It is shown that the negative diffusion coefficients exhibited by the current approaches to the Fokker-Planck equation for non-Markovian and bistable processes result from assuming the diffusion process to be stationary. The rejection of this assumption leads to predictions on the escape over a barrier agreeing with the physical arguments pertaining to the extremely colored-noise regime.

A theory of self-diffusion in concentrated dispersions is presented, in which the effects of hydrodynamic and direct interactions between macroparticles are virtually decoupled. The long-time self-diffusion coefficient $D_s^L$ is written in terms of the radial distribution function of the suspended particles and of their short-time self-diffusion coefficient $D_s^S$. The predictions for the volume fraction dependence of $D_s^L$ in hard-sphere suspensions are in excellent agreement with available experimental results.

For a general isotropic two-dimensional theory near criticality, the universal singular free energy per correlation volume $f_s{\xi }^2$ is equal to $-\left(\frac{c}{12}\pi \right)(2-\alpha ){(1-\alpha )}^{-1\mathrm{}}$, where $c$ is the central charge of the theory at criticality, $\alpha$ is the usual specific-heat exponent, and $\xi$ is defined (for $\alpha >0\mathrm{}$) in terms of the second moment of the energy correlations. Some generalizations of this result are also noted.

We present a theory to study external non-Markovian noise in diffusion processes by using the multistate continuous-time random walk. We show that if the external noise is anomalous the effective Green's function of the walker has no long-time tails. We study the internal fluctuations in presence of non-Markovian noise.

The statistical hypothesis underlying the "strange attractor" explanation of fluid turbulence is suspect. Spatially extended systems generically exhibit long transients that preclude observation of the behavior governed by the asymptotic invariant measure. Even if the local dynamics is periodic, when it is coupled into a spatial system complex "turbulent" behavior can exist for times that grow faster than exponentially with increasing system volume. The nature of the attractor is irrelevant to the observed behavior when such systems are of even moderate size.

We present measurements of the phase diagram for a nonlinear electronic oscillator circuit which has three intrinsic frequencies. The persistence of chaos to almost zero intermode coupling and the measured structure of the chaotic zones in parameter space provide unprecedented detail in the interpretation of the Ruelle-Takens scenario for the onset of chaos.

The nucleation rate of interacting thermal kinks and antikinks in a sine-Gordon chain is evaluated within the dilute-gas approximation. Our novel result describes both the diffusive and nondiffusive limits. In comparison with the nondiffusive and noninteracting nucleation rate calculated previously by Büttiker and Landauer, interactions modify the temperature dependence of the prefactor and, in the diffusive limit, the Arrhenius factor as well.

A theoretical result for correlated quantum systems is presented which leads to a noise commutation relation and a generalized Heisenberg uncertainty relation. These relations imply an inherent and unavoidable extra noise in quantum measurements beyond that already included in the Heisenberg lower bound. These relations lead directly to model-independent lower bounds on inherent noise, useful in a variety of applications, including balanced homodyne detection and quantum optical linear amplifiers.

We develop a scaling theory for linear fragmentation processes, for general breakup kernels characterized by a homogeneity index $\lambda >0\mathrm{}$. We discuss the existence of scaling, and show that the scaled cluster-size distribution $\phi \mathrm{}\left(x\right)\mathrm{}$ generally decays with the scaled mass $x$ as $x^{-2\mathrm{}}\exp (-a{x}^{\lambda })$ as $x\to \infty$. For small $x$, $\phi \mathrm{}\left(x\right)\mathrm{}$ approaches the log-normal form, $\exp (-a{\ln }^{2}x)$, if the kernel has a small-size cutoff, and a power-law form in the absence of a cutoff. We also show that $\lambda <0\mathrm{}$ leads to a shattering transition. Finally, we outline the essential features of a nonlinear fragmentation process.

It is shown that Berry's phase appears in a more general context than realized so far. The evolution of the quantum system need be neither unitary nor cyclic and may be interrupted by quantum measurements. A key ingredient in this generalization is the use of some ideas introduced by Pancharatnam in his study of the interference of polarized light, which, when carried over to quantum mechanics, allow a meaningful comparison of the phase between any two nonorthogonal vectors in Hilbert space.

We derive the exact probability distribution of the winding angle of a two-dimensional self-avoiding walk around its extremities, using Coulomb-gas techniques and conformal invariance. This gives a first example of an exact full distribution in critical geometrical models. It is extended to dense and $\Theta$ self-avoiding walks.

Critical indices $\beta$, $\gamma$ $\delta$, $\nu$, etc. are defined and calculated for self-organized critical phenomena. Scaling relations are derived and checked numerically. The order-parameter exponent $\beta$ describes the spontaneous current and the relaxation to the critical point. The power spectrum has "$\frac{1}{f}$" behavior with exponent $\phi =\frac{\gamma }{\nu z}$, where $z$ is the dynamical critical exponent.

The long-time tail in the diffusive behavior in a strongly chaotic system (stadium billiard) is studied from the point of view of "stickiness" of invariant lines. Diffusion near the invariant line in the phase space can be described by a one-dimensional continuous random-walk problem with pausing time which is inversely proportional to the distance from the line. The problem is studied by numerical methods and it is shown that the first-passage-time distribution has an algebraic decay tail in the form $\sim t^{-v}$ with the estimated exponent $v\simeq 2.933\pm 0.005$.

Wigner statistics for correlations of matrix eigenvalues are shown to be a property of any matrix ensemble with a density of levels and probability distributions for matrix elements that are smooth. This justifies the universality of level correlations in generic quantum systems, while suggesting that level widths and other eigenvector-dependent statistics are system dependent.

A computer simulation of a representative continuum system, the inverted random-void model in three and six dimensions, is reported. It is the first such simulation where the local geometry of the conducting particles is taken into account. The results show that the critical behaviors of both the electrical resistivity and the resistance noise, near the percolation threshold, are well described by the recently suggested models of links in the nodes-links-blobs picture.

A dilute nonsymmetric ferromagnet and a version of the Hopfield model are solved in the limit where the average number of inputs per spin is finite. The solution is based on the fact that the magnetization of each site is a function of the number and the type of inputs and their connection strengths. Extension of this solution to the symmetric case is discussed. The solution can also be extended to any other distribution of interactions.

The average eigenvalue distribution $\rho \left(\lambda \right)$ of $N\times N$ real random asymmetric matrices $J_{\mathrm{ij}} \left(J_{\mathrm{ji}}\ne J_{\mathrm{ij}}\right)$ is calculated in the limit of $N\to \infty$. It is found that $\rho \left(\lambda \right)$ is uniform in an ellipse, in the complex plane, whose real and imaginary axes are $1+\tau$ and $1-\tau$, respectively. The parameter $\tau$ is given by $\tau =N{\left[J_{\mathrm{ij}}{J}_{\mathrm{ji}}\right]}_J$ and $N{\left[J_{\mathrm{ij}}^2\right]}_J$ is normalized to 1. In the $\tau =1\mathrm{}$ limit, Wigner's semicircle law is recovered. The results are extended to complex asymmetric matrices.

Spatiotemporal chaos can be produced by complicated local dynamics in a small spatial region and observed globally through a process we call information transport. Information transport can be detected by computation of an information-theoretic quantity, the time-delayed mutual information, between measurements of the system at separate spatial points.

The conditions for a macroscopic segregation of A and B in a steady-state A+B→0 reaction are studied via a finite continuum model. Segregation occurs in d=1, is marginal for d=2 (critical dimension), and is absent for d≥3. Results for infinite systems are also mentioned.

The assumptions underlying the revised Enskog kinetic theory are used to determine the dynamics of correlations between time-displaced fluctuations in an equilibrium hard-sphere fluid, leading to a new expression for the spectrum observed in the depolarized interaction-induced light scattering.