An analytical approach to determining the mean avalanche size in a broad class of dynamical models on random networks is introduced. Previous results on percolation transitions and epidemic sizes are shown to be special cases of the method. The time-dependence of cascades and extensions to networks with community structure or degree-degree correlations are discussed. Analytical results for the rate of spread of innovations in a modular network and for the size of $k$ cores in networks with degree-degree correlations are confirmed with numerical simulations.

The classical problem of the diffusion of heat in a homogeneous medium is reexamined, the medium being confined by fixed boundaries maintained at a fixed temperature. When the thermal diffusivity is small, the relaxation of the temperature of the medium to that of the boundary proceeds on two time scales, one associated with a lightly damped high-frequency acoustic mode and the other with an aperiodically damped diffusive mode. Considering for simplicity a spherical configuration, it is shown that the latter does not obey the classical linear heat conduction equation.

We have verified some results of Nana and Woafo [Phys. Rev. E 74, 046213 (2006)] in the area of the complete synchronization. We have found that the motion of the van der Pol network is quasiperiodic, not chaotic as the authors have written.

The validity of desynchronization in Fig. 2 [B. Nana and P. Woafo, Phys. Rev. E 74, 046213 (2006)] is justified by the fact that the oscillators are nonidentical (instead of identical as quoted in Perlikowski et al.) and by the role of phase differences in synchronization of self-sustained oscillators.

The effects of random magnetic fields are considered in an Ising spin-glass model defined in the limit of infinite-range interactions. The probability distribution for the random magnetic fields is a double Gaussian, which consists of two Gaussian distributions centered, respectively, at $+H_0$ and $-H_0$, presenting the same width $\sigma$. It is argued that such a distribution is more appropriate for a theoretical description of real systems than its simpler particular two well-known limits, namely, the single Gaussian distribution $\left(\sigma ⪢H_0\right)$ and the bimodal one $\left(\sigma =0\right)$. The model is investigated by means of the replica method, and phase diagrams are obtained within the replica-symmetric solution. Critical frontiers exhibiting tricritical points occur for different values of $\sigma$, with the possibility of two tricritical points along the same critical frontier. To our knowledge, it is the first time that such a behavior is verified for a spin-glass model in the presence of a continuous-distribution random field, which represents a typical situation of a real system. The stability of the replica-symmetric solution is analyzed, and the usual Almeida-Thouless instability is verified for low temperatures. It is verified that the higher-temperature tricritical point always appears in the region of stability of the replica-symmetric solution; a condition involving the parameters $H_0$ and $\sigma$, for the occurrence of this tricritical point only, is obtained analytically. Some of our results are discussed in view of experimental measurements available in the literature.

We show that semiclassical formulas such as the Gutzwiller trace formula can be implemented on a quantum computer more efficiently than on a classical device. We give explicit quantum algorithms which yield quantum observables from classical trajectories, and which alternatively test the semiclassical approximation by computing classical actions from quantum evolution. The gain over classical computation is in general quadratic, and can be larger in some specific cases.

The Wigner time delay of a classically chaotic quantum system can be expressed semiclassically either in terms of pairs of scattering trajectories that enter and leave the system or in terms of the periodic orbits trapped inside the system. We show how these two pictures are related on the semiclassical level. We start from the semiclassical formula with the scattering trajectories and derive from it all terms in the periodic orbit formula for the time delay. The main ingredient in this calculation are correlations between scattering trajectories which are due to trajectories that approach the trapped periodic orbits closely. The equivalence between the two pictures is also demonstrated by considering correlation functions of the time delay. A corresponding calculation for the conductance gives no periodic orbit contributions in leading order.

We apply an analytical signal analysis to strange nonchaotic dynamics. Through this technique it is possible to obtain the spectrum of instantaneous intrinsic mode frequencies that are present in a given signal. We find that the second-mode frequency and its variance are good order parameters for dynamical transitions from quasiperiodic tori to strange nonchaotic attractors (SNAs) and from SNAs to chaotic attractors. Phase fluctuation analysis shows that SNAs and chaotic attractors behave identically within short time windows as a consequence of local instabilities in the dynamics. In longer time windows, however, the globally stable character of SNAs becomes apparent. This methodology can be of great utility in the analysis of experimental time series, and representative applications are made to signals obtained from Rössler and Duffing oscillators.

We study the classical and quantum mechanics of a three-dimensional stadium billiard. It consists of two quarter cylinders that are rotated with respect to each other by 90° and it is classically chaotic. The billiard exhibits only a few families of nongeneric periodic orbits. We introduce an analytic method for their treatment. The length spectrum can be understood in terms of the nongeneric and unstable periodic orbits. For unequal radii of the quarter cylinders the level statistics agree well with predictions from random matrix theory. For equal radii the billiard exhibits an additional symmetry. We investigated the effects of symmetry breaking on spectral properties. Moreover, for equal radii, we observe a small deviation of the level statistics from random matrix theory. This led to the discovery of stable and marginally stable orbits, which are absent for unequal radii.

We describe a scheme of molecular tagging velocimetry in air in which nitric oxide (NO) molecules are created out of ${\text{O}}_2$ and ${\text{N}}_2$ molecules in the focus of a strong laser beam. The NO molecules are visualized a while later by laser-induced fluorescence. The precision of the molecular tagging velocimetry of gas flows is affected by the gradual blurring of the written patterns through molecular diffusion. In the case of turbulent flows, molecular diffusion poses a fundamental limit on the resolution of the smallest scales in the flow. We study the diffusion of written patterns in detail for our tagging scheme which, at short $\left(\mu \text{s}\right)$ delay times is slightly anomalous due to local heating by absorption of laser radiation. We show that our experiments agree with a simple convection-diffusion model that allows us to estimate the temperature rise upon writing. Molecular tagging can be a highly nonlinear process, which affects the art of writing. We find that our tagging scheme is (only) quadratic in the intensity of the writing laser.

Spurious velocities are unphysical currents that appear close to curved interfaces in diffuse interface methods. We analyze the causes of these spurious velocities in the free-energy lattice Boltzmann algorithm. By making a suitable choice of the equilibrium distribution, and by finding the best way to numerically calculate derivatives, we show that these velocities may be decreased by an order of magnitude compared to previous models. Furthermore, we propose a momentum conserving forcing method that reduces spurious velocities by another factor of $\sim 5$. In three dimensions we find that 19 velocity vectors is the minimum number necessary.

When the invasion percolation model is applied as a simplified model for the displacement of a viscous fluid by a less viscous one, the distribution of displaced mass follows two distinct universality classes, depending on the criteria used to stop the displacement. Here we study the distribution of mass for this process, in the case where four extraction wells are placed around a single injection well in the middle of a square lattice. Our analysis considers the limit where the pressure of the extraction well $P_e$ is zero; in other words, an extraction well is capped as soon as less viscous fluid reaches that extraction well. Our results show that, as expected, the probability of stopping the production with small amounts of displaced mass is greatly reduced. We also investigate whether or not creating extra extraction wells is an efficient strategy. We show that the probability of increasing the amount of displaced fluid by adding an extra extraction well depends on the total recovered mass obtained before adding this well. The results presented here could be relevant to determine efficient strategies in oil exploration.

We consider two spherical, roughened crystals with approximately isotropic surface free energy, which are brought into contact and begin to sinter. We argue that the geometry immediately postcontact is two dimensional and Cartesian and can be approximated by the evolution of a slot-shaped cavity. On this basis, we construct traveling wave solutions for the crystal shape in the limits of bulk- and surface-diffusion-limited kinetics. These solutions are then used to calculate scalings for the neck size as a function of time $t$ after contact: We predict that neck size is proportional to $t^{1/4}$ for the bulk-diffusion-limited case and (following a single pinch-off event) approximately proportional to $t^{1/3}$ for the surface-diffusion-limited case.

The dynamical behavior of a class of randomly assembled networks of binary threshold units subject to random deletion of connections is studied based on the annealed approximation suitable in the thermodynamic limit. The dynamical phase diagram is constructed for several forms of the probability density distribution of nonvanishing connection strengths. The family of power-law distribution functions ${\rho }_0\left(x\right)=\left(1-\alpha \right)/\left(2{\left|x\right|}^{\alpha }\right)$ is found to play a special role in expanding the domain of stable, ordered dynamics at the expense of the disordered, “chaotic” phase. Relationships with other recent studies of the dynamics of complex networks allowing for variable in-degree of the units are explored. The relevance of the pruning of network connections to neural modeling and developmental neurobiology is discussed.

The universality split in absorbing phase transition between the conserved lattice gas (CLG) model and the conserved threshold transfer process (CTTP) is investigated on a checkerboard fractal and on a Sierpinski gasket. The critical exponents $\theta$, $\beta$, ${\nu }_{\parallel }$, and $z$, which are associated with, respectively, the density of active particles in time, the order parameter, the temporal correlation length, and the dynamics of active particles, are elaborately measured for two models on selected fractal lattices. The exponents for the CLG model are found to be distinctly different from those of the CTTP model on a checkerboard fractal, whereas the two models exhibit the same critical behavior on a Sierpinski gasket, indicating that the universality split between the two models occurs only on a checkerboard fractal. Such a universality split is attributed from the dominant hopping mechanisms caused by the intrinsic properties of the underlying fractal lattice.

Two-dimensional ferromagnetic $N$-state clock models are studied on a hyperbolic lattice represented by tessellation of pentagons. The lattice lies on the hyperbolic plane with a constant negative scalar curvature. We observe the spontaneous magnetization, the internal energy, and the specific heat at the center of sufficiently large systems, where fixed boundary conditions are imposed, for the cases $N\ge 3$ up to $N=30$. The model with $N=3$, which is equivalent to the three-state Potts model on the hyperbolic lattice, exhibits a first-order phase transition. A mean-field-like phase transition of second order is observed for the cases $N\ge 4$. When $N\ge 5$ we observe a Schottky-type specific heat below the transition temperature, where its peak height at low temperatures scales as $N^{-2}$. From these facts we conclude that the phase transition of the classical $XY$ model deep inside hyperbolic lattices is not of the Berezinskii-Kosterlitz-Thouless type.

Time-resolved small-angle x-ray scattering (SAXS) is used to follow the formation of resorcinol-formaldehyde (RF) gels. An existing morphological model based on Gaussian random fields, and validated on RF aerogels, is generalized to analyze the data. The generalization is done in two different ways, one being relevant to colloid aggregation and the other to microphase separation. The SAXS data do not enable discrimination between the two mechanisms of gel formation, which shows that aggregation and microphase separation can generate very similar morphologies at the length scales explored by SAXS. Furthermore, physical arguments suggest that, in the case of RF gels, aggregation and microphase separation can be regarded as two idealizations of the same complex physical process.

Shear localization is a generic feature of flows in yield stress fluids and soft glassy materials but is incompletely understood. In the classical picture of yield stress fluids, shear banding happens because of a stress heterogeneity. Using recent developments in magnetic resonance imaging velocimetry, we show here for a colloidal gel that even in a homogeneous stress situation shear banding occurs, and that the width of the flowing band is uniquely determined by the macroscopically imposed shear rate rather than the stress. We present a simple physical model for flow of the gel showing that shear banding (localization) is a flow instability that is intrinsic to the material, and confirm the model predictions for our system using rheology and light scattering.

Mixtures of different compositions of an antiferroelectric liquid crystal compound that exhibits direct smectic-$A^{\ast }\left(\text{Sm-}{A}^{\ast }\right)$-smectic-$C_A^{\ast }\left(\text{Sm-}{C}_A^{\ast }\right)$ transition with a ferroelectric liquid crystal compound that exhibits $\text{Sm-}{A}^{\ast }$-smectic-$C^{\ast }\left(\text{Sm-}{C}^{\ast }\right)$ transition are studied using electro-optics and dielectric spectroscopy. The results of optical texture, birefringence, and the tilt angle suggest that a part of the $\text{Sm-}{A}^{\ast }$ phase is of de Vries type, since an increase in the tilt angle with decreasing temperature results in a reduction in the value of the birefringence in the $\text{Sm-}{A}^{\ast }$ phase, whereas the birefringence at $\text{Sm-}{A}^{\ast }$ to $\text{Sm-}{C}^{\ast }$ transition goes up by 12.7%. The soft mode relaxation strength, the Landau coefficient of the temperature dependent term, and the other related parameters of the de Vries–type $\text{Sm-}{A}^{\ast }\text{-Sm-}{C}_A^{\ast }$ and $\text{Sm-}{A}^{\ast }\text{-Sm-}{C}^{\ast }$ transitions are determined using the Landau theory of the second-order phase transition. For the $\text{Sm-}{A}^{\ast }\text{-Sm-}{C}^{\ast }$ transition, we find that the soft mode relaxation strength decreases, the Landau coefficient increases, and the Curie-Weiss temperature range decreases with an increased ferroelectric composition in the mixture. These observations can be explained by assuming that with increased ferroelectric composition in the mixture, the layer shrinkage at the de Vries $\text{Sm-}{A}^{\ast }\text{-Sm-}{C}^{\ast }$ transition increases. On comparing the results of de Vries–type $\text{Sm-}{A}^{\ast }$ to $\text{Sm-}{C}_A^{\ast }$ and $\text{Sm-}{C}^{\ast }$ transitions, we find that the soft mode dielectric strength and the other related Landau parameters of the de Vries $\text{Sm-}{A}^{\ast }$ phase are of the same order of magnitude for transitions from $\text{Sm-}{A}^{\ast }$ to $\text{Sm-}{C}^{\ast }$ and to $\text{Sm-}{C}_A^{\ast }$ except for the composition of the mixture where both $\text{Sm-}{C}^{\ast }$ and $\text{Sm-}{C}_A^{\ast }$ transitions are stable and the phase diagram shows phase sequence $\text{Sm-}{A}^{\ast }$ to $\text{Sm-}{C}^{\ast }$ to $\text{Sm-}{C}_A^{\ast }$.