A new method is proposed to locate and analyze phase transitions in a thermodynamic formalism for the description of fractal sets. By studying order parameters appropriate to the transitions we get both an efficient numerical tool for locating phase transitions and an understanding of the structure of the ordered phase. With this method, we examine fractal sets generated by a class of maps of the interval close to the map x→4x(1-x). We show that the existence of phase transitions is a persistent phenomenon and remains when the map is perturbed although the structure of the entropy function changes drastically. For strong perturbations the transition disappears and the entropy function becomes nonsingular. The phase transitions describe transitions in the distribution of the characteristic Lyapunov exponents.

We report measurements on the reversible and irreversible kinetics of aggregation in a colloid of polystyrene particles, performed by quasielastic light scattering. The experiment was made with small to moderate concentrations of the simple electrolyte inducing the aggregation in order to observe both the reversible flocculation and the slow, irreversible diffusion-limited coagulation. In the latter case we observe deviations from asymptotic scaling in time. The experimental results are well described by a numerical solution of the von Smoluchowski equation for aggregation limited by Brownian diffusion.

Neural networks contain, very often, asymmetric bonds. The interactions $J_{\mathrm{ij}}$ and $J_{\mathrm{ji}}$ between the ith and the jth neurons are not identical. In this paper we study the Langevin dynamics of fully connected spin systems whose interaction matrix contains a random antisymmetric part. The symmetric part consists of independent random bonds whose mean is either zero or ferromagnetic. We also consider a more general class of systems such as the asymmetric Hopfield model and other neural-network models. Within the framework of mean-field theory, the spin fluctuations are viewed as local, thermally averaged, time-dependent magnetic moments. These moments are induced by excess (i.e., nonthermal) internal noise which, in the presence of asymmetry, is time dependent and does not vanish even in the high-temperature phase. The mean-field equations are solved using a simplified, spherical model, in which the spins are linear variables except for a global constraint on the total level of their fluctuations. Random asymmetry of arbitrary strength destroys spin-glass freezing. Ferromagnetic phases, as well as ‘‘retrieval’’ states in neural networks, are affected only slightly by weak random asymmetry, in agreement with the conclusions of Hertz et al.

The dynamical behavior of a system with weak asymmetry is studied in some detail. In the spin-glass case at low temperatures, when the strength of the asymmetry decreases, the internal excess noise does not vanish but slows down with a characteristic correlation time τ∝$k^{\mathrm{-}6}$. The parameter k denotes the relative strength of the antisymmetric components of the bonds. The system behaves as a frozen symmetric spin glass on time scales t≪τ and as a paramagnet on scales t≫τ. The thermal fluctuations decay with a characteristic time ${\tau }_T$∝$k^{\mathrm{-}4}$. The spherical model exhibits a completely frozen spin-glass state at zero temperature. As T→0, fluctuations exhibit a critical slowing down with time τ∝$T^{\mathrm{-}1}$ for all values of k>0. This T=0 spin-glass transition is probably an artifact of the spherical model and is not expected to exist in nonlinear systems. The relevance of the results to the performance of neural networks is discussed.

An expression for the Fokker-Planck equation governing the velocity distribution function of particles or heavy molecules immersed in a host light gas valid for arbitrary mean velocities of the heavy component is given. This expression generalizes previous results which were limited to small differences between the mean velocities of the heavy and light components compared with the thermal velocity of the light gas. The derivation assumes a Maxwellian velocity distribution function for the light gas, elastic heavy-light collisions, and makes use of integrals computed by Riesco-Chueca, Fernández-Feria, and Fernández de la Mora in Ref. 1. The stochastic Langevin equation associated with this Fokker-Planck collision operator is also obtained. More in general, we derive the Langevin equation corresponding to the general form of the Fokker-Planck collision operator, and particularize it to the present case.

In this paper a generalization of the Onsager reciprocity theorem (ORT) is presented to include fluctuations of the regression variables around an arbitrary initial state. The arbitrariness of the initial state is constrained only by the content of Mazur’s lemma. The physical implications of this result, as well as its relationship with previous work reported some years ago on this question, are thoroughly discussed. The connection with other attempts to generalize the ORT is also reported.

We have studied the effect of fluctuations on multifractals. Averaging over the fluctuations leads to an additive correction to the measure, i.e., to a metadimension. This effect is particularly important at the crossover point where the generalized dimension changes sharply.

The effect of noise on the period-doubling scenario in Hamiltonian systems is considered. The escape time from islands is found to exhibit asymptotically universal scaling. With use of renormalization theory, it is shown that there is a universal noise distribution and a scale factor associated with it, responsible for self-similarity with noise. The fast transport within chaotic regions enhances diffusion, which obeys an anomalous power law.

We investigate geometric and algebraic properties of binary vectors within the context of Hopfield models for orthogonal binary vectors. We define a generalized Hamming distance h which is an invariant property of even sets of vectors of the N-dimensional hypercube and vanishes for an orthogonal set which forms a basis. We present an algorithm for the construction of orthogonal distant sets of size N/2, in which no subset of size divisible by 4 has h=0, and argue that these sets are faithful, i.e., they are exactly and exclusively reproduced by the model.

We propose a curvature correction to the heat-balance equation at the solid-melt interface that should be included in any realistic model of dendritic growth. Numerical evidences are presented to show that this term allows for steady-state solutions in the absence of anisotropy in the two-dimensional nonlocal symmetric model of solidification. It is shown that these solutions can be conjectured analytically in the large undercooling limit. Steady-state solutions are also obtained when anisotropy is added.

Reaction kinetics was developed in special and general relativity. The covariant rate equation and the special-relativistic rate equation which expresses an effect of a relative velocity are identical in form to the classical equation. The general-relativistic rate equation expressing a gravitational effect no longer has the same form. Many outstanding relativistic effects have been elucidated here. The relativistic law of mass action was reformulated through dynamic-equilibrium concept on the basis of these theories. Subsequent thermodynamical expression of rate constant gave an excellent model system for introducing temperature transformation. Relativistic-enzyme kinetics may realize an experimental detection of the relativistic effects.

A method based upon Pearson-type approximations from statistics is developed for approximating a symmetric probability density function from its moments. The extended Fokker-Planck equation for non-Markov processes is shown to be the underlying foundation for the approximations. The approximation is shown to be exact for the beta probability density function. The applicability of the general method is illustrated by numerous pithy examples from linear and nonlinear filtering of both Markov and non-Markov dichotomous noise. New approximations are given for the probability density function in two cases in which exact solutions are unavailable, those of (i) the filter-limiter-filter problem and (ii) second-order Butterworth filtering of the random telegraph signal. The approximate results are compared with previously published Monte Carlo simulations in these two cases.

We show that a dislocation-loop theory of the smectic-A–nematic transition predicts the existence of both nematic regions surrounded by smectic regions and a smectic–$A_1$-$A_d$ critical point in the phase diagram, in agreement with recent observations. Fluctuation-corrected mean-field theory leads to a remarkable topology for the isolated nematic region.

A reversible-growth model is built by modifying the cluster-cluster aggregation model with a finite interparticle attraction energy -E. When E is ∞, the aggregation is described by the ordinary cluster-cluster aggregation model. Within our model, particles as well as clusters are performing Brownian motion according to the rate 1/${\tau }_D$, and the unbinding takes place according to (1/${\tau }_R$)$e^{\mathrm{-}\Delta E/T}$, where ΔE is the energy change due to the unbinding, T is the room temperature, and ${\tau }_R$ is the time constant associated with the unbinding. The Boltzmann constant is taken to be unity. By changing E and ${\tau }_R$/${\tau }_D$, we are able to change the aggregation behavior over a wide range from ramified clusters to compact ones. Moreover, due to a finite E, ramified aggregates may become compact at a later time. We show that the initially fractal aggregates can remain fractal objects during restructuring while the fractal dimension D increases with time. At large E, D can stay at some value that is larger than the value of the cluster-cluster aggregation model and can remain unchanged for a long time. At a given time, D increases drastically with decreasing E from the value of the cluster-cluster aggregation model when E≤3T. The curve of the estimated sedimentation density versus E resembles that of D versus E and agrees with the experiments.

The existence of self-similar solutions of the one-dimensional Fokker-Planck equation ∂P/∂t=(∂/∂x)(RP+D ∂P/∂x) poses specific constraints on the shape of the drag and diffusion coefficients R(x) and D(x) as functions of the space variable x. It is shown that these constraints can be formulated in terms of a single Riccati equation which can be solved exactly. The form of the functions R(x) and D(x) leading to self-similar evolution of one-dimensional systems governed by the above Fokker-Planck equation is therefore determined in full generality.

The asymptotic rate of dispersion in generalized Taylor diffusion is characterized by a longitudinal diffusion coefficient K. On the basis of the analogy with the kinetic theory of gases, and following a general prescription to derive variational principles, we obtain two lower bounds, or rather, infimum principles for K. Their usefulness is illustrated on simple examples.

If V(r) is homogeneous in the position coordinates, then the Schrödinger equation with H(t)=T+α(t)V(r) is separable in time if α(t) is properly chosen. This fact makes the time evolution of such systems much easier to compute than it would otherwise be. Examples of such systems are the hydrogen atom with time-varying nuclear charge, the harmonic oscillator with a changing spring constant, and the infinite square well with moving walls. If α(t) is made periodic by concatenating ‘‘properly chosen’’ segments, then a unitary operator U can be defined which evolves the system through one period. If the wave function is expressed as an N-term truncated expansion of eigenfunctions of H(t), then U is an N×N matrix which can be computed straightforwardly. The properties of U are studied by examining the statistical properties of its eigenvectors and eigenvalues, whose phases are the quasienergies. Numerical examples are given for the Fermi-Ulam cosmic accelerator (FUCA), which is a particle in a one-dimensional box with a moving wall, with various values for relevant parameters, which can lead to Poisson, Gaussian orthogonal ensemble, Gaussian unitary ensemble, or other quasienergy statistics, corresponding to inte- grable, chaotic, and non-time-reversible chaotic situations. Poincaré sections for the classical FUCA are also displayed. They show regimes of regular and of chaotic motion, whose correspondence with the statistics of U is sought with limited success.

A recent new microscopic, molecular statistical-physics theory for the intrachain constraints on the n-alkyl tail-chain flexibility for molecules composed of rigid rodlike cores and n-alkyl tail chains in the smectic-$A_1$, smectic-$A_d$, and nematic (including reentrant nematic) liquid-crystal phases and the isotropic liquid phase is used to calculate relative stabilities as well as various thermodynamic and molecular ordering properties (including odd-even effects) for these phases. The molecules can interact via these site-site intermolecular interactions: hard (steric) repulsions, various Lennard-Jones (LJ) potentials for soft repulsions and London dispersion attractions, and/or dipolar forces (dipole-dipole and dipole–induced-dipole). The theoretical results in this paper are in significantly better agreement with experiment than are results in earlier papers using older, more approximate theories. The effects of varying pressure, tail-chain flexibility, and intermolecular interactions on the relative stabilities of the multiple smectic-A phases and the reentrant nematic phase are presented. These relative phase stabilities are sensitive to subtle changes in these variables. Predictions and accompanying physical explanations are also made for various systems that have not yet been chemically synthesized and/or studied experimentally.

We study the node-avoiding (NALF) and path-avoiding (PALF) extensions of the Lévy flights in one dimension both numerically and analytically. The asymptotic behavior of the PALF is determined exactly while that of NALF has been available from the mapping to a spin model. Monte Carlo results for both types of self-avoiding Lévy flights are used to study the convergence toward the asymptotic behavior. We find very large corrections to asymptotic scaling in NALF for a wide range of the Lévy index μ and also, surprisingly, that the moments of the end-to-end distance of the NALF are greater than those of the PALF when they both exist. Based on these observations we conclude that the morphology of the NALF is far more complex than that of the PALF or the random Lévy flights, and that the NALF and PALF are certainly in different universality classes in one dimension.

We examine the general kink-antikink (bion) collision process in the driven damped sine-Gordon equation. A collective-coordinate method is used to study several aspects of the bion dissociation and annihilation. Thresholds for the driving term are numerically obtained in both cases. The limitations and reliability of our reducing approach are discussed.

It is shown by calculations on several model systems of physical importance that exponential time-dependent perturbation theory (the Magnus expansion and similar expansions) does not converge in the Schrödinger representation for time intervals larger than the natural period of the system. This result has serious implications for the application of exponential perturbation theory in the Schrödinger representation to scattering and adiabatic turn-on problems. It is argued that this result also applies generally to more realistic systems.