New diffusion-limited growth models are described in which growth occurs at the site which is first contacted by k (k is a small integer) random walkers launched from outside the growing cluster and adsorbed on contact with the cluster. This model is closely related to the dielectric-breakdown model in which the growth probability is proportional to the kth power of the field gradient at the surface of the growing cluster. For the case k=2 our results for the radius-of-gyration exponent β are consistent with the theoretical predictions of Turkevich and Sher. However, our results do indicate that the main assumption of the theory of Turkevich and Sher (the formation of a diamond shape on the square lattice) is not correct in this case. The clusters grow into a crosslike shape in which the increase in length and width of the arms of the cross with increasing cluster size are described by different exponents. A similar anisotropy is not observed from the largest clusters which can be grown on hexagonal lattices for the cases k=2, 3, and 4. For k=2 the radius-of-gyration exponent has effective values of about 0.73 and 0.70 for the largest clusters which can be grown on square and hexagonal lattices, respectively.

The noncanonical Hamiltonian formalism is based upon a generalization of the Poisson bracket, a particular form of which is possessed by continuous media fields. Associated with this generalization are special constants of motion called Casimir invariants. These are constants that can be viewed as being built into the phase space, for they are invariant for all Hamiltonians. Casimir invariants are important because when added to the Hamiltonian they yield an effective Hamiltonian that produces equilibrium states upon variation. The stability of these states can be ascertained by a second variation. Goldstone’s theorem, in its usual context, determines zero eigenvalues of the mass matrix for a given vacuum state, the equilibrium with minimum energy. Here, since for fluids and plasmas the vacuum state is uninteresting, we examine symmetry breaking for general equilibria. Broken symmetries imply directions of neutral stability. Two examples are presented: the nonlinear Alfvén wave of plasma physics and the Korteweg–de Vries soliton.

A detailed analysis of the plasma-maser effect is shown. Coupled equations for the Langmuir and ion sound waves are reduced to the Lotka-Volterra type. A relaxation oscillation around a steady state is analyzed. The finite-amplitude effect of the ion sound pump field on the excitation of the Langmuir wave is qualitatively discussed.

The dynamical structure of systems of coupled discrete-space, discrete-time oscillators is described. The behavior of these systems is considered for different forms of the coupling term, and parallels with cellular automaton models are pointed out. An analysis of the dynamics of dislocation structures is made and the mechanisms for their creation and destruction are discussed. Chaotic spatial patterns may arise near intermittency transitions and the mechanisms which give rise to these structures are studied. External noise can have important effects on the character of spatial patterns and their evolution; such effects are also briefly considered.

New approximate integral equations for the three- and four-body correlation functions and structure functions are derived. These approximations are based on the hypernetted-chain resummation. The equation for the three-body correlation function satisfies a self-consistency condition, the Born-Green-Yvon equation, and has the Kirkwood superposition approximation as the leading or nonhomogeneous term. Similarly, the new approximation for the third-order structure function has the convolution approximation as its leading term. In addition to the formalism, numeric solutions are presented for a Lennard-Jones system corresponding to argon at its triple point and at room temperature (with its triple-point density). At this high density, the integral equations give results significantly different from their leading terms.

The recently developed theory for tunneling in dissipative systems is rederived using quantal transition-state theory. As predicted by Caldeira and Leggett, we find an exponential damping of the tunneling rate at 0 K. The exponential rate enhancement at low temperatures as well as the crossover temperature are also obtained with this approach. Moreover, the rate enhancement is given explicitly in terms of energy transfer from the bath to the dissociative mode. The present derivation includes memory effects.

Nosé has developed many-body equations of motion designed to reproduce Gibbs’s canonical phase-space distribution. These equations of motion have a Hamiltonian basis and are accordingly time reversible and deterministic. They include thermodynamic temperature control through a single deterministic friction coefficient, which can be thought of as a control variable or as a memory function. We apply Nosé’s ideas to a single classical one-dimensional harmonic oscillator. This relatively simple system exhibits both regular and chaotic dynamical trajectories, depending on the initial conditions. We explore here the nature of these solutions by estimating their fractal dimensionality and Lyapunov instability. The Nosé oscillator is a borderline case, not sufficiently chaotic for a fully statistical description. We suggest that the behavior of only slightly more complicated systems is considerably simpler and in accord with statistical mechanics.

The growth of correlations between velocity and magnetic fields in fully developed magnetohydrodynamic turbulence, conjectured by Dobrowolny, Mangeney, and Veltri [Phys. Rev. Lett. 45, 144 (1980)], is observed in both two- and three-dimensional numerical simulations with resolution up to 512×512 and 64×64×64, respectively. In all cases the correlation spectrum has for long times a simple structure with opposite signs at large and small scales.

The structure of a fluid subjected to a plane Couette flow is calculated starting from a simple model kinetic equation. After the presentation of the general formal solution of this kinetic equation, an expansion in powers of the shear rate is investigated in detail; terms up to order 6 are taken into account. As a specific example, the shear-flow-induced distortion of the structure is analyzed and displayed graphically in the vicinity of its first peak. The pertinent equilibrium structure was chosen to correspond to a soft-sphere fluid where computer simulation data are available for the radial distribution function. Two distinct scattering geometries are considered.

The cold-fluid Maxwell equations are used to investigate detailed equilibrium properties (∂/∂t=0) of relativistic non-neutral electron flow in cylindrical diode geometry with applied magnetic field. There is considerable latitude in specifying the equilibrium profiles. In particular, the profiles for any one of the density $n_b^0$(r), axial magnetic field $B_z^0$(r), radial electric field $E_r^0$(r), or azimuthal flow velocity $V_{\mathrm{thetab}}^0$(r)=${\beta }_b$(r)c can be specified, and the remaining three profiles calculated self-consistently from Poisson’s equation, the ∇×${\mathrm{B}}^0$ Maxwell equation, and the radial force balance equation. As a first example, assuming that the functional form of ${\beta }_b$(r) is specified, formal expressions are derived for $B_z^0$(r), $n_b^0$(r), and $E_r^0$(r). As a second example, it is assumed that the radial electric field is prescribed by $E_r^0$(r)=const×sinh[κ(r-a)] over the radial extent of the electron layer extending from r=a (the cathode) to r=$r_b$ (the outer edge of the electron layer). In this case, a closed-form expression for the density profile $n_b^0$(r) is obtained, but the profiles for the magnetic field $B_z^0$(r) and flow velocity ${\beta }_b$(r)c must generally be determined numerically when the layer aspect ratio A=a/($r_b$-a) is finite. The influence of cylindrical effects (e.g., centrifugal effects) and finite aspect ratio A on detailed equilibrium properties is calculated, treating the planar diode limit (A=∞) as a reference case where the equilibrium profiles are known in closed analytic form. It is found that cylindrical effects and finite aspect ratio can have a large influence on profile shape, particularly when the electron flow is relativistic.

The effective ionization and recombination rate coefficients for Ti xvii–xx were obtained by studying the modulation of line emission by sawtooth oscillations in the TEXT tokamak at the University of Texas. Experimental data are matched with the simulation of titanium-ion emission by a one-dimensional (1D) transport code which includes ionization and recombination processes. The rate coefficients were determined for $T_e$≊600–1100 eV and $N_e$∼4×${10}^{13}$ ${\mathrm{cm}}^{\mathrm{-}3}$. The ionization rates were about 20% higher than values calculated by Lotz’s formula. The recombination rates were about 20% less than theoretical values. The uncertainty of ionization and recombination rate coefficients is estimated to be less than 20% and 60%, respectively. The sensitivity of our method to various parameters, i.e., the diffusion coefficient, convective velocity, and electron temperature, is also discussed.

An exact relationship between the equations for the dynamics of average variables and those for their fluctuations is established for Markov processes. The result includes the general case of nonlinear and nonstationary dynamics and provides an extension of Onsager’s regression hypothesis to a broad class of systems far from equilibrium.

We present theoretical and experimental results on quasistatic-electric- and optical-field-induced birefringence as well as quasistatic-electric-field-enhanced optical diffraction effects in a nematic-liquid-crystal film. For an inclined-incident pump beam, the enhancement of the diffraction effect and birefringence values due to the quasistatic electric field does not saturate at high quasistatic electric field strength. Instead, it decreases from the maximum, through zero (i.e., no effect), to a region of negative coupling between the fields. Locally this novel zero-crossing phenomenon occurs at a quasistatic electric field strength for which the nematic director has been reoriented by the electric field to align with the polarization direction of the optical field. The zero-crossing effect manifested by the sample as a whole is the net result of the field-coupling effect on the local director along the beam path, which locally could be positive, zero, or negative. This novel effect is also independent of the pump-laser intensity. Our experimental results are in quantitative agreement with theoretical predictions.

A model for the phase separation of binary mixtures in porous media is analyzed within the mean-field approximation. The calculation is based upon a Landau-Ginzburg free energy, modified to include interactions between voids in the medium which are separated by narrow capillaries. These quench fluctuations. Two effects are observed: (1) a lowering of the transition temperature and (2) a finite wave vector phase transition if the correlations of the porous medium are sufficiently long ranged or if the transfer of material between different voids is difficult.

A fluid-filled horizontal annulus is formed by two coaxial circular cylinders of nearly equal radii. When subjected to heating, there is one convective state that is uniform in the axial coordinate and so is equivalent to the steady loop convection of the Lorenz equations. This loop convection is susceptible to axially periodic instabilities analogous to the zigzag or cross-roll instabilities on two-dimensional Bénard convection. The loop convection bifurcates from the conductive state at Rayleigh number R=1 and is itself unstable to axially periodic steady convection at a critical value R=$R_s$(k) at a wave number k where $R_s$→1 as k→0 so that asymptotically the loop solution is always unstable to steady long-wave instabilities. We analyze the long-wave structure by deriving a coupled system of nonlinear evolution equations that give the interaction of the k=0 loop solution and the k→0 spatially periodic convection.

We investigate the quantum energy-level sequence of two coupled quartic oscillators. The classical system is a scaling one and has been shown elsewhere to undergo a transition from regular to irregular motion for increasing coupling strength. The distribution of spacings between adjacent energy levels P(S) and the ${\Delta }_3$ statistic indicate a corresponding transition from an uncorrelated spectrum where successive levels arrive randomly to a spectrum described by the Gaussian orthogonal ensemble of random matrices. For P(S) we use Berry and Robnik’s semiclassical formulas which depend on the volumes of regions in phase space filled with chaotic trajectories and the extension of this ansatz to the ${\Delta }_3$ statistic. We investigate systematic deviations from this picture due to quantum effects and discuss their universality. We find that P(S) is mainly sensitive to the total irregular fraction of phase space, whereas ${\Delta }_3$ strongly reflects its partitioning into several parts. For vanishing coupling strength, where the system becomes integrable, we find extremely slow convergence to the semiclassical limit.

The Wigner-Kirkwood expansion for matrix-valued potentials is derived by first discussing the scalar problem within a framework that directly admits a matrix generalization. Attention is confined to the space-diagonal problem for which explicit calculations are presented up to fourth order in derivatives of the potential.

The effects of a self-generated magnetic field on the density-profile modification in laser-plasma interactions are investigated by means of calculating the field structure with the assumption of plane electromagnetic waves propagating into a one-dimensional inhomogeneous magnetized plasma. The computed results are in good agreement not only with laser-target experiments using a Nd-glass laser, but also with those using a ${\mathrm{CO}}_2$ laser. In addition, the calculations give the density dip also observed in experiments.

The phenomenon of anomalous fluctuations associated with the decay of an unstable state is analyzed in the presence of multiplicative noise. A theory is presented and compared with a numerical simulation. Our results allow us to distinguish the roles of additive and multiplicative noise in the nonlinear relaxation process. We suggest the use of experiments on transient dynamics to understand the effect of these two sources of noise in problems in which parametric noise is thought to be important, such as dye lasers.

In experiments, solitons emerge from arbitrary initial data even when the Hamiltonian perturbations are quite large. In this paper, it is shown explicitly through lowest order that the perturbations which appear in ion acoustic wave and shallow-channel water-wave experiments do not destroy, but merely renormalize, their velocity and shape. Comparisons with other perturbation approaches are made.