A method whereby electron-electron collisions can be included in Monte Carlo simulations of electron swarms in low-temperature partially ionized plasmas is presented. In this method, electron-electron (e-e) collisions are treated as being functionally equivalent to electron–neutral-species collisions. In doing so, individual pseudoelectron particles in the simulation collide with an energy-resolved electron fluid. Energy is exchanged with this fluid using the Coulomb cross section. When e-e collisions dominate, the electron-energy distribution evolves to being Maxwellian. The method is made computationally tractable by using a modified null-cross-section technique that eliminates the need for recalculating collision probabilities as the electron-energy distribution evolves.

The quadratic density response function has been shown to be determined by the three-point density correlations at equilibrium. Consequently, the conservation laws enjoyed by the density fluctuation are expected to have their manifestations in the quadratic density response function. New frequency-moment sum rules are established along these lines. We show that the lowest-order quadratic response function exhausts the quadratic equivalent of the f-sum rule.

The Hamiltonian wave fronts, i.e., the configuration-space surfaces with a constant value of the action, are numerically constructed for conservative dynamical systems. The fronts are orthogonal to the trajectories and can be used as an effective tool for the investigation of some properties of the trajectories’ families. The method is applied to the investigation of the counterparts, in the configuration space, of the ergodic properties in the phase space. The results, for the considered examples, indicate that in the chaotic regime the wave fronts are characterized by a very fast proliferation of singularities, which agrees with previous observations by other authors for area-preserving maps.

We investigate, in detail, the learning process in a feedforward Boolean network. Such a network transforms a set of n binary inputs into a binary output, i.e., it implements an n-variable Boolean function. If the network is constructed at random, the various possible Boolean functions are implemented with widely different probabilities. This probability (or entropy) landscape does not change very much, once a minimal size of the network is reached. Moreover, it has a hierarchical structure with the number of Boolean function scaling as an inverse power law of their probability (Zipf’s law). This indicates the existence of an underlying fractal structure. An amazing property of these networks is their ability to generalize, i.e., to learn some of the Boolean functions on the basis of a restricted number of teaching examples. We suggest that the fractal and hierarchical nature of the entropy landscape is the key to this ability. We identify the Boolean functions, which undergo such a learning transition, as those corresponding to a local entropic maximum in the Hamming distance space. We give evidence for the fact that the learning transition shifts to lower fractions of teaching, and becomes sharper, suggesting the existence of a genuine phase transition, as more complex problems are considered. The results on the entropy landscape are obtained by Monte Carlo simulations, while the effect of teaching is discussed using mean-field analysis, by an exact enumeration of the Monte Carlo data, and by actually training the network with a simulated annealing type of learning process.

We give an interpretation of the entropy of feed-forward neural networks based on information theory and we related it to the thermodynamics of the learning algorithm. As expected, this entropy allows one to classify neural networks as special-purpose or general-purpose computational devices. These concepts are extended to the task of learning an archetype of the class of Boolean functions generated by the symmetries of the architecture of the network. The entropy associated with the learning of an archetype is smaller than or equal to that corresponding to learning a function. In the former case, less bits are needed to be used as examples during the training of the network. Some numerical examples are presented. The results suggest that the entire space of synaptic matrices is evenly partitioned, in agreement with the principle of maximum entropy.

It is shown through numerical experiments that the linear combination of Poisson and Gaussian orthogonal random matrices yields level spectra whose nearest-neighbor spacing statistics do not have the scaling property when the dimension of the matrix is changed. It is argued that this type of random matrix is unsuitable as a model of chaotic quantum systems.

We have studied critical circle maps of the type θ→f(θ), with f(θ)=Ω+θ‖2θ${\mathrm{\Vert }}^{\mathit{z}\mathrm{-}1}$/2, -1/2<θ<1/2, and f(θ+1)=1+f(θ), where z is the order of the inflection point of the angular variable θ. Such maps are believed to be useful in the study of the two-frequency quasiperiodic route to chaos. Using a variety of numerical approaches, we have calculated the fractal dimension D associated with such maps as a function of z. The different approaches yield consistent values for D and the completeness of the staircase has also been checked at each order. By comparing 1-D with the width of the mode-locked interval ΔΩ(0/1) (which may be analytically determined as a function of z for this class of maps), we have established that the ratio appears to be roughly independent of z with a value of 1/3. This leads to the empirical formula D≊1-[(z-1)/3](1/z${)}^{\mathit{z}/\left(\mathit{z}\mathrm{-}1\right)}$, which predicts for z=3 that D≊0.872, in good agreement with previous precise direct numerical determinations. This general result suggests that the average width of mode-locked intervals of cycle length Q declines at the rate of ${\mathit{Q}}^{\mathrm{-}2/\mathit{D}}$ with the 0/1 interval setting the scale for arbitrary intervals for all z and thereby governing the fractal dimension of the set complementary to the devil’s staircase.

We derive nonlinear diffusion equations and equations containing corrections due to fluctuations for a coarse-grained concentration field. To deal with diffusion coefficients with an explicit dependence on the concentration values, we generalize the Van Kampen method of expansion of the master equation to field variables. We apply these results to the derivation of equations of phase-separation dynamics and interfacial growth instabilities.

The complex Ginzburg-Landau equation is considered in the weak-dissipation regime. The equation is supplemented by a periodic boundary condition, admitting a minimum wave number close to the threshold of the Benjamin-Feir instability. We demonstrate that the original equation can be consistently approximated by a three-dimensional dynamical system, which, depending on values of parameters, either coincides with the Lorenz model or differs from it in the sign of one coefficient. For the latter case, a diagram of dynamical regimes is constructed by numerical methods, and a region of chaos is found.

The phase stability of a Lennard-Jones mixture of particles of different size has been investigated with use of the hypernetted-chain integral equation. The dependence of the critical point on the strength of the unlike-particle interaction has been calculated, and the results have been compared with those regarding Lennard-Jones mixtures of equal-size particles [G. Malescio, Phys. Rev. A 42, 2211 (1990)].

Using heuristic arguments, we obtain asymptotic scaling behavior for aggregation due to the anomalous transport of particles, both in the dispersive and in the enhanced diffusion cases.

A simple theory for multi-ionic transport, nonequilibrium water dissociation, and space-charge effects in bipolar membranes is developed on the basis of some of the concepts used in the solid-state n-p junction. Ion transport is described in terms of the Nernst-Planck flux equation and nonequilibrium water dissociation is accounted for by the Onsager theory of the second Wien effect. The model is expected to be of interest for biological and synthetic membranes, and can explain a number of observed effects.

Andrews and Blake [Phys. Rev. A 38, 3113 (1988)] have presented an analysis suggesting that no coherent optical second-harmonic radiation is produced to any order in a multipole expansion by a fluid of noninteracting, randomly oriented molecules under excitation by a plane-wave electric field. We demonstrate to all orders of nonlocality that the correct description of any isotropic centrosymmetric medium in the field of a plane wave includes a longitudinal (generalized) nonlinear source polarization, which was not treated by Andrews and Blake. This bulk polarization does not give rise to a second-harmonic wave with a growing intensity in the medium; however, as has been recognized in the literature, it cannot be neglected in measurements of second-harmonic generation from surfaces and interfaces, since a longitudinal polarization is capable of exciting transverse electromagnetic waves at a discontinuity.

We explore the onset of delocalization as a function of coupling strength for time-dependent wave packets started on specific phase-space structures in the quantum standard map. Coherent-state wave packets, initially placed on fixed points of the classical map, are propagated for a long time and the delocalization is quantitatively measured. In most cases the delocalization occurs rather abruptly as a function of the kicking strength. The relative quantum stability of various phase-space structures is determined.

An alternative formulation of the stochastic representation of the Feynman path-integral theory is introduced. Stationary (imaginary-time) paths are generated as solutions of suitable Zwanzig-Ramshaw augmented Langevin equations. The required equilibrium distribution is obtained for a freely chosen functional form of the fluctuations and its associated dissipative term, in the presence of a generalized reversible drift. The method is both theoretically and numerically advantageous with respect to the standard Langevin and hybrid algorithms.

Sufficient conditions are given for the emergence of asymptotically synchronous chaotic orbits in ensembles of identical dynamical systems with all-to-all interaction.

We study the statistical property of eigenvalues for the stadium billiard to clarify what is universal and what is a reflection of the classical entropy on level statistics by changing the length of straight segments of the boundary. Irrespective of the underlying classical entropy, the nearest-neighbor level spacings tend to the Wigner characteristic; however, the convergence rate strongly depends on the classical entropy. Nonuniversal behavior is amplified in the property of the long-range fluctuation, and a robust structure in the energy sequence is found by Fourier transform of the density of states of wave numbers.

The effects of lattice anisotropy on the dynamics of an interface between coexisting phases is studied using a Ginzburg-Landau model in two dimensions that incorporates an anisotropic surface tension. It is shown that the anisotropy does not change growth exponents, but the kinetic coefficient in the Langevin equation, which describes the dynamics of the interface, turns out to be anisotropic. Thus the shapes of growing domains are different from the equilibrium crystal shape.

It is shown that Laplacian growth models, such as dielectric breakdown, diffusion-limited aggregation, and viscous fingering evolve in a stationary, self-similar state that can be well described by the self-organized criticality hypothesis of Bak, Tang, and Wisenfeld [Phys. Rev. Lett. 59, 381 (1987); Phys. Rev. A 38, 364 (1988)]. The pattern developing suffers rearrangements at all length scales in a power-law fashion.

We find that the fluctuations of the probability density P(r,t) of random walks on random fractals, for fixed distance r and time t, have a broad logarithmic distribution. The average moments 〈${\mathit{P}}^{\mathit{q}}$〉 scale is a multifractal way, 〈${\mathit{P}}^{\mathit{q}}$〉∼〈P${\mathrm{〉}}^{\mathrm{\tau }\left(\mathit{q}\right)}$, where τ(q)∼${\mathit{q}}^{\gamma }$, γ<1. The exponent γ characterizes the structural disorder in the fractal. In chemical l space, the distribution of P(l,t) is narrow and 〈${\mathit{P}}^{\mathit{q}}$〉∼〈P${\mathrm{〉}}^{\mathit{q}}$.