condition. As a counterexample, we exhibit a pathological irrational number which does not verify this condition and show its associated dynamic exponent only has ballistic bound. Moreover, we establish a global lower bound for the lower box counting dimension of the spectrum that is used to obtain a dynamical lower bound for bounded density irrational numbers.

Quantitative analysis of time-series fluorescence microscopy using a spot tracking method: application to Min protein dynamics

The dynamics of MinD protein has been recognized as playing an important role in the accurate positioning of the septum during cell division. In this work, spot tracking technique (STT) was applied to track the motion and quantitatively characterize the dynamic behavior of green fluorescent protein-labeled MinD (GFP-MinD) in an Escherichia coli system. We investigated MinD dynamics on the level of particle ensemble or cluster focusing on the position and motion of the maximum in the spatial distribution of MinD proteins. The main results are twofold: (i) a demonstration of how STT could be an acceptable tool for MinD dynamics studies; and (ii) quantitative findings with parametric and non-parametric analyses. Specifically, experimental data monitored from the dividing E. coli cells (typically 4.98 ± 0.75 µm in length) has demonstrated a fast oscillation of the MinD protein between the two poles, with an average period of 54.6 ± 8.6 s. Observations of the oscillating trajectory and velocity show a trapping or localized behavior of MinD around the polar zone, with average localization velocity of 0.29 ± 0.06 µm/s; and flight switching was observed at the pole-to-pole leading edge, with an average switching velocity of 2.95 ± 0.31 µm/s. Subdiffusive motion of MinD proteins at the polar zone was found and investigated with the dynamic exponent, α of 0.34 ± 0.16. To compare with the Gaussian-based analysis, non-parametric statistical analysis and noise consideration were also performed.