A theoretical and experimental study on the Rayleigh-Taylor instability of a thin liquid film is presented. The experiments were performed by coating the underside of a flat beam with uniform liquid film. The size of the beam is such that a linear arrangement of drops is formed allowing the measurement of the drops growth. The experimental growth of the drops has been compared to the results obtained from a stability analysis performed on the equation that controls the thickness evolution and an excellent agreement is found. In addition, both time and length scales of the phenomenon are established.

The validity of modified finite-size scaling above the upper critical dimension is demonstrated for the quantum phase transition whose dynamical critical exponent is $z=2$. We consider the $N$-component Bose-Hubbard model, which is exactly solvable and exhibits mean-field type critical phenomena in the large-$N$ limit. The modified finite-size scaling holds exactly in that limit. However, the usual procedure, taking the large system-size limit with fixed temperature, does not lead to the expected (and correct) mean-field critical behavior because of the limited range of applicability of the finite-size scaling form. By quantum Monte Carlo simulation, it is shown that the modified finite-size scaling holds in the case of $N=1$.

We discuss the relation between the fluctuation-dissipation relation derived by Chatelain and Ricci-Tersenghi [C. Chatelain, J. Phys. A 36, 10739 (2003); F. Ricci-Tersenghi, Phys. Rev. E 68, 065104(R) (2003)] and that by Lippiello-Corberi-Zannetti [E. Lippiello, F. Corberi, and M. Zannetti, Phys. Rev. E 71, 036104 (2005)]. In order to do that, we rederive the fluctuation-dissipation relation for systems of discrete variables evolving in discrete time via a stochastic nonequilibrium Markov process. The calculation is carried out in a general formalism comprising the Chatelain, Ricci-Tersenghi, result and that by Lippiello-Corberi-Zannetti as special cases. The applicability, generality, and experimental feasibility of the two approaches are thoroughly discussed. Extending the analytical calculation to the variance of the response function, we show the advantage of field-free numerical methods with respect to the standard method, where the perturbation is applied. We also show that the signal-to-noise ratio is better (by a factor $\sqrt{2}$) in the algorithm of Lippiello-Corberi-Zannetti with respect to that of Chatelain-Ricci Tersenghi.

We report numerically and analytically estimated values for the Hurst exponent for a recently proposed non-Markovian walk characterized by amnestically induced persistence. These results are consistent with earlier studies showing that log-periodic oscillations arise only for large memory losses of the recent past. We also report numerical estimates of the Hurst exponent for non-Markovian walks with diluted memory. Finally, we study walks with a fractal memory of the past for a Thue-Morse and Fibonacci memory patterns. These results are interpreted and discussed in the context of the necessary and sufficient conditions for the central limit theorem to hold.

The morphological scaling properties of linear polymer films grown by vapor deposition polymerization are studied by $1+1\text{D}$ Monte Carlo simulations. The model implements the basic processes of random angle ballistic deposition $\left(F\right)$, free-monomer diffusion $\left(D\right)$ and monomer adsorption along with the dynamical processes of polymer chain initiation, extension, and merger. The ratio $G=D/F$ is found to have a strong influence on the polymer film morphology. Spatial and temporal behavior of kinetic roughening has been extensively studied using finite-length scaling and height-height correlations $H\left(r,t\right)$. The scaling analysis has been performed within the no-overhang approximation and the scaling behaviors at local and global length scales were found to be very different. The global and local scaling exponents for morphological evolution have been evaluated for varying free-monomer diffusion by growing the films at $G=10$, ${10}^2$, ${10}^3$, and ${10}^4$ and fixing the deposition flux $F$. With an increase in $G$ from 10 to ${10}^4$, the average growth exponent $\beta \approx 0.50$ was found to be invariant, whereas the global roughness exponent ${\alpha }_g$ decreased from 0.87 (1) to 0.73 (1) along with a corresponding decrease in the global dynamic exponent $z_g$ from 1.71(1) to 1.38(2). The global scaling exponents were observed to follow the dynamic scaling hypothesis, $z_g={\alpha }_g/\beta$. With a similar increase in $G$ however, the average local roughness exponent ${\alpha }_l$ remained close to 0.46 and the anomalous growth exponent ${\beta }_{\ast }$ decreased from 0.23(4) to 0.18(8). The interfaces display anomalous scaling and multiscaling in the relevant height-height correlations. The variation in $H\left(r,t\right)$ with deposition time $t$ indicates nonstationary growth. A comparison has been made between the simulational findings and the experiments wherever applicable.

X-ray diffraction and proton NMR relaxation measurements were carried out on two liquid-crystalline organosiloxane tetrapodes with side-on mesogenic groups, exhibiting nematic and smectic-$C$ phases, and on a monomeric analog. Packing models for the mesophases exhibited by these systems are proposed on the basis of x-ray diffraction data. As a consequence of microsegregation, the aromatic cores are packed in between two sublayers formed by a mixture of interdigitated aliphatic and siloxane chains. The mixed sublayers are characteristic for the tetrapodes with side-on mesogenic groups presented in this work and have not been observed in tetrapodes with terminally attached mesogens. The tilt angle in the smectic-$C$ phase is found very large, i.e., $\sim 61°–62°$. Notably, smectic-$C$ clusters are present also in the whole temperature range of the nematic phase. NMR relaxometry yields $T_1^{-1}$ dispersions clearly different from those of conventional calamitics. The influence of molecular tendency to form interdigitated structures is evidenced by frequency-dependent relaxation rate in the isotropic phase—indicating the presence of ordered clusters far above the phase transition—and by the diminished role of molecular self-diffusion in ordered phases. Nematiclike director fluctuations are the dominating relaxation mechanism whereas the translational displacements are strongly hindered by the interdigitation of dendrimer arms.

We introduce an external control to reduce the size of avalanches in some sandpile models exhibiting self-organized criticality. This rather intuitive approach seems to be missing in the vast literature on such systems. The control action, which amounts to triggering avalanches in sites that are near to become critical, reduces the probability of very large events, so that energy dissipation occurs most locally. The control is applied to a directed Abelian sandpile model driven by both uncorrelated and correlated depositions. The latter is essential to design an efficient and simple control heuristic, but has only small influence in the uncontrolled avalanche probability distribution. The proposed control seeks a trade-off between control cost and large event risk. Preliminary results hint that the proposed control works also for an undirected sandpile model.

Two-dimensional turbulence generated in a finite box produces large-scale coherent vortices coexisting with small-scale fluctuations. We present a rigorous theory explaining the $\eta =1/4$ scaling in the $V\propto r^{-\eta }$ law of the velocity spatial profile within a vortex, where $r$ is the distance from the vortex center. This scaling, consistent with earlier numerical and laboratory measurements, is universal in its independence of details of the small-scale injection of turbulent fluctuations and details of the shape of the box.

We study the stability of unstable steady states in scalar retarded time-delayed systems subjected to a variable-delay feedback control. The important aspect of such a control problem is that time-delayed systems are already infinite-dimensional before the delayed feedback control is turned on. When the frequency of the modulation is large compared to the system’s dynamics, the analytic approach consists of relating the stability properties of the resulting variable-delay system with those of an analogous distributed-delay system. Otherwise, the stability domains are obtained by a numerical integration of the linearized variable-delay system. The analysis shows that the control domains are significantly larger than those in the usual time-delayed feedback control, and that the complexity of the domain structure depends on the form and the frequency of the delay modulation.

We focus on two approaches that have been proposed in recent years for the explanation of the so-called Fermi-Pasta-Ulam (FPU) paradox, i.e., the persistence of energy localization in the “low-$q$” Fourier modes of Fermi-Pasta-Ulam nonlinear lattices, preventing equipartition among all modes at low energies. In the first approach, a low-frequency fraction of the spectrum is initially excited leading to the formation of “natural packets” exhibiting exponential stability, while in the second, emphasis is placed on the existence of “$q$ breathers,” i.e., periodic continuations of the linear modes of the lattice, which are exponentially localized in Fourier space. Following ideas of the latter, we introduce in this paper the concept of “$q$-tori” representing exponentially localized solutions on low-dimensional tori and use their stability properties to reconcile these two approaches and provide a more complete explanation of the FPU paradox.

The linearized R13 and R26 moment equations are used to study Kramers’ problem. Analytical solutions for the defect velocity and slip coefficient are derived and compared with numerical results from the kinetic theory. It is found that the linearized R26 equations can capture the Knudsen layer fairly accurately in terms of the defect velocity and slip coefficient, while the linearized R13 equations underpredict the kinetic data. At the wall, however, the kinetic models predict a slightly higher value for the defect velocity than the linearized R26 equations. In general, the linearized R26 equations perform well for both specular and diffusive walls.

We have performed molecular-dynamics simulations of dynamic compression waves propagating through amorphous carbon using the Tersoff potential and find that a variety of dynamic compression features appear for two different initial densities. These features include steady elastic shocks, steady chemically reactive shocks, unsteady elastic waves, and unsteady chemically reactive waves. We show how these features can be distinguished by analyzing time-dependent propagation speeds, time-dependent sound speeds, and comparison to multiscale shock technique (MSST) simulations. Understanding such features is a key challenge in quasi-isentropic experiments involving phase transformations. In addition to direct simulations of dynamic compression, we employ the MSST and find agreement with the direct method for this system for the shocks observed. We show how the MSST can be extended to include explicit material viscosity and demonstrate on an amorphous Lennard-Jones system.

We discuss convergence and coupling of Markov chains, and present general relations between the transfer matrices describing these two processes. We then analyze a recently developed local-patch algorithm, which computes rigorous upper bound for the coupling time of a Markov chain for nontrivial statistical-mechanics models. Using the “coupling from the past” protocol, this allows one to exactly sample the underlying equilibrium distribution. For spin glasses in two and three spatial dimensions, the local-patch algorithm works at lower temperatures than previous exact-sampling methods. We discuss variants of the algorithm which might allow one to reach, in three dimensions, the spin-glass transition temperature. The algorithm can be adapted to hard-sphere models. For two-dimensional hard disks, the algorithm allows us to draw exact samples at higher densities than previously possible.

It is known that a single product shock measure in some of the one-dimensional driven-diffusive systems with nearest-neighbor interactions might evolve in time quite similarly to a random walker moving on a one-dimensional lattice with reflecting boundaries. The nonequilibrium steady state of the system in this case can be written in terms of a linear superposition of such uncorrelated shocks. Equivalently, one can write the steady state of this system using a matrix-product approach with two-dimensional matrices. In this paper, we present an equilibrium two-dimensional one-transit walk model and find its partition function using a transfer matrix method. We will show that there is a direct connection between the partition functions of these two systems. We will explicitly show that in the steady state, the transfer matrix of the one-transit walk model is related to the matrix representation of the algebra of the driven-diffusive model through a similarity transformation. The physical quantities are also related through the same transformation.

The fluctuation-dissipation theorem (FDT) is very general and applies to a broad variety of different physical phenomena in condensed matter physics. With the help of the FDT and following the famous work of Caldeira and Leggett, we show that, whenever linear response theory applies, any generic bosonic or fermionic system at finite temperature $T$ can be mapped onto a fictitious system of free-harmonic oscillators. To the best of our knowledge, this is the first time that such a mapping is explicitly worked out. This finding provides further theoretical support to the phenomenological harmonic oscillator models commonly used in condensed matter. Moreover, our result helps in clarifying an interpretation issue related to the presence and physical origin of the Bose-Einstein factor in the FDT.

We study the nonlinear driven response and sliding friction behavior of the phase-field-crystal (PFC) model with pinning including both thermal fluctuations and inertial effects. The model provides a continuous description of adsorbed layers on a substrate under the action of an external driving force at finite temperatures, allowing for both elastic and plastic deformations. We derive general stochastic dynamical equations for the particle and momentum densities including both thermal fluctuations and inertial effects. The resulting coupled equations for the PFC model are studied numerically. At sufficiently low temperatures, we find that the velocity response of an initially pinned commensurate layer shows hysteresis with dynamical melting and freezing transitions for increasing and decreasing applied forces at different critical values. The main features of the nonlinear response in the PFC model are similar to the results obtained previously with molecular dynamics simulations of particle models for adsorbed layers.

Populations represented by collections of points scattered randomly on the real line are ubiquitous in science and engineering. The statistical modeling of such populations leads naturally to Poissonian populations—Poisson processes on the real line with a distinguished maximal point. Poissonian populations are infinite objects underlying key issues in statistical physics, probability theory, and random fractals. Due to their infiniteness, measuring the diversity of Poissonian populations depends on the lower-bound cut-off applied. This research characterizes the classes of Poissonian populations whose diversities are invariant with respect to the cut-off level applied and establishes an elemental connection between these classes and extreme-value theory. The measures of diversity considered are variance and dispersion, Simpson’s index and inverse participation ratio, Shannon’s entropy and Rényi’s entropy, and Gini’s index.

Three recently proposed hyperelastic models for granular materials are compared with experiment data. Though all three are formulated to give elastic moduli that are power law functions of the mean stress, they have rather different dependencies on individual stresses, and generally differ from well established experimental forms. Predicted static stress distributions are in qualitative agreement with experiments, but do not differ greatly from isotropic linear elasticity, and similarly fail to account for variability in experiment data that presumably occurs due to a preparation dependence of granular materials.

We report surface shear rheological measurements on monolayers of silica nanoparticles at the air-water interface. We have used the method of strain-rate frequency superposition (SRFS) to characterize the structural relaxation. We show that the rheological properties of the layers have the same universal linear and nonlinear behavior as three-dimensional soft materials. We also discuss the original healing properties of these monolayers.