Extending our previous work on two-dimensional growth for the Laplace equation [M. B. Mineev, Physica D 43, 288 (1990)] we study here multidimensional growth for arbitrary elliptic equations, describing inhomogeneous and anisotropic pattern-formation processes. We find that these nonlinear processes are governed by an infinite number of conservation laws. Moreover, in many cases all 2 dynamics of the interface can be reduced to the linear time dependence of only one ‘‘moment’’ ${\mathit{M}}_0$, which corresponds to the changing volume, while all higher moments ${\mathit{M}}_{\mathit{l}}$ are constant in time. These moments have a purely geometrical nature, and thus carry information about the moving shape. These conserved quantities [Eqs. (7) and (8) of this article] are interpreted as coefficients of the multipole expansion of the Newtonian potential created by the mass uniformly occupying the domain enclosing the moving interface. Thus the question of how to recover the moving shape using these conserved quantities is reduced to the classical inverse potential problem of reconstructing the shape of a body from its exterior gravitational potential. Our results also suggest the possibility of controlling a moving interface by appropriately varying the location and strength of sources and sinks.

We show that the wall-localized convection states observed in Rayleigh-Bénard convection in rotating cylindrical cells can be explained in terms of a geometry-independent traveling-wave wall state. We calculate the onset Rayleigh number, frequency, and wave number of such a state, as well as its amplitude equation. We also study the large-rotation-rate asymptotic behavior and the small-rotation-rate limit.

We introduce a lattice Boltzmann computational scheme capable of modeling thermohydrodynamic flows of monatomic gases. The parallel nature of this approach provides a numerically efficient alternative to traditional methods of computational fluid dynamics. The scheme uses a small number of discrete velocity states and a linear, single-time relaxation collision operator. Numerical simulations in two dimensions agree well with exact solutions for adiabatic sound propagation and Couette flow with heat transfer.

Measurements of temperature and velocity profiles in a convection cell of aspect ratio 1 at a Rayleigh number of 1.1×${10}^9$ and a Prandtl number (Pr) of 6.6 are presented. A large-scale flow persists in the cell. The temperature boundary layer is entirely contained in the viscous boundary layer and the velocity field is smooth on the characteristic length scale of temperature variations. The core of the cell is stably stratified. Most of the heat transported between the plates is carried by the general circulation and does not traverse the core of the cell. Comparison is made with observations at Pr=0.7.

The problem of wave-number selection by a ramp, i.e., a region smoothly matching sub- and supercritical domains, is considered within the framework of the cubic and quintic Ginzburg-Landau (GL) equations with a sign-changing overcriticality parameter. A local frequency is also allowed to be a smooth function of the spatial coordinate. For the cubic model, a unique value of the selected wave number is found by means of an asymptotic procedure valid when the imaginary parts of coefficients in the GL equation are small, while the group velocity is arbitrary. Under certain conditions, the wave number may lie outside the stability band, which is expected to give rise to a dynamical chaos. In the quintic model, which describes a system with the inverted bifurcation, the selection scenario is much simpler: A front separating a traveling wave and the trivial state is expected to be pinned at the point of the ramp where its velocity, regarded as a function of the local overcriticality, vanishes. Eventually, the wave-number selection is performed by the pinned front. Experimentally, the selection may be realized in the traveling-wave convection, or in a self-oscillatory chemical system.

We present an approximate analytical calculation of the time-dependent structure factor for the phase-ordering dynamics of nematic liquid crystals. The structure factor is found to exhibit a ${\mathit{k}}^{\mathrm{-}5}$ tail, due to the presence of 1/2 integrally charged string defects. Our theoretical results are in good agreement with simulation data, and are consistent with recent experiments.

The results of molecular-dynamics simulations of the phase separation of binary fluids in narrow channels are presented. Two channel widths allowing for one or two fluids layers are considered. The one-layer case does not exhibit good scaling. The two-layer case does and the growth exponents are the same as that found earlier for bulk phase separation. Our results point to the importance of conservation laws and hydrodynamic modes in determining the growth dynamics.

A transient optical Kerr measurement is used to study molecular reorientation about the long molecular axis of chiral and nonchiral smectic-A liquid crystals. Two relaxation components were identified, associated with individual and collective molecular motions. In contradiction with a recent report [J. R. Lalanne et al., Phys. Rev. Lett. 62, 3046 (1989); Phys. Rev. A 44, 6632 (1991)], no critical behavior of this reorientation was observed as the transition to the ferroelectric smectic-${\mathit{C}}^{\mathrm{*}}$ phase was approached.

We analyze diffusion-limited aggregation (DLA) [T. A. Witten and L. M. Sander, Phys. Rev. Lett. 47, 1400 (1981); Phys. Rev. B 23, 5686 (1983)] with a branchless needle model. We modify the growth rules of our needles by assigning them a fractal dimension of ${\mathit{D}}_{\mathit{b}}$≊1.7, the fractal dimension of DLA. We then construct a mean-field theory of the evolution of the number of needles having particular heights. Our model accounts for the correlations within a needle. We argue that DLA is an isotropic fractal with a scaling density profile and that the fractal dimension of the individual branches should be the same as the dynamical dimension of the aggregate.

Laser-driven electron acceleration by means of two counterpropagating waves is investigated, where the comoving field is an intense laser light beam, the other is a rf wave beam. A striking feature of the interaction is the presence of a ponderomotive potential propagating with phase velocity slower than the speed of light, allowing one to achieve wave-particle matching. Electron trajectories in the combined field are derived analytically, following the plane-wave and single-particle approximations. Special attention has been given to analyzing the phase slippage of an electron in the field, and the potential of accelerating electrons by means of this scheme. The results show that a neodymium-doped yttrium aluminum garnet laser beam with wavelength 1.06 μm and electric field intensity 4×${10}^{10}$ V/cm is capable of accelerating electrons from 43 to 258 MeV over 0.27 m by using a rf field of wavelength 10.6 cm and an electric field of ${10}^5$ V/cm. The averaged accelerating gradient is 1038 MeV/m.

We develop a critical finite-force-range scaling theory for D-dimensional scalar ${\mathrm{\phi }}^{\mathit{n}}$ field theories that is based on a scaling ansatz equivalent to a Ginzburg criterion. To investigate its relationship to other scaling theories we derive equivalent results from renormalization groups and from finite-size crossover scaling for systems with weak long-range forces. By comparing our finite-range scaling relations with finite-size scaling relations for hypercylindrical systems above the upper critical dimension ${\mathit{D}}_{\mathit{c}}$, we arrive at a criterion of critical equivalence that provides an asymptotic mapping between the two kinds of systems. We apply our scaling relations to a ${\mathrm{\phi }}^4$ Ginzburg-Landau Hamiltonian, to the one-dimensional Kac model with exponentially decaying interactions, and to the N×∞ quasi-one-dimensional Ising (Q1DI) model, in which each spin interacts with O(N) others. Near the Gaussian mean-field critical point the Ginzburg-Landau Hamiltonians for all three models become identical, but for the Q1DI model this requires a length rescaling. For the Kac model the resulting scaling relations are those of a D=1 quartic field theory, and for the Q1DI model they are those of a cylindrical Ising system above ${\mathit{D}}_{\mathit{c}}$. Results of specialized numerical scaling techniques applied to transfer-matrix calculations for the Q1DI model with N≤1024 strongly support our theoretically obtained scaling relations.

Using the field-theoretical formalism, nonequilibrium thermodynamics is discussed where the volume V(t) is changing with time. The time-dependent boundary condition on the wave function is handled by introducing the time-independent mode variables, thus leading to the time-independent Fock space. The correct representation of the second quantized Hamiltonian is derived which depends explicitly on both V(t) and V̇(t). This leads to two kinds of force operators, ${\mathit{X}}_1^0$(t) (pressure) and an extra term ${\mathit{X}}_2$(t). The adiabatic expansion of the increase ΔE(t) of internal energy is performed and it is shown that ${\mathit{X}}_2$(t) has no effect up to the leading nonequilibrium correction. The resulting form of ΔE(t) ensures that the correction is positive, thus proving the principle of maximum-minimum work.

The growth kinetics following a quench from high temperatures to zero temperature is studied using the time-dependent Ginzburg-Landau model. We investigate d-dimensional systems with n-component order parameter and assume that the interactions decay with distance r as V^(r)∼${\mathit{r}}^{\mathrm{-}\mathit{d}\mathrm{-}\mathrm{\sigma }}$ with 0<σ<2. The spherical limit (n=∞) is solved for both conserved and nonconserved order-parameter dynamics and the scaling properties of the structure factor are calculated. We find scaling features (including multiscaling in the conserved case) that are similar to those of systems with short-range interactions. The essential difference is that the short-range value of the dynamic critical exponent ${\mathit{z}}_{\mathit{s}}$ is replaced by z=${\mathit{z}}_{\mathit{s}}$-2+σ and the form of the scaling function is modified. We also study the general n case for nonconserved order-parameter dynamics and calculate the structure factor in an approximate scheme with the results that (i) the spherical-limit value of z remains unchanged as n is decreased down to n=1 and (ii) the spatial correlations decay at large distances as ${\mathit{r}}^{\mathrm{-}\mathit{d}\mathrm{-}\mathrm{\sigma }}$.

Meandering spiral waves in the light-sensitive excitable Belousov-Zhabotinsky reaction with tips proceeding along epicycles were forced to rigid rotation. This spiral anchoring was induced by an argon laser beam creating an unexcitable disk close to the tip. The stabilized spirals rotate with frequencies above the low secondary frequency of meandering. Simulations using a simple reaction-diffusion model demonstrate that anchoring is only possible up to a critical radius of the disk. For this critical value anchored spirals obey the unstable primary parameters (radius and frequency) of compound motion.

We illustrate a derivation of a standard fluctuation-dissipation process from a discrete deterministic dynamical model. This model is a three-dimensional mapping, driving the motion of three variables, w, ξ, and π. We show that for suitable values of the parameters of this mapping, the motion of the variable w is indistinguishable from that of a stochastic variable described by a Fokker-Planck equation with well-defined friction γ and diffusion D. This result can be explained as follows. The bidimensional system of the two variables ξ and π is a nonlinear, deterministic, and chaotic system, with the key property of resulting in a finite correlation time for the variable ξ and in a linear response of ξ to an external perturbation. Both properties are traced back to the fully chaotic nature of this system. When this subsystem is coupled to the variable w, via a very weak coupling guaranteeing a large-time-scale separation between the two systems, the variable w is proven to be driven by a standard fluctuation-dissipation process. We call the subsystem a booster whose chaotic nature triggers the standard fluctuation-dissipation process exhibited by the variable w. The diffusion process is a trivial consequence of the central-limit theorem, whose validity is assured by the finite time scale of the correlation function of ξ. The dissipation affecting the variable w is traced back to the linear response of the booster, which is evaluated adopting a geometrical procedure based on the properties of chaos rather than the conventional perturbation approach.

A d=2 dimensional kinetic Ising model that evolves by a combination of spin flips and spin exchanges is investigated. The spin flips satisfy detailed balance for the equilibrium state of the Ising model at temperature T while the spin exchanges are random Lévy flights of dimension σ=1.5. Our Monte Carlo (MC) simulations show that the steady state of this system displays a second-order phase transition as T is lowered. Comparing the critical fluctuations of the magnetization to those of an Ising model in which the interaction decays with distance as ${\mathit{r}}^{\mathrm{-}3.5}$, we find that, within the resolution of the MC data, the critical exponents and the scaling functions of the two systems coincide. We argue that this coincidence indicates that a recent conjecture about the random Lévy flights generating long-range interaction of the form ${\mathit{V}}_{\mathrm{eff}}$(r)∼${\mathit{r}}^{\mathrm{-}\mathit{d}\mathrm{-}\mathrm{\sigma }}$ is valid not only in the spherical limit and in d=1 but also in d=2.

It is the conventional wisdom that the correlation length for the XY model with linear damping should asymptotically grow diffusively as the square root of time after a quench into the ordered phase. This implies that the defect density ρ should decay with time as ρ∝${\mathit{t}}^{\mathrm{-}\nu }$ with the scaling exponent ν=1. We present evidence, by numerically integrating the equations of motion for a two-dimensional XY model, for a logarithmic correction to this scaling which makes it difficult to reach the asymptotic regime ν=-d(lnρ)/d(lnt)=1. Even after the defect density has decayed by three orders of magnitude ν=0.91, which still deviates by 10% from the asymptotic value.

In order to investigate the different aspects of the transport of heat in the absence of gravity, we performed a thermal cycle close to and around the critical point of ${\mathrm{CO}}_2$ at critical density. Reduced gravity was provided during a 6-min flight of a sounding rocket. A cell has been designed which allows surface and bulk phenomena to be distinguished. Special optical devices are used to observe the sample and measure locally temperature and density. We also present experiments under Earth’s gravity at a few mK from the critical temperature (${\mathit{T}}_{\mathit{c}}$). Convection patterns are observed which correspond to Grashof numbers as large as a few ${10}^4$. The thermal cycle of the experiment in weightlessness starts at ${\mathit{T}}_{\mathit{c}}$+2.5 mK, where we study the relaxation of the perturbations caused by liftoff: fluid flows, density and temperature gradients. We then investigate the effect of a quench from ${\mathit{T}}_{\mathit{c}}$+2.3 to ${\mathit{T}}_{\mathit{c}}$+1.3 mK, and the expected mechanism of heat transport by the ‘‘Piston effect’’ (PE) is experimentally evidenced. In particular, we observe homogeneous thermalization, with a time of thermalization less than a few seconds. We compare this time with a number of theoretical estimations. During thermalization, density and temperature gradients are not affected by the PE and are seen to relax diffusively. We perform then a quench at ${\mathit{T}}_{\mathit{c}}$-0.8 mK. We observe the onset of phase separation, making a clear visualization of the PE. Thereafter, spinodal decomposition occurs, followed by the growth of a bicontinuous pattern of gas and liquid domains. After a last quench back to ${\mathit{T}}_{\mathit{c}}$+2.0 mK, we still observe the PE in spite of the presence of density gradients. These gradients are seen to relax by diffusion.

The long-time limit of the solutions of a master equation with time-dependent transition rates is analyzed, and the existence of a special (normal) solution, that all the other solutions tend to approach, is shown under quite general conditions. In general, the normal solution will be quite different from the time-dependent equilibrium distribution, although for not too fast continuous heating processes, it asymptotically tends to the equilibrium curve. The results seem to be relevant for the explanation of what is observed in real experiments. The general theory is applied to a very simple model, for which the normal solution is exactly found.

We study, qualitatively and quantitatively, the forms of the asymptotic curves from an unstable periodic orbit in a simple Hamiltonian for various values of the energy. The asymptotic curves define two resonance areas and form infinite elongated ‘‘lobes.’’ We give the exact (not schematic) forms of such lobes over long times, and formulate certain rules followed by them. The lengths of the lobes of order n are of the order of ${\lambda }^{\mathit{n}}$, where λ is the largest eigenvalue of the periodic orbit. The lobes surround the resonance areas, spiraling outwards, before going into the large stochastic region outside the resonances. This explains the ‘‘stickiness’’ property of the resonance areas over long times. As the energy increases, the number of rotations of the lobes decreases and the onset of chaos is faster. The lengths and the areas of the lobes increase considerably. The number of intersections of the lobes increases, and we find how new tangencies between the various lobes are formed. If the energy goes beyond the escape energy, certain lobes terminate at ‘‘limiting asymptotic curves’’ corresponding to asymptotic curves of the Lyapunov orbits at the various escape channels.