Synchronization of neural networks has been used for public channel protocols in cryptography. In the case of tree parity machines the dynamics of both bidirectional synchronization and unidirectional learning is driven by attractive and repulsive stochastic forces. Thus it can be described well by a random walk model for the overlap between participating neural networks. For that purpose transition probabilities and scaling laws for the step sizes are derived analytically. Both these calculations as well as numerical simulations show that bidirectional interaction leads to full synchronization on average. In contrast, successful learning is only possible by means of fluctuations. Consequently, synchronization is much faster than learning, which is essential for the security of the neural key-exchange protocol. However, this qualitative difference between bidirectional and unidirectional interaction vanishes if tree parity machines with more than three hidden units are used, so that those neural networks are not suitable for neural cryptography. In addition, the effective number of keys which can be generated by the neural key-exchange protocol is calculated using the entropy of the weight distribution. As this quantity increases exponentially with the system size, brute-force attacks on neural cryptography can easily be made unfeasible.

A model for mode I fracture in brittle materials is used to elucidate the relationship between characteristics of the fracture process, such as crack roughness, fractal dimension, and fragment size distributions. It is shown that different roughness in local regions of the crack path leads to different mechanisms for the subsequent fracture of those regions. Formation of two robust power laws for the distribution of formed fragments is observed, governing the size distribution of smaller and larger fragments. We connect measurements in fragment size distribution with the local roughness of cracks in the region of fragment formation.

A two-state system (TSS) under time-periodic perturbations (to be regarded as input signals) is studied in connection with self-tuning (ST) of threshold and stochastic resonance (SR). By ST, we observe an improvement of the signal-to-noise ratio (SNR) in a weak-noise region. An analytic approach to a tuning equation reveals that SNR improvement is possible also for the large-noise region and this is demonstrated by Monte Carlo simulations of hopping processes in a TSS. ST and SR are discussed from a little more physical point of view of the energy transfer (dissipation) rate, which behaves in a similar way as the SNR. Finally ST is considered briefly for a double-well potential system, which is closely related to the TSS.

Nonlocally coupled oscillator systems can exhibit an exotic spatiotemporal structure called a chimera, where the system splits into two groups of oscillators with sharp boundaries, one of which is phase locked and the other phase randomized. Two examples of chimera states are known: the first one appears in a ring of phase oscillators, and the second is associated with two-dimensional rotating spiral waves. In this paper, we report yet another example of the chimera state that is associated with the so-called Ising walls in one-dimensional spatially extended systems. This chimera state is exhibited by a nonlocally coupled complex Ginzburg-Landau equation with external forcing.

The ubiquity of scale-free networks in nature and technological applications and the finding that such networks may be more difficult to synchronize than homogeneous networks pose an interesting phenomenon for study in network science. We argue and demonstrate that, in the presence of some proper gradient fields, scale-free networks can be more synchronizable than homogeneous networks. The gradient structure can in fact arise naturally in any weighted and asymmetrical networks; based on this we propose a coupling scheme that permits effective synchronous dynamics on the network. The synchronization scheme is verified by eigenvalue analysis and by direct numerical simulations using networks of nonidentical chaotic oscillators.

A pair-ion plasma without electrons consisting of $\mathrm{C}_{60}{}^{+}$ and $\mathrm{C}_{60}{}^{-}$ is generated through the processes of electron-impact ionization, electron attachment, and magnetic filtering. Properties of electrostatic modes propagating along magnetic-field lines are experimentally investigated by externally exciting them with two types of electrodes. It is found that four kinds of wave modes exist and a frequency spectrum of phase lag between the density fluctuations of $\mathrm{C}_{60}{}^{+}$ and $\mathrm{C}_{60}{}^{-}$ is unique in comparison with ordinary electron-ion plasmas. One of the modes is an ion acoustic wave which is divided into two branches at around the ion cyclotron frequency in the presence of a backwardlike mode joining them. The phase lag of the ion acoustic wave strongly depends on the frequency, while those for the other ion plasma and intermediate-frequency waves are constant at $\pi$ independent of the frequency.

We investigate fundamental localized modes in two-dimensional lattices with an edge (surface). The interaction with the edge expands the stability area for fundamental solitons, and induces a difference between dipoles oriented perpendicular and parallel to the surface. On the contrary, lattice vortex solitons cannot exist too close to the border. We also show, analytically and numerically, that the edge supports a species of localized patterns, which exists too but is unstable in the uniform lattice, namely, a horseshoe-shaped soliton, whose “skeleton” consists of three lattice sites. Unstable horseshoes transform themselves into a pair of ordinary solitons.

The transfer matrix method has been widely used to calculate the scattering of electromagnetic waves. In this paper, we develop the conventional transfer matrix method to analyze the problem of second harmonic generation in a one-dimensional multilayer nonlinear optical structure. In the designed nonlinear photonic crystal structure, the linear and nonlinear optical parameters are both periodically modulated. We have taken into account the multiple reflection and interference effects of both the linear and nonlinear optical waves during the construction of the transfer matrix for each composite layer. Application of this method to multilayer nonlinear photonic crystal structures with different refractive indices indicates that the proposed method is an exact approach and can simulate the generation of the second harmonic field precisely. In an optimum structure, the second harmonic generation efficiency can be several orders of magnitude larger than in a conventional quasi-phase-matched nonlinear structure with the same sample length. The reason is that, due to the presence of photonic band gap edges, the density of states of the electromagnetic fields is large, the group velocity is small, and the local field is enhanced. All three factors contribute to significant enhancement of the nonlinear optical interactions.

A modified lattice Boltzmann model based on the two-dimensional, nine-velocity lattice-Bhatnagar-Gross-Krook fluid is presented for axisymmetric flows. A spatially and temporally varying source term is incorporated into the evolution equation for the momentum distribution function on a two-dimensional Cartesian lattice. The precise form of the source term is derived through a Chapman-Enskog analysis, so that the additional axisymmetric contributions in the Navier-Stokes equations are furnished when written in the cylindrical polar coordinate system.

We study the effect of a small cutoff $ϵ$ on the velocity of a pulled front in one dimension by means of a variational principle. We obtain a lower bound on the speed dependent on the cutoff, for which the two leading order terms correspond to the Brunet-Derrida expression. To do so we cast a known variational principle for the speed of propagation of fronts in different variables which makes it more suitable for applications.

We have investigated the nature and experimental consequences of vapor-liquid asymmetry in near-critical fluids within the framework of “complete scaling” [M. E. Fisher and G. Orkoulas, Phys. Rev. Lett. 85, 696 (2000); Y. C. Kim et al., Phys. Rev. E 67, 061506 (2003)]. We used the thermodynamic freedom for a choice of the critical-entropy value to simplify “complete scaling” to a form with only two independent parameters, responsible for two different sources of the asymmetry. We then developed a procedure to obtain these two parameters from mean-field equations of state. By combining accurate liquid-vapor coexistence and heat-capacity data, we have unambiguously separated two nonanalytic contributions from the two sources of vapor-liquid asymmetry and proved the validity of “complete scaling.” Since the nonanalytic asymmetry effects in the critical region are fully determined by the Ising critical exponents for the symmetric lattice-gas model, there is no need for a special renormalization-group theoretical treatment of “non-Ising” asymmetry in fluid criticality.

Kauffman nets, also known as $N\text{−}K$ models, have been studied extensively because their dynamics can be used to model a variety of interesting dynamical processes. This paper investigates the properties of the problem of determining whether or not a given configuration of a Kauffman net has a predecessor. Here it is shown that when the parameter $K$ that governs the number of connections grows as $\ln \left(N\right)$, where $N$ is the number of elements, the problem of finding a solution is extremely sensitive to small changes in the problem statement. This result has implications for studies of the physics of random systems and also may have applications for questions in computational complexity theory.

We have proposed a generalized Langevin-type rate-code model subjected to multiplicative noise, in order to study stationary and dynamical properties of an ensemble containing a finite number $N$ of neurons. Calculations using the Fokker-Planck equation have shown that, owing to the multiplicative noise, our rate model yields various kinds of stationary non-Gaussian distributions such as $\Gamma$, inverse-Gaussian-like, and log-normal-like distributions, which have been experimentally observed. The dynamical properties of the rate model have been studied with the use of the augmented moment method (AMM), which was previously proposed by the author from a macroscopic point of view for finite-unit stochastic systems. In the AMM, the original $N$-dimensional stochastic differential equations (DEs) are transformed into three-dimensional deterministic DEs for the means and fluctuations of local and global variables. The dynamical responses of the neuron ensemble to pulse and sinusoidal inputs calculated by the AMM are in good agreement with those obtained by direct simulation. The synchronization in the neuronal ensemble is discussed. The variabilities of the firing rate and of the interspike interval are shown to increase with increasing magnitude of multiplicative noise, which may be a conceivable origin of the observed large variability in cortical neurons.

A model-based control methodology for suppressing chaos for nonlinear systems is introduced. The proposed methodology generates new periodic orbits or steady states, which are not the solutions of the free system, using output feedback and state observers. The design uses the certainty equivalence principle to construct a linear closed loop system that can follow desired outputs with zero steady state offsets via using a pole-placement-like approach. The combined dynamics of both the controller and the state observer are carefully studied using the well-known Rössler system. The similarities and differences between the proposed control technique and both the double-notch filter feedback and the time-delay autosynchronization are investigated. The effect of parameter uncertainties is studied and robustness of the proposed controller is analyzed.

We consider the propagation of an exothermic chemical front toward an unstable steady state. The hydrodynamic equations are solved numerically for increasing values of the activation energy of the reaction which controls the reaction front speed. For a large speed, the marginal stability criterion of the isothermal case is recovered. For a small speed, we observe two well-separated traveling waves: a heat front is preceding the reaction front. We find analytically a forbidden speed interval where the hydrodynamical system does not admit stationary traveling solutions.

Models of screened ions in equilibrium plasmas with all quantum electrons are important in opacity and equation of state calculations. Although such models have to be derived from variational principles, up to now existing models have not been fully variational. In this paper a fully variational theory respecting virial theorem is proposed—all variables are variational except the parameters defining the equilibrium, i.e., the temperature $T$, the ion density $n_i$ and the atomic number $Z$. The theory is applied to the quasiclassical Thomas-Fermi (TF) atom, the quantum average atom (QAA), and the superconfigurations (SC) in plasmas. Both the self-consistent-field (SCF) equations for the electronic structure and the condition for the mean ionization $Z^{*}$ are found from minimization of a thermodynamic potential. This potential is constructed using the cluster expansion of the plasma free energy from which the zero and the first-order terms are retained. In the zero order the free energy per ion is that of the quantum homogeneous plasma of an unknown free-electron density $n_0=Z^{*}{n}_i$ occupying the volume $1∕n_i$. In the first order, ions submerged in this plasma are considered and local neutrality is assumed. These ions are considered in the infinite space without imposing the neutrality of the Wigner-Seitz (WS) cell. As in the Inferno model, a central cavity of a radius $R$ is introduced, however, the value of $R$ is unknown a priori. The charge density due to noncentral ions is zero inside the cavity and equals $e{n}_0$ outside. The first-order contribution to free energy per ion is the difference between the free energy of the system “central $\text{ion}+\text{infinite}$ plasma” and the free energy of the system “infinite plasma.” An important part of the approach is an “ionization model” (IM), which is a relation between the mean ionization charge $Z^{*}$ and the first-order structure variables. Both the IM and the local neutrality are respected in the minimization procedure. The correct IM in the TF case is found to be $Z-Z^{*}=\int d^{3}r[n\left(\vec{r}\right)-n_0]$, where $n\left(\vec{r}\right)$ is the first-order electron density. It is shown that in the QAA case the same IM has to be used and that other IMs lead to unphysical solutions. With this IM $R$ becomes in both cases (TF and QAA) equal to the WS radius and the variational calculation leads to SCF equations in an infinite plasma while $n_0$ (or equivalently $Z^{*}$) is to be found from the condition $\int d^{3}r\theta (r-R)V_{\mathit{el}}\left(\vec{r}\right)=0$, where $\theta$ denotes Heaviside function and $V_{\mathit{el}}\left(\vec{r}\right)$ is the SCF electrostatic potential. In the SC case results are similar except that averages over all superconfigurations appear. In the TF case the condition for $n_0$ gives the neutrality of the WS sphere and one gets the classical TF ion-in-cell average atom. The situation is different in the QAA and in the SC cases in which the cavity is not neutral and the SCF potential $V_{\mathit{el}}\left(\vec{r}\right)$ is not zero outside the cavity. Due to the fully variational character of our approach the expression for the thermodynamic pressure in all cases does not require any numerical differentiation and is consistent with the virial theorem.

The Jarzynski identity [C. Jarzynski, Phys. Rev. Lett. 78, 2690 (1997)] and the Crooks equation [G. E. Crooks, J. Stat. Phys. 90, 1481 (1998)] relate thermodynamic free energy differences to the work done on a system during a collection of nonequilibrium transformations. In the present Rapid Communication we provide generalized versions of these nonequilibrium work theorems, which hold for dissipative transformations where the system may undergo simultaneously mechanical work and pressure-temperature or volume-temperature changes. The proof is valid in the context of dynamic systems that evolve with $NPT$-based equations of motion according to the Martyna-Tobias-Klein algorithm [Martyna et al. J. Chem. Phys. 101, 4177 (1994)]. An extension of the proof to dynamic systems that evolve through $NVT$-based equations of motion is also provided. The theorems may be effectively used in non-Hamiltonian molecular dynamics simulations for evaluating Helmholtz or Gibbs free energy differences, or the ratio of partition functions at different temperatures to be eventually used in thermodynamic cycles.

We report measurements of the nonlinear micromechanics of strongly flocculated model colloidal aggregates. Linear aggregates directly assembled using laser tweezers are subjected to bending loads until a critical bending moment is reached, which is identified by a stictionlike rearrangement of a single colloidal bond. This nanoscale phenomenon provides a quantitative basis for understanding the macroscopic shear yield stresses of strongly flocculated polystyrene latex gels, based on the maximum bending moment exceeding the critical moment of the constituent colloidal bonds of the gel microstructure. These mechanics are consistent with the local bending moment overcoming the static friction force between neighboring adhesive particles. This results in a direct relationship between the rheology of these gels and the boundary friction between Brownian particles.

Recently, the experimentally observed asymmetric properties of ion transport in charged conical nanopores (CCNs) that resemble those in biological ion channels have attracted a lot of attention in theoretical studies in nanotechnology research. In this paper, we report several tactics to study this effect by directly solving the Poisson-Nernst-Planck (PNP) equations. The result shows that PNP equations can indeed quantitatively describe the properties of these nanopores. Based on our numerical solutions, we contribute the rectification effect to ion-enrichment and ion-depletion. A detailed study of length dependence of current indicates that a relatively long length is indispensable for the CCNs to have rectification effect. We suggest that PNP equations and the calculation method could be further used to study other shapes of nanopores.