We consider mean first-passage times (MFPT’s) for systems driven by non-Markov gamma and McFadden dichotomous noises. A simplified derivation is given of the underlying integral equations and the theory for ordinary renewal processes is extended to modified and equilibrium renewal processes. The exact results are compared with the MFPT for Markov dichotomous noise and with the results of Monte Carlo simulations.

We study the probability distribution of a passive scalar undergoing turbulent mixing in the presence of a mean scalar gradient. Kerstein’s model, which describes the turbulent mixing process as a collection of instantaneous local rearrangements of the scalar, is argued to be a plausible approximation when the size of the system is much larger than the velocity correlation length, and after the description has been coarse grained over a correlation volume. In the physical range of parameters, we find numerically that the fluctuations of the scalar θ are close to exponentially distributed when a linear mean scalar gradient is imposed. The phenomenological mean-field-like theory of Pumir, Shraiman, and Siggia [Phys. Rev. Lett. 66, 2984 (1991)] is derived heuristically beginning with the Kerstein model. This theory predicts strictly exponential tails for the probability distribution of the scalar fluctuations P(θ), i.e., P(θ)∼exp(‖θ‖). We also consider a simplified version of the Kerstein model which is analytically fully tractable and gives qualitatively similar results numerically. Under conditions of spatial homogeneity and imposed linear mean scalar gradient, this simplified model and the full Kerstein model are described by the same mean-field theory. However, for the simplified model, the large-‖θ‖ asymptotic behavior of P(θ) is Poisson-like, i.e., P(θ)∼exp[-‖θ‖ln‖θ/(const)‖].

In recent numerical experiments on series arrays of overdamped Josephson junctions, Nichols and Wiesenfeld [Phys. Rev. A 45, 8430 (1992)] discovered that the periodic states known as splay states are neutrally stable in all but four directions in phase space. We present a theory that accounts for this enormous degree of neutral stability. The theory also predicts the four non-neutral Floquet multipliers to within 0.1% of their numerically computed values. The analytical approach used here may be appli- cable to other globally coupled systems of oscillators, such as multimode lasers, electronic oscillator circuits, and solid-state laser arrays.

The correlation functions ${\mathit{C}}_{\mathrm{\phi }}$ and ${\mathit{C}}_{\mathrm{\phi }}^2$, associated with the order-parameter field φ→(r,t) and its square, respectively, are discussed using heuristic arguments and an approximate analytical approach. Topological defects (walls, strings, monopoles) in the field, seeded by a quench from the high- to the low-temperature phase, lead to singular short-distance behavior in the scaling functions, and power-law tails in the corresponding structure factors. For superfluid helium, the structure factor ${\mathit{S}}_{\mathrm{\phi }}^2$(k,t) is measurable in principle using small-angle scattering (whereas ${\mathit{S}}_{\mathrm{\phi }}$ is inaccessible). It is predicted to exhibit a power-law tail, ∼[${\mathit{a}}^4$/L(t${)}^2$](lnka${)}^2$/k, where L(t) is the characteristic scale at time t after the quench and a is the core size of a vortex line. Correlation functions for the defect density are also discussed.

The motion of a proton in a double-well potential created by the potentials of two heavy ions in a molecular chain is considered. The topology of the potential changes depending on the distance between the molecules. Both forms, double well and single well, are possible. Individual chaotic proton motion is triggered by the oscillations of the lattice. Depending on the system parameters, both the one-well and the cross-well attractors can be either periodic or chaotic. This has some interesting consequences for the interpretation and understanding of propagation of ionic defects in hydrogen-bonded chains in the presence of external oscillating fields. A frequency-locked propagating kink is found.

Dynamic light scattering is applied to the glass phase of nonaqueous suspensions of sterically stabilized colloidal spheres. The short-ranged steric repulsion ensures that the particle interactions are close to hard sphere. This is supported by the observation that the equilibrium phase behavior of these suspensions agrees with that predicted for the hard-sphere atomic system. We verify a model for a nonergodic medium, which assumes that the particles are localized during an experiment and which allows the intermediate scattering function to be calculated from a single measurement of the time-averaged intensity autocorrelation function. Intermediate scattering functions are obtained for several concentrations over a range of wave vectors around the main diffraction peak. The measured nonergodicity parameters are in good agreement with the predictions of mode-coupling theory for the hard-sphere glass. The comparison involves no adjustable parameters. At long times the intermediate scattering functions can be scaled to a single curve for over 2.5 decades in time. This, combined with the results that the nonergodicity parameters and critical amplitudes required for the scaling are in quantitative agreement with mode-coupling theory, provides a convincing verification of the predicted factorization property of the β process in the glass phase.

The self-avoiding random walk on lattices with quenched random site energies is studied using exact enumeration in d=2 and 3. For each configuration we compute the size R and energy E of the minimum-energy self-avoiding walk (SAW). Configuration averages yield the exponents ν and χ, defined by ${\mathit{R}}^2$¯∼${\mathit{N}}^{2\nu }$ and δ${\mathit{E}}^2$¯∼${\mathit{N}}^{2\mathrm{\chi }}$. These calculations indicate that ν is significantly larger than its value in the pure system. Finite-temperature studies support the notion that the system is controlled by a zero-temperature fixed point. Consequently, exponents obtained from minimum-energy SAW’s characterize the properties of finite temperature SAW’s on disordered lattices.

The unstable periodic orbits of a chaotic NMR laser have been identified and classified according to their stability properties. The dynamical control method by Ott, Grebogi, and Yorke (OGY) [Phys. Rev. Lett. 64, 1196 (1990)] has been successfully applied in a six-dimensional embedding space, as required for a proper reconstruction of the attractors. An improved method has then been implemented in order to circumvent the occurrence of several, possibly complex, unstable eigenvalues in the standard OGY procedure and has shown superior performance. Finally, an extension to several control stations per Poincaré time is discussed.

The Mori-Lee formalism for solving the Liouville (or Heisenberg) equation of motion for Hermitian systems demonstrates that the Laplace-transformed dynamical correlations in canonical ensembles can be written as (in)finite continued fractions. We show that a model-independent direct summation method allows accurate numerical evaluation of all known classes of these (in)finite continued fractions that arise in dynamics problems and thus provides a powerful technique to study the dynamics of many-body and few-body systems. Some studies on dynamical correlations in s=1/2 quantum spin chains are cited as applications of the method presented.

In a recent Letter [Phys. Rev. Lett. 67, 664 (1991)] we found semiclassical propagation to be remarkably accurate in the chaotic stadium billiard long after classical fine structure had developed on a scale much smaller than ħ. We give a complete account of that work and derive an approximate time scale for the validity of the semiclassical approximation as a function of ħ.

We extend the master-equation treatment of dynamical evolution of a system-plus-reservoir configuration including the propagation of initial correlations as a noise source. Specializing into the quantum harmonic oscillator coupled to a fermionic heat bath, we develop a model for the diffusion matrix in the space of diagonal density operators. It can be shown that mean values of observables undergo Langevin-like motion and, in particular, that the mean value and dispersion of the oscillator quanta approach the canonical equilibrium values. A final interpretation of the characteristics and role of the noise source is given.

This paper describes a procedure to steer rapidly successive iterates of an initial condition on a chaotic attractor to a small target region about any prespecified point on the attractor using only small controlling perturbations. Such a procedure is called ‘‘targeting.’’ Previous work on targeting for chaotic attractors has been in the context of one- and two-dimensional maps. Here it is shown that targeting can also be done in higher-dimensional cases. The method is demonstrated with a mechanical system described by a four-dimensional mapping whose attractor has two positive Lyapunov exponents and a Lyapunov dimension of 2.8. The target is reached by making very small successive changes in a single control parameter. In one typical case, 35 iterates on average are required to reach a target region of diameter ${10}^{\mathrm{-}4}$, as compared to roughly ${10}^{11}$ iterates without the use of the targeting procedure.

We consider the fully developed chaos in a class of driven one-degree-of-freedom nonlinear systems. In analogy to Kubo relations in statistical mechanics, we have quantitatively related the maximal positive Lyapunov exponent, which is characteristic of divergence of trajectories, to the spectral density of fluctuations of the appropriate dynamical variable. A numerical experiment is carried out to confirm the qualitative validity of the theoretical prediction. A generalization of the relationship for N-dimensional Hamiltonian system has been given.

For a variety of ‘‘random configurations’’ of equal disks, the probability distributions of the Voronoi-cell areas are obtained from a computer simulation and a comparison is made in terms of the relation between the regularity of the cell areas and the mean bulk fraction of the area occupied by the disks.

The relationship between the stochastic and deterministic approaches to the description of the dynamic behavior for a nonlinear reacting system is studied. A self-propagating exothermal reaction in a heterogeneous medium has been chosen as an example. A stochastic model for this phenomenon is developed. The system is composed of cells characterized by temperature and degree of conversion, their transformation probability being dependent on temperature. Computer simulation showed that stochasticity reveals itself in the generation of disturbances that are absent in the deterministic model. For a well-developed steady-state regime, the distributions of the mean temperature and degree of conversion are close to those given by the deterministic model. Under the conditions of planar-wave-front instability, the stochastic model possesses a mechanism for spontaneous transfer to a stable regime from arbitrary initial conditions (temperature distribution, etc.) due to origination, propagation, and disintegration of disturbances. Such behavior agrees with experimental data and is not predicted by the deterministic model.

The Boltzmann equation for reacting charged-particle swarms in neutral gases in the presence of both electric and magnetic fields is decomposed into a hierarchy of kinetic equations by expanding the velocity dependence of the phase-space distribution function in terms of spherical harmonics. No limit is set on the number of spherical harmonics and no approximation is made concerning the mass of the charged particles related to that of the neutrals species. The space-time dependence is treated by making the hydrodynamic assumption which is taken to second order in density gradients. Spherical tensors are used throughout. The resulting heirarchy of equations has universal validity and is amenable to a range of numerical solutions. The structure of these equations is discussed and the inadequacies of a Legendre-polynomial expansion are pointed out. The special configurations of the magnetic field parallel and perpendicular to the electric field are discussed in detail.

Algorithms for molecular dynamics (MD) at constant temperature and pressure are investigated. The ability to remain in a regular orbit in an intermittent chaotic regime is used as a criterion for long-time stability. A simple time-centered algorithm (leap frog) is found to be the most stable of the commonly used algorithms in MD. A model of N one-dimensional dimers with a double-well intermolecular potential, for which the distribution functions at constant temperature T and pressure P can be calculated, is used to investigate MD-NPT dynamics. A time-centered NPT algorithm is found to sample correctly and to be very robust with respect to volume scaling.

Anomalous viscoelastic, diffusion, and heat-transfer phenomena observed in spatially inhomogeneous simple and complex fluids are analyzed in this paper in the setting of weakly nonlocal hydrodynamics. Governing equations of this generalized hydrodynamics involve higher-order derivatives with respect to the position coordinate. The governing equations are obtained on the basis of the following consideration. The time evolution in both local and weakly nonlocal hydrodynamics is generated by a thermodynamic potential in a state space equipped with a Poisson structure and a dissipative potential. The Poisson structure is an expression of kinematics in the chosen state space. In weakly nonlocal hydrodynamics the Poisson structure remains the same as in local hydrodynamics but the potentials are generalized. The potentials are allowed to depend on higher-order derivatives of the hydrodynamic fields that are chosen as state variables. This way of introducing the governing equations guarantees that the equations are intrinsically compatible and that their solutions agree with certain fundamental macroscopic observations (e.g., the observations constituting the basis of equilibrium thermodynamics).

Evolution of slow magnetoacoustic perturbations is used to demonstrate a new mechanism of plane Couette flow shear energy transformation. The initial uniform magnetic field is considered to be directed along the unperturbed flow velocity. We find that the amplification of waves is nonordinary—it occurs in a limited time interval and is anomalously strong. The unusual behavior is manifested also by the fact that the characteristic length scale of perturbation diminishes down to the dissipative scales not only due to nonlinear cascade processes, but due to a linear drift in the k space as well.

We consider the effect of viscosity on Rayleigh-Taylor (RT) and Richtmyer-Meshkov (RM) instabilities by deriving a moment equation for fluids with arbitrary density and viscosity profiles, including surface tension. We apply our result to the classical case of two semi-infinite fluids with densities ${\mathrm{\rho }}_1$ and ${\mathrm{\rho }}_2$ and viscosities ${\mathrm{\mu }}_1$ and ${\mathrm{\mu }}_2$. Treating a shock as an instantaneous acceleration we find that perturbations at the interface undergo damped oscillations when viscosity and surface tension are both present. For pure viscosity the amplitude η(t) evolves according to η(t)/η(0)=1+(ΔvA/2kν)(1-${\mathit{e}}^{\mathrm{-}2\mathit{k}2}$νt) where Δv is the jump velocity imparted by the shock, A=(${\mathrm{\rho }}_2$-${\mathrm{\rho }}_1$)/(${\mathrm{\rho }}_2$+${\mathrm{\rho }}_1$), ν=(${\mathrm{\mu }}_1$+${\mathrm{\mu }}_2$)/(${\mathrm{\rho }}_1$+${\mathrm{\rho }}_2$), k=2π/λ is the wave number of the perturbation, and t is time. We also consider the turbulent energy in accelerating fluids and calculate the reduction in ${\mathit{E}}_{\mathrm{turbulent}}$ as a function of ν, and propose experiments to measure the effect of viscosity on RT and RM instabilities.