We investigate properties of the particle distribution near the tip of one-dimensional branching random walks at large times t, focusing on unusual realizations in which the rightmost lead particle is very far ahead of its expected position, but still within a distance smaller than the diffusion radius ∼sqrt[t]. Our approach consists in a study of the generating function G_Δx(λ)=∑_nλ^np_n(Δx) for the probabilities p_n(Δx) of observing n particles in an interval of given size Δx from the lead particle to its left, fixing the position of the latter. This generating function can be expressed with the help of functions solving the Fisher-Kolmogorov-Petrovsky-Piscounov (FKPP) equation with suitable initial conditions. In the infinite-time and large-Δx limits, we find that the mean number of particles in the interval grows exponentially with Δx, and that the generating function obeys a nontrivial scaling law, depending on Δx and λ through the combined variable [Δx-f(λ)]^3/Δx^2, where f(λ)≡-ln(1-λ)-ln[-ln(1-λ)]. From this property, one may conjecture that the growth of the typical particle number with the size of the interval is slower than exponential, but, surprisingly enough, only by a subleading factor at large Δx. The scaling we argue is consistent with results from a numerical integration of the FKPP equation.Classical quasi-integrable systems are known to have Lyapunov times much shorter than their ergodicity time-the clearest example being the Solar System-but the situation for their quantum counterparts is less well understood. As a first example, we examine the quantum Lyapunov exponent, defined by the evolution of the four-point out-of-time-order correlator (OTOC), of integrable systems which are weakly perturbed by an external noise, a setting that has proven to be illuminating in the classical case. In analogy to the tangent space in classical systems, we derive a linear superoperator equation which dictates the OTOC dynamics. (1) We find that in the semiclassical limit the quantum Lyapunov exponent is given by the classical one it scales as ε^1/3, with ε being the variance of the random drive, leading to short Lyapunov times compared to the diffusion time (which is ∼ε^-1). (2) We also find that in the highly quantal regime the Lyapunov instability is suppressed by quantum fluctuations, and (3) for sufficiently small perturbations the ε^1/3 dependence is also suppressed-another purely quantum effect which we explain. https://www.selleckchem.com/products/4egi-1.html These essential features of the problem are already present in a rotor that is kicked weakly but randomly. Concerning quantum limits on chaos, we find that quasi-integrable systems are relatively good scramblers in the sense that the ratio between the Lyapunov exponent and kT/ℏ may stay finite at a low temperature T.The boson peak is a largely unexplained excitation found universally in the terahertz vibrational spectra of disordered systems; the so-called fracton is a vibrational excitation associated with the self-similar structure of monomers in polymeric glasses. We demonstrate that such excitations can be detected using terahertz spectroscopy. In the case of fractal structures, we determine the infrared light-vibration coupling coefficient for the fracton region and show that information concerning the fractal and fracton dimensions appears in the exponent of the absorption coefficient. Finally, using terahertz time-domain spectroscopy and low-frequency Raman scattering, we experimentally observe these universal excitations in a protein (lysozyme) system that has an intrinsically disordered and fractal structure and argue that the system should be considered a single supramolecule. These findings are applicable to amorphous and fractal objects in general and will be valuable for understanding universal dynamics of disordered systems via terahertz light.In this work, we propose a two-dimensional extension of a previously defined one-dimensional version of a model of particles in counterflowing streams, which considers an adapted Fermi-Dirac distribution to describe the transition probabilities. In this modified and extended version of the model, we consider that only particles of different species can interact, and they hop through the cells of a two-dimensional rectangular lattice with probabilities taking into account diffusive and scattering aspects. We show that for a sufficiently low level of randomness (α≥10), the system can relax to a mobile self-organized steady state of counterflow (lane formation) or to an immobile state (clogging) if the system has an average density near a certain crossover value (ρ_c). We also show that in the case of imbalance between the species, we can simultaneously have three different situations for the same density value set (i) an immobile phase, (ii) a mobile pattern organized by lanes, and (iii) a profile with mobility but without lane formation, which essentially is the coexistence of situations (i) and (ii). All of our results were obtained by performing Monte Carlo simulations.The present study is devoted to the investigation of surface anchoring and finite-size effects on nematic-smectic-A-smectic-C (N-Sm-A-Sm-C) phase transitions in free-standing films. Using an extended version of the molecular theory for smectic-C liquid crystals, we analyze how surface anchoring and film thickness affect the thermal behavior of the order parameters in free-standing smectic films. In particular, we determine how the transition temperature depends on the surface ordering and film thickness. We show that the additional orientational order imposed by the surface anchoring may lead to a stabilization of order parameters in central layers, thus modifying the nature of the phase transitions. We compare our results with experimental findings for typical thermotropic compounds presenting a N-Sm-A-Sm-C phase sequence.We study the low-temperature out-of-equilibrium Monte Carlo dynamics of the disordered Ising p-spin Model with p=3 and a small number of spin variables. We focus on sequences of configurations that are stable against single spin flips obtained by instantaneous gradient descent from persistent ones. We analyze the statistics of energy gaps, energy barriers, and trapping times on subsequences such that the overlap between consecutive configurations does not overcome a threshold. We compare our results to the predictions of various trap models finding the best agreement with the step model when the p-spin configurations are constrained to be uncorrelated.