We investigate the effect of noise on random Boolean networks. Noise is implemented as a probability $p$ that a node does not obey its deterministic update rule. We define two order parameters, the long-time average of the Hamming distance between a network with and without noise, and the average frozenness, which is a measure of the extent to which a node prefers one of the two Boolean states. We evaluate both order parameters as function of the noise strength, and of the number of inputs per node $K$ finding a smooth transition from deterministic $(p=0)$ to fully stochastic $(p=1∕2)$ dynamics for networks with $K⩽2$, and a first-order transition at $p=0$ for $K>2$. Most of the results obtained by computer simulation are also derived analytically. The average Hamming distance can be evaluated using the annealed approximation. In order to obtain the distribution of frozenness as function of the noise strength, more sophisticated self-consistent calculations had to be performed. This distribution is a collection of $\delta$ peaks for $K=1$, and it has a fractal sructure for $K>1$, approaching a continuous distribution in the limit $K⪢1$.

The accurate estimation of scaling exponents is central in the observational study of scale-invariant phenomena. Natural systems unavoidably provide observations over restricted intervals; consequently, a stationary stochastic process (time series) can yield anomalous time variation in the scaling exponents, suggestive of nonstationarity. The variance in the estimates of scaling exponents computed from an interval of $N$ observations is known for finite variance processes to vary as $\sim 1∕N$ as $N\to \infty$ for certain statistical estimators; however, the convergence to this behavior will depend on the details of the process, and may be slow. We study the variation in the scaling of second-order moments of the time-series increments with $N$ for a variety of synthetic and “real world” time series, and we find that in particular for heavy tailed processes, for realizable $N$, one is far from this $\sim 1∕N$ limiting behavior. We propose a semiempirical estimate for the minimum $N$ needed to make a meaningful estimate of the scaling exponents for model stochastic processes and compare these with some “real world” time series.

Dust-particles charging in a low-pressure glow discharge was investigated theoretically. The dust-particle charge was found on the basis of a developed self-consistent model taking into account the nonequilibrium character of electron distribution function and the formation of an ionic coat composed of bound or trapped ions around the dust particle. The dust-particle charge, the radial distributions of electron density, free and trapped ions densities, and the distribution of electrostatic potential were found. It was shown that the non-Maxwellian electron distribution function and collisional flux of trapped ions both reduce the dust-particle charge in comparison with that received with the help of the conventional orbital motion limited (OML) model. However, in rare collisional regimes in plasma when the collisional flux is negligible, the formation of ionic coat around a particle leads to a shielding of the proper charge of a dust particle. In low-pressure experiments, it is only possible to detect the effective charge of a dust particle that is equal to the difference between the proper charge of the particle and the charge of trapped ions. The calculated effective dust particle charge is in fairly good agreement with the experimental measurements of dust-particle charge dependence on gas pressure.

Generalized Maxwell distribution function is derived analytically for the lattice Boltzmann (LB) method. All the previously introduced equilibria for LB are found as special cases of the generalized Maxwellian. The generalized Maxwellian is used to derive a different class of multiple relaxation-time LB models and prove the $H$ theorem for them.

We discovered a narrow band of chaos close to the grazing condition for a simple soft impact oscillator. The phenomenon was observed experimentally for a range of system parameters. Through numerical stability analysis, we argue that this abrupt onset to chaos is caused by a dangerous bifurcation in which two unstable period-3 orbits, created at “invisible” grazings, take part.

We simulate numerically the time evolution of 1000 interacting hard spheres in a finite box with periodic boundary conditions, and repeat the simulations many times for a selected random distribution of initial conditions. We use the resulting data to compute, directly, the smallest nonzero eigenvalue of the collision operator for this gas. We also give exact expressions for the transport coefficients and compare them to approximate expressions commonly seen in the literature.

A hypothetical phase diagram of a crystalline polymer/photoreactive monomer mixture has been calculated on the basis of phase field (PF) free energy of crystal solidification in conjunction with Flory-Huggins (FH) free energy of liquid-liquid demixing to guide the morphology development during photopolymerization of poly(ethylene oxide)/triacrylate blend. The self-consistent solution of the combined PF-FH theory exhibits a crystalline-amorphous phase diagram showing the coexistence of $\text{solid}+\text{liquid}$ gap bound by the liquidus and solidus lines, followed by an upper critical solution temperature at a lower temperature. When photopolymerization was triggered in the isotropic region, i.e., slightly above the crystal melting transition temperatures, the depressed melting transition line moves upward. When it surpasses the reaction temperature, both crystallization and phase separation occur. The temporal evolution of phase morphology is examined in the context of time-dependent Ginzburg-Landau equations coupled with the energy balance (heat conduction) equation using the aforementioned PF-FH free-energy densities. Of particular interest is that the emerged morphology in the crystalline blends depends on the competition between dynamics of liquid-liquid phase separation and/or liquid-solid phase transition (i.e., crystallization) and photopolymerization rates.

DNA aptamers are molecular biosensors consisting of single functionalized DNA molecules, which can bind to specific targets or complementary DNA sequences. The binding kinetics of DNA aptamers is studied by fluorescence quenching at $23°\text{C}$. A kinetic model for the binding reaction of DNA aptamer, antisense DNA, and ATP target is developed to describe experimental observations. The approach leads to a simple procedure to deduce relevant kinetic reactions and their rate constants. A comparison between theory and experiments indicates that the previously established bimolecular DNA-ATP binding does not provide a complete description of the experimental data. Side reactions such as trimolecular complexation are proposed. Rate constants of the model are determined by comparing the model predictions and experiments. Good agreements between the model and experiments have been obtained. Possible blocking reactions by the misfolded DNA aptamer are also discussed.

We present an x-ray diffraction study on $n$-hexane in tubular silicon channels of approximately 10 nm diameter both as a function of the filling fraction $f$ of the channels and as a function of temperature. Upon cooling, confined $n$-hexane crystallizes in a triclinic phase typical of the bulk crystalline state. However, the anisotropic spatial confinement leads to a preferred orientation of the confined crystallites, where the $⟨001⟩$ crystallographic direction coincides with the long axis of the channels. The magnitude of this preferred orientation increases with the filling fraction, which corroborates the assumption of a Bridgman-type crystallization process being responsible for the peculiar crystalline texture. This growth process predicts for a channel-like confinement an alignment of the fastest crystallization direction parallel to the long channel axis. It is expected to be increasingly effective with the length of solidifying liquid parcels and thus with increasing $f$. In fact, the fastest solidification front is expected to sweep over the full silicon nanochannel for $f=1$, in agreement with our observation of a practically perfect texture for entirely filled nanochannels.

Fontanari et al. introduced [Phys. Rev. Lett. 91, 218101 (2003)] a model for studying Muller’s ratchet phenomenon in growing asexual populations. They studied two situations, either including a death probability for each newborn or not, but were able to find analytical (recursive) expressions only in the no-decay case. In this Brief Report a branching process formalism is used to find recurrence equations that generalize the analytical results of the original paper besides confirming the interesting effects their simulations revealed.

This work considers an Ising model on the Apollonian network, where the exchange constant $J_{i,j}\sim 1/{\left(k_i{k}_j\right)}^{\mu }$ between two neighboring spins $\left(i,j\right)$ is a function of the degree $k$ of both spins. Using the exact geometrical construction rule for the network, the thermodynamical and magnetic properties are evaluated by iterating a system of discrete maps that allows for very precise results in the thermodynamic limit. The results can be compared to the predictions of a general framework for spin models on scale-free networks, where the node distribution $P\left(k\right)\sim k^{-\gamma }$, with node-dependent interacting constants. We observe that, by increasing $\mu$, the critical behavior of the model changes from a phase transition at $T=\infty$ for a uniform system $\left(\mu =0\right)$ to a $T=0$ phase transition when $\mu =1$: in the thermodynamic limit, the system shows no true critical behavior at a finite temperature for the whole $\mu \ge 0$ interval. The magnetization and magnetic susceptibility are found to present noncritical scaling properties.

Recently increased accessibility of large-scale digital records enables one to monitor human activities such as the interevent time distributions between two consecutive visits to a web portal by a single user, two consecutive emails sent out by a user, two consecutive library loans made by a single individual, etc. Interestingly, those distributions exhibit a universal behavior, $D\left(\tau \right)\sim {\tau }^{-\delta }$, where $\tau$ is the interevent time, and $\delta \simeq 1$ or $3∕2$. The universal behaviors have been modeled via the waiting-time distribution of a task in the queue operating based on priority; the waiting time follows a power-law distribution $P_{\mathrm{w}}\left(\tau \right)\sim {\tau }^{-\alpha }$ with either $\alpha =1$ or $3∕2$ depending on the detail of queuing dynamics. In these models, the number of incoming tasks in a unit time interval has been assumed to follow a Poisson-type distribution. For an email system, however, the number of emails delivered to a mail box in a unit time we measured follows a power-law distribution with general exponent $\gamma$. For this case, we obtain analytically the exponent $\alpha$, which is not necessarily 1 or $3∕2$ and takes nonuniversal values depending on $\gamma$. We develop the generating function formalism to obtain the exponent $\alpha$, which is distinct from the continuous time approximation used in the previous studies.

The influence of circularly polarized electric fields (CPEFs) on the stability of multiarmed spiral waves is investigated. It is shown that CPEFs can change the period of the multiarmed spirals. The average period is an important quantity of multiarmed spiral and it must be larger than a threshold for stable multiarmed spiral. After a counter-rotating CPEF with suitable amplitude and period is applied, the average period of the multiarmed spiral may increase, which stabilizes the multiarmed spiral.

The general theory of motion in the vicinity of a moving quantum nodal point (vortex) is studied in the framework of the de Broglie–Bohm trajectory method of quantum mechanics. Using an adiabatic approximation, we find that near any nodal point of an arbitrary wave function $\psi$ there is an unstable point (called the $X$ point) in a frame of reference moving with the nodal point. The local phase portrait forms always a characteristic pattern called the “nodal-point–$X$-point complex.” We find general formulas for this complex as well as necessary and sufficient conditions of validity of the adiabatic approximation. We demonstrate that chaos emerges from the consecutive scattering events of the orbits with nodal-point–$X$-point complexes. The scattering events are of two types (called type I and type II). A theoretical model is constructed yielding the local value of the Lyapunov characteristic numbers in scattering events of both types. The local Lyapunov characteristic number scales as an inverse power of the speed of the nodal point in the rest frame, implying that it scales proportionally to the size of the nodal-point–$X$-point complex. It is also an inverse power of the distance of a trajectory from the $X$ point’s stable manifold far from the complex. This distance plays the role of an effective “impact parameter.” The results of detailed numerical experiments with different wave functions, possessing one, two, or three moving nodal points, are reported. Examples are given of regular and chaotic trajectories, and the statistics of the Lyapunov characteristic numbers of the orbits are found and compared to the number of encounter events of each orbit with the nodal-point–$X$-point complexes. The numerical results are in agreement with the theory, and various phenomena appearing at first as counterintuitive find a straightforward explanation.

We study the work fluctuations of a particle, confined to a moving harmonic potential, under the influence of friction and external asymmetric Poissonian shot noise. This type of noise generalizes the usual Gaussian noise and induces an effective interaction between the noise and the potential, leading to an effectively nonlinear system with singular features. On the basis of an analytic solution we investigate the roles of time scales, symmetries, and singularities in the context of nonequilibrium fluctuations. Our results highlight the nonuniversality of the steady-state fluctuation theorem in stochastic systems.

Generally, the threshold of percolation in complex networks depends on the underlying structural characterization. However, what topological property plays a predominant role is still unknown, despite the speculation of some authors that degree distribution is a key ingredient. The purpose of this paper is to show that power-law degree distribution itself is not sufficient to characterize the threshold of bond percolation in scale-free networks. To achieve this goal, we first propose a family of scale-free networks with the same degree sequence and obtain by analytical or numerical means several topological features of the networks. Then, by making use of the renormalization-group technique we determine the threshold of bond percolation in our networks. We find an existence of nonzero thresholds and demonstrate that these thresholds can be quite different, which implies that power-law degree distribution does not suffice to characterize the percolation threshold in scale-free networks.

Snow on the ground is a complex three-dimensional porous medium consisting of an ice matrix formed by sintered snow crystals and a pore space filled with air and water vapor. If a temperature gradient is imposed on the snow, a water vapor gradient in the pore space is induced and the snow microstructure changes due to diffusion, sublimation, and resublimation: the snow metamorphoses. The snow microstructure, in turn, determines macroscopic snow properties such as the thermal conductivity of a snowpack. We develop a phase-field model for snow metamorphism that operates on natural snow microstructures as observed by computed x-ray microtomography. The model takes into account heat and mass diffusion within the ice matrix and pore space, as well as phase changes at the ice-air interfaces. Its construction is inspired by phase-field models for alloy solidification, which allows us to relate the phase-field to a sharp-interface formulation of the problem without performing formal matched asymptotics. To overcome the computational difficulties created by the large difference between diffusional and interface-migration time scales, we introduce a method for accelerating the numerical simulations that formally amounts to reducing the heat- and mass-diffusion coefficients while maintaining the correct interface velocities. The model is validated by simulations for simple one- and two-dimensional test cases. Furthermore, we perform qualitative metamorphism simulations on natural snow structures to demonstrate the potential of the approach.

Using Monte Carlo simulation, we investigate the structural phase behavior of a continuum molecular model for self-assembling semiflexible equilibrium polymers in two dimensions. Particle-particle interaction is modeled via a Lennard-Jones potential with tunable anisotropic attraction. Depending on the strength of the anisotropy, we find the formation of reversible networks as well as stiff rodlike aggregates. The phase transition observed in the presence of the network structures is compared to predictions of the Tlusty-Safran defect model.

Ultrathin polymer films that are produced, e.g., by spin coating are believed to be stressed since polymers are “frozen in” into out-of-equilibrium configurations during this process. In the framework of a viscoelastic thin-film model, we study the effects of lateral residual stresses on the dewetting dynamics of the film. The temporal evolution of the height profiles and the velocity profiles inside the film as well as the dissipation mechanisms are investigated in detail. Both the shape of the profiles and the importance of frictional dissipation vs viscous dissipation inside the film are found to change in the course of dewetting. The interplay of the nonstationary profiles, the relaxing initial stress, and the changes in the dominance of the two dissipation mechanisms caused by nonlinear friction with the substrate is responsible for the rich behavior of the system. In particular, our analysis sheds a different light on the occurrence of the unexpected maximum in the rim width obtained recently in experiments on polystyrene-polydimethylsiloxane systems.

The density of synaptic receptors in front of presynaptic release sites is stabilized in the presence of scaffold proteins, but the receptors and scaffold molecules have local exchanges with characteristic times shorter than that of the receptor-scaffold assembly. We propose a mesoscopic model to account for the regulation of the local density of receptors as quasiequilibrium. It is based on two zones (synaptic and extrasynaptic) and multilayer (membrane, submembrane, and cytoplasmic) topological organization. The model includes the balance of chemical potentials associated with the receptor and scaffold protein concentrations in the various compartments. The model shows highly cooperative behavior including a “phase change” resulting in the formation of well-defined postsynaptic domains. This study provides theoretical tools to approach the complex issue of synaptic stability at the synapse, where receptors are transiently trapped yet rapidly diffuse laterally on the plasma membrane.