Effects of additive noise at the onset of Rayleigh-Bénard convection

Phase-field crystal modeling of graphene/hexagonal boron nitride interfaces

Two-dimensional (2D) materials such as graphene and hexagonal boron nitride (h-BN) are an essential class of materials with enhanced structural and electronic properties compared to their bulk counterparts. The phase-field crystal (PFC) model can reach diffusive time scales to study nucleation, growth of crystallites, and relaxation of strain-driven 2D monolayers that are much larger in comparison to molecular dynamics (MD) and quantum mechanical density functional theory (QMDFT) methods while retaining atomic resolution. The model also naturally incorporates an atomic length scale and elastic and plastic deformations. We simulate the morphological transition of the crystal growth of various equilibrium crystal shapes. In this work, we generalize the one-mode PFC model to study the graphene/h-BN heterostructure interface by using conserved dynamics to describe the dynamics of the model. The model was used to find the equilibrium shape of the crystal of the h-BN crystal embedded in a graphene monolayer.

Collisions of localized patterns in a nonvariational Swift-Hohenberg equation

The cubic-quintic Swift-Hohenberg equation (SH35) has been proposed as an order parameter description of several convective systems with reflection symmetry in the layer midplane, including binary fluid convection. We use numerical continuation, together with extensive direct numerical simulations (DNSs), to study SH35 with an additional nonvariational quadratic term to model the effects of breaking the midplane reflection symmetry. The nonvariational structure of the model leads to the propagation of asymmetric spatially localized structures (LSs). An asymptotic prediction for the drift velocity of such structures, derived in the limit of weak symmetry breaking, is validated numerically. Next, we present an extensive study of possible collision scenarios between identical and nonidentical traveling structures, varying a temperaturelike control parameter. These collisions are inelastic and result in stationary or traveling structures. Depending on system parameters and the types of structures colliding, the final state may be a simple bound state of the initial LSs, but it can also be longer or shorter than the sum of the two initial states as a result of nonlinear interactions. The Maxwell point of the variational system, where the free energy of the global pattern state equals that of the trivial state, is shown to have no bearing on which of these scenarios is realized. Instead, we argue that the stability properties of bound states are key. While individual LSs lie on a modified snakes-and-ladders structure in the nonvariational SH35, the multipulse bound states resulting from collisions lie on isolas in parameter space, disconnected from the trivial solution. In the gradient SH35, such isolas are always of figure-eight shape, but in the present nongradient case they are generically more complex, although the figure-eight shape is preserved in a small subset of cases. Some of these complex isolas are shown to terminate in T-point bifurcations. A reduced model is proposed to describe the interactions between the tails of the LSs. The model consists of two coupled ordinary differential equations (ODEs) capturing the oscillatory nature of SH35 profiles at the linear level. It contains three parameters: two interaction amplitudes and a phase, whose values are deduced from high-resolution DNSs using gradient descent optimization. For collisions leading to the formation of simple bound states, the reduced model reproduces the trajectories of LSs with high quantitative accuracy. When nonlinear interactions lead to the creation or deletion of wavelengths, the model performs less well. Finally, we propose an effective signature of a given interaction in terms of net attraction or repulsion relative to free propagation. It is found that interactions can be attractive or repulsive in the net, irrespective of whether the two closest interacting extrema are of the same or opposite signs. Our findings highlight the rich temporal dynamics described by this bistable nonvariational SH35, and show that the interactions in this system can be quantitatively captured, to a significant extent, by a highly reduced ODE model.

A Novel Treatment of Fuzzy Fractional Swift–Hohenberg Equation for a Hybrid Transform within the Fractional Derivative Operator

This article investigates the semi-analytical method coupled with a new hybrid fuzzy integral transform and the Adomian decomposition method via the notion of fuzziness known as the Elzaki Adomian decomposition method (briefly, EADM). In addition, we apply this method to the time-fractional Swift–Hohenberg equation (SHe) with various initial conditions (IC) under gH-differentiability. Some aspects of the fuzzy Caputo fractional derivative (CFD) with the Elzaki transform are presented. Moreover, we established the general formulation and approximate findings by testing examples in series form of the models under investigation with success. With the aid of the projected method, we establish the approximate analytical results of SHe with graphical representations of initial value problems by inserting the uncertainty parameter 0≤℘≤1 with different fractional orders. It is expected that fuzzy EADM will be powerful and accurate in configuring numerical solutions to nonlinear fuzzy fractional partial differential equations arising in physical and complex structures.

Fourier-Spectral Method for the Phase-Field Equations

In this paper, we review the Fourier-spectral method for some phase-field models: Allen–Cahn (AC), Cahn–Hilliard (CH), Swift–Hohenberg (SH), phase-field crystal (PFC), and molecular beam epitaxy (MBE) growth. These equations are very important parabolic partial differential equations and are applicable to many interesting scientific problems. The AC equation is a reaction-diffusion equation modeling anti-phase domain coarsening dynamics. The CH equation models phase segregation of binary mixtures. The SH equation is a popular model for generating patterns in spatially extended dissipative systems. A classical PFC model is originally derived to investigate the dynamics of atomic-scale crystal growth. An isotropic symmetry MBE growth model is originally devised as a method for directly growing high purity epitaxial thin film of molecular beams evaporating on a heated substrate. The Fourier-spectral method is highly accurate and simple to implement. We present a detailed description of the method and explain its connection to MATLAB usage so that the interested readers can use the Fourier-spectral method for their research needs without difficulties. Several standard computational tests are done to demonstrate the performance of the method. Furthermore, we provide the MATLAB codes implementation in the Appendix A.

Large-Scale Pattern Formation in the Presence of Small-Scale Random Advection

Despite the presence of strong fluctuations, many turbulent systems such as Rayleigh-Bénard convection and Taylor-Couette flow display self-organized large-scale flow patterns. How do small-scale turbulent fluctuations impact the emergence and stability of such large-scale flow patterns? Here, we approach this question conceptually by investigating a class of pattern forming systems in the presence of random advection by a Kraichnan-Kazantsev velocity field. Combining tools from pattern formation with statistical theory and simulations, we show that random advection shifts the onset and the wave number of emergent patterns. As a simple model for pattern formation in convection, the effects are demonstrated with a generalized Swift-Hohenberg equation including random advection. We also discuss the implications of our results for the large-scale flow of turbulent Rayleigh-Bénard convection.

Numerical Simulation of Pattern Formation on Surfaces Using an Efficient Linear Second-Order Method

We present an efficient linear second-order method for a Swift–Hohenberg (SH) type of a partial differential equation having quadratic-cubic nonlinearity on surfaces to simulate pattern formation on surfaces numerically. The equation is symmetric under a change of sign of the density field if there is no quadratic nonlinearity. We introduce a narrow band neighborhood of a surface and extend the equation on the surface to the narrow band domain. By applying a pseudo-Neumann boundary condition through the closest point, the Laplace–Beltrami operator can be replaced by the standard Laplacian operator. The equation on the narrow band domain is split into one linear and two nonlinear subequations, where the nonlinear subequations are independent of spatial derivatives and thus are ordinary differential equations and have closed-form solutions. Therefore, we only solve the linear subequation on the narrow band domain using the Crank–Nicolson method. Numerical experiments on various surfaces are given verifying the accuracy and efficiency of the proposed method.

Noisy-flow-induced instability in a reaction-diffusion system

We consider a generic reaction-diffusion-advection system where the flow velocity of the advection term is subjected to dichotomous noise with zero mean and Ornstein-Zernike correlation. A general condition for noisy-flow-induced instability is derived in the flow velocity-correlation rate parameter plane. Full numerical simulations on Gierer-Meinhardt model with activator-inhibitor kinetics have been performed to show how noisy differential flow can lead to symmetry breaking of a homogeneous stable state in the presence of noise resulting in traveling waves.

Confinement effect on the dynamics of non-equilibrium concentration fluctuations far from the onset of convection

In a recent letter (C. Giraudet et al., EPL 111, 60013 (2015)) we reported preliminary data showing evidence of a slowing-down of non-equilibrium fluctuations of the concentration in thermodiffusion experiments on a binary mixture of miscible fluids. The reason for this slowing-down was attributed to the effect of confinement. Such tentative explanation is here experimentally corroborated by new measurements and theoretically substantiated by studying analytically and numerically the relevant fluctuating hydrodynamics equations. In the new experiments presented here, the magnitude of the temperature gradient is changed, confirming that the system is controlled solely by the solutal Rayleigh number, and that the slowing-down is dominated by a combined effect of the driving force of buoyancy, the dissipating force of diffusion and the confinement provided by the vertical extension of the sample cell. Moreover, a compact phenomenological interpolating formula is proposed for easy analysis of experimental results.

Metallic-thin-film instability with spatially correlated thermal noise

We study the effects of stochastic thermal fluctuations on the instability of the free surface of a flat liquid metallic film on a solid substrate. These fluctuations are represented by a stochastic noise term added to the deterministic equation for the film thickness within the long-wave approximation. Unlike the case of polymeric films, we find that this noise, while remaining white in time, must be colored in space, at least in some regimes. The corresponding noise term is characterized by a nonzero correlation length, ℓ_{c}, which, combined with the size of the system, leads to a dimensionless parameter β that accounts for the relative importance of the spatial correlation (β∼ℓ_{c}^{-1}). We perform the linear stability analysis (LSA) of the film both with and without the noise term and find that for ℓ_{c} larger than some critical value (depending on the system size), the wavelength of the peak of the spectrum is larger than that corresponding to the deterministic case, while for smaller ℓ_{c} this peak corresponds to smaller wavelength than the latter. Interestingly, whatever the value of ℓ_{c}, the peak always approaches the deterministic one for larger times. We compare LSA results with the numerical simulations of the complete nonlinear problem and find a good agreement in the power spectra for early times at different values of β. For late times, we find that the stochastic LSA predicts well the position of the dominant wavelength, showing that nonlinear interactions do not modify the trends of the early linear stages. Finally, we fit the theoretical spectra to experimental data from a nanometric laser-melted copper film and find that at later times, the adjustment requires smaller values of β (larger space correlations).

Aging of orientation fluctuations in stripe phases

Stripe patterns, observed in a large variety of physical systems, often exhibit a slow nonequilibrium dynamics because ordering is impeded by the presence of topological defects. Using computer simulations based on a well-established model for stripe formation, we show that a slow dynamics and aging occur also in stripe patterns free of topological defects. For a wide range of noise strengths, the two-time orientation correlation function follows a scaling form that is typical for systems exhibiting a growing length scale. In our case, the underlying mechanism is the coarsening of orientation fluctuations, ultimately leading to power-law spatial correlations perpendicular to the stripes. Our results show that even for the smallest amount of noise, stripe phases without topological defects do not reach equilibrium. This constitutes an important aspect of the dynamics of modulated phases.

Optimal Variational Asymptotic Method for Nonlinear Fractional Partial Differential Equations

We propose optimal variational asymptotic method to solve time fractional nonlinear partial differential equations. In the proposed method, an arbitrary number of auxiliary parameters γ₀, γ₁, γ₂,… and auxiliary functions H₀(x), H₁(x), H₂(x),… are introduced in the correction functional of the standard variational iteration method. The optimal values of these parameters are obtained by minimizing the square residual error. To test the method, we apply it to solve two important classes of nonlinear partial differential equations: (1) the fractional advection-diffusion equation with nonlinear source term and (2) the fractional Swift-Hohenberg equation. Only few iterations are required to achieve fairly accurate solutions of both the first and second problems.

Exploding dissipative solitons in reaction-diffusion systems

We show that exploding dissipative solitons can arise in a reaction-diffusion system for a range of parameters. As a function of a vorticity parameter, we observe a sequence of transitions from oscillatory localized states via meandering dissipative solitons to exploding dissipative solitons propagating in one direction for long times followed by the reverse cascade back to oscillatory localized states. While exploding dissipative solitons are known from the cubic-quintic complex Ginzburg-Landau (CGL) equation, propagating exploding dissipative solitons appear to require for their existence a system of lower symmetry such as the reaction-diffusion model studied here.

Nucleation, growth, and coarsening of crystalline domains in order-order transitions between lamellar and hexagonal phases

Using the numerical solution of the time-dependent Ginzburg-Landau equation, we study the entire process of transformation between the lamellar and the hexagonal phases from the early-stage nucleation and growth to the late-stage coarsening regime. The metastable crystalline structure that nucleates first is identified in terms of the mean-field theory under the single-wave-number approximation. This has been borne out by the numerically efficient preparation of single-crystal structure developed via the noise-induced self-organization. We also present results for the scaling of the late-time domain growth, which is quantified by two measures: the structure factor and the orientational correlation function. In particular, the growth exponent is shown to be robust and indifferent to conservation of the order parameter.

Capillary waves of compressible fluids

The interplay of thermal noise and molecular forces is responsible for surprising features of liquids on sub-micrometer lengths-in particular at interfaces. Not only does the surface tension depend on the size of an applied distortion and nanoscopic thin liquid films dewet faster than would be expected from hydrodynamics, but also the dispersion relation of capillary waves differ at the nanoscale from the familiar macroscopic behavior. Starting with the stochastic Navier-Stokes equation we study the coupling of capillary waves to acoustic surface waves which is possible in compressible fluids. We find propagating 'acoustic-capillary waves' at nanometer wavelengths where in incompressible fluids capillary waves are overdamped.

Phase-field-crystal dynamics for binary systems: Derivation from dynamical density functional theory, amplitude equation formalism, and applications to alloy heterostructures

The dynamics of phase field crystal (PFC) modeling is derived from dynamical density functional theory (DDFT), for both single-component and binary systems. The derivation is based on a truncation up to the three-point direct correlation functions in DDFT, and the lowest order approximation using scale analysis. The complete amplitude equation formalism for binary PFC is developed to describe the coupled dynamics of slowly varying complex amplitudes of structural profile, zeroth-mode average atomic density, and system concentration field. Effects of noise (corresponding to stochastic amplitude equations) and species-dependent atomic mobilities are also incorporated in this formalism. Results of a sample application to the study of surface segregation and interface intermixing in alloy heterostructures and strained layer growth are presented, showing the effects of different atomic sizes and mobilities of alloy components. A phenomenon of composition overshooting at the interface is found, which can be connected to the surface segregation and enrichment of one of the atomic components observed in recent experiments of alloying heterostructures.

Subharmonic wave transition in a quasi-one-dimensional noisy fluidized shallow granular bed

We present an experimental and theoretical study of the pattern formation process of standing subharmonic waves in a fluidized quasi-one-dimensional shallow granular bed. The fluidization process is driven by means of a time-periodic air flow, analogous to a tapping type of forcing. Measurements of the amplitude of the critical mode close to the transition are in quite good agreement with those inferred from a universal stochastic amplitude equation. This allows us to determine both the bifurcation point of the deterministic system and the corresponding noise intensity. We also show that the probability density distribution is well described by a generalized Rayleigh distribution, which is the stationary solution of the corresponding Fokker-Planck equation of the universal stochastic amplitude equation that describes our system.

Stripe domain coarsening in geographical small-world networks on a Euclidean lattice

We study phase ordering dynamics of spatially periodic striped patterns on the small-world network that is derived from a two-dimensional regular lattice with distance-dependent random connections. It is demonstrated numerically that addition of spatial disorder in the form of shortcuts makes the growth of domains much slower or even frozen at late times.

Universal shape law of stochastic supercritical bifurcations: theory and experiments

A universal analytical expression for the supercritical bifurcation shape of transverse one-dimensional (1D) systems in the presence of additive noise is given. The stochastic Langevin equation of such systems is solved by using a Fokker-Planck equation, leading to the expression for the most probable amplitude of the critical mode. From this universal expression, the shape of the bifurcation, its location, and its evolution with the noise level are completely defined. Experimental results obtained for a 1D transverse Kerr-type slice subjected to optical feedback are in excellent agreement.

Generalization of the reaction-diffusion, Swift-Hohenberg, and Kuramoto-Sivashinsky equations and effects of finite propagation speeds

The work proposes and studies a model for one-dimensional spatially extended systems, which involve nonlocal interactions and finite propagation speed. It shows that the general reaction-diffusion equation, the Swift-Hohenberg equation, and the general Kuramoto-Sivashinsky equation represent special cases of the proposed model for limited spatial interaction ranges and for infinite propagation speeds. Moreover, the Swift-Hohenberg equation is derived from a general energy functional. After a detailed validity study on the generalization conditions, the three equations are extended to involve finite propagation speeds. Moreover, linear stability studies of the extended equations reveal critical propagation speeds and unusual types of instabilities in all three equations. In addition, an extended diffusion equation is derived and studied in some detail with respect to finite propagation speeds. The extended model allows for the explanation of recent experimental results on non-Fourier heat conduction in nonhomogeneous material.

Spatiotemporal chaos in electroconvection of a homeotropically aligned nematic liquid crystal

We present patterns of electroconvection (EC) for the homeotropically aligned nematic liquid crystal MBBA. A voltage V = square root of 2V0 sin(2pift) was applied. With increasing V0, the bend Freedericksz transition at VF was followed by the onset of EC at Vc > VF. We found four distinct pattern types. First, a primary supercritical Hopf bifurcation to traveling waves (TW's) of convection rolls occurred. The structure factor S(k) of this state reflected the azimuthal anisotropy of the underlying Freedericksz state. For f < fL approximately 75 Hz there was a superposition of two oblique-roll modes (pattern I). These patterns were chaotic in space and time. For larger f the patterns consisted of chaotic TW normal rolls (pattern II). Here the chaos was attributable to the motion of dislocations and domain walls between left- and right-traveling waves. A secondary bifurcation yielded pattern III; it had no dominant TW frequency but had broadband chaotic dynamics dominated by the motion of dislocations. This pattern type had been referred to by others as a "chevron pattern;" its structure factor still revealed azimuthal anisotropy. Finally, at somewhat larger identical with epsilon = V2/Vc2 -1 a highly disordered pattern IV with defect dynamics was found. This state had been studied before by Kai and co-workers and was referred to by them as "phase turbulence." It had a structure factor that was (within our resolution) invariant under rotation. For patterns I, II, and III, S(k) contained crescent-shaped peaks. The peak shape was qualitatively different from the case of planar EC where the structure factor has an elliptical cross section. We present measurements of the widths 1/xik and 1/xitheta in the radial (k) and the azimuthal (theta) directions. For small epsilon (patterns I and II) we found that xik was consistent with the usual Ginzburg-Landau scaling xik approximately epsilon(-nuk) with nuk approximately 1/2. However, for xitheta we found xitheta approximately epsilon(-nutheta) with nutheta approximately 3/4. Presumably this anomalous scaling of xitheta is associated with the Goldstone mode of homeotropic EC. We also show data for the height S0 of the structure factor that are consistent with S0 approximately epsilonbeta with beta approximately -0.5, implying that S0 diverges at onset. This differs from the case of domain chaos in rotating Rayleigh-Bénard convection where experiment is consistent with beta = 1/2 and thus with a vanishing S0. The difference between the shape of the structure-factor cross section and between the exponents, for the present case, for planar EC, and for domain chaos suggests that there are different universality classes for spatiotemporal chaos.

Hydrodynamic coarsening in striped pattern formation with a conservation law

We observed in numerical simulations that the interaction of striped-pattern-forming instability and a neutrally stable zero mode induces patterns of domains of upflow hexagons coexisting with domains of downflow hexagons. They appear only when hydrodynamic flow is present.

Dynamics of fluctuations below a stationary bifurcation to electroconvection in the planar nematic liquid crystal N4

We fitted C(k,tau,epsilon) proportional to exp([-sigma(k,epsilon)tau] to time-correlation functions C(k,tau,epsilon) of structure factors S(k,t,epsilon) of shadowgraph images of fluctuations below a supercritical bifurcation at V(0)=V(c) to electroconvection of a planar nematic liquid crystal in the presence of a voltage V=sqrt[2]V(0)cos((2pift) [k=(p,q) is the wave vector and epsilon identical with V(2)(0)/V(2)(c)-1]. There were stationary oblique (normal) rolls at small (large) f. Fits of a modified Swift-Hohenberg form to sigma(k,epsilon) gave f-dependent critical behavior for the minimum decay rates sigma(0)(epsilon) and the correlation lengths xi(p,q)(epsilon).

Noise-induced instability: an approach based on higher-order moments

Noise-induced transitions in the organization of systems far from equilibrium have been of vital interest. Although the effects of additive and multiplicative noise have been widely studied, it is only the multiplicative noise that can be dealt with within the scope of a linear analysis of first moments of the spatiotemporal perturbations, by the application of Novikov's theorem. For the case of additive noise, the corresponding straightforward linear analysis of the first moment throws no light on the effect of the noise on stability conditions. We propose here a simple approach based on higher-order moments to show how additive noise can give rise to noise-induced instability in spatially extended systems, at times leading to pattern formation. Our theoretical analysis is corroborated by numerical simulations on two simple one-component reaction-diffusion systems in two dimensions.

Onset of patterns in an oscillated granular layer: continuum and molecular dynamics simulations

We study the onset of patterns in vertically oscillated layers of frictionless dissipative particles. Using both numerical solutions of continuum equations to Navier-Stokes order and molecular dynamics (MD) simulations, we find that standing waves form stripe patterns above a critical acceleration of the cell. Changing the frequency of oscillation of the cell changes the wavelength of the resulting pattern; MD and continuum simulations both yield wavelengths in accord with previous experimental results. The value of the critical acceleration for ordered standing waves is approximately 10% higher in molecular dynamics simulations than in the continuum simulations, and the amplitude of the waves differs significantly between the models. The delay in the onset of order in molecular dynamics simulations and the amplitude of noise below this onset are consistent with the presence of fluctuations which are absent in the continuum theory. The strength of the noise obtained by fit to Swift-Hohenberg theory is orders of magnitude larger than the thermal noise in fluid convection experiments, and is comparable to the noise found in experiments with oscillated granular layers and in recent fluid experiments on fluids near the critical point. Good agreement is found between the mean field value of onset from the Swift-Hohenberg fit and the onset in continuum simulations. Patterns are compared in cells oscillated at two different frequencies in MD; the layer with larger wavelength patterns has less noise than the layer with smaller wavelength patterns.

Giant fluctuations at a granular phase separation threshold

We investigate a phase separation instability that occurs in a system of nearly elastically colliding hard spheres driven by a thermal wall. If the aspect ratio of the confining box exceeds a threshold value, granular hydrostatics predict phase separation: the formation of a high-density region coexisting with a low-density region along the wall that is opposite to the thermal wall. Event-driven molecular dynamics simulations confirm this prediction. The theoretical bifurcation curve agrees with the simulations quantitatively well below and well above the threshold. However, in a wide region of aspect ratios around the threshold, the system is dominated by fluctuations, and the hydrostatic theory breaks down. Two possible scenarios of the origin of the giant fluctuations are discussed.

Dynamics of fluctuations in a fluid below the onset of Rayleigh-Bénard convection

We present experimental data and their theoretical interpretation for the decay rates of temperature fluctuations in a thin layer of a fluid heated from below and confined between parallel horizontal plates. The measurements were made with the mean temperature of the layer corresponding to the critical isochore of sulfur hexafluoride above but near the critical point where fluctuations are exceptionally strong. They cover a wide range of temperature gradients below the onset of Rayleigh-Bénard convection, and span wave numbers on both sides of the critical value for this onset. The decay rates were determined from experimental shadowgraph images of the fluctuations at several camera exposure times. We present a theoretical expression for an exposure-time-dependent structure factor which is needed for the data analysis. As the onset of convection is approached, the data reveal the critical slowing down associated with the bifurcation. Theoretical predictions for the decay rates as a function of the wave number and temperature gradient are presented and compared with the experimental data. Quantitative agreement is obtained if allowance is made for some uncertainty in the small spacing between the plates, and when an empirical estimate is employed for the influence of symmetric deviations from the Oberbeck-Boussinesq approximation which are to be expected in a fluid with its density at the mean temperature located on the critical isochore.

Thermal-noise effect on the transition to Rayleigh-Bénard convection

We report measurements of fluctuation and roll patterns near the transition to Rayleigh-Bénard convection which are consistent with a fluctuation-induced first-order transition, as predicted by Swift and Hohenberg. Above onset, we find convection rolls with noise-induced fluctuations, time-dependent amplitude modulation and roll undulation, and homogeneous dislocation nucleation.

Competition of spiral-defect chaos and rolls in Rayleigh-Bénard convection under shear flow

Computer simulations of domain coarsening of Rayleigh-Bénard convective patterns under horizontal shear flow are carried out. The model calculations reported here explicitly include the hydrodynamic interaction of the order parameter field and provide a description of the spiral-defect chaos which competes with the roll pattern. We observe shear banding at moderate strain rates.

Boundary effects on the nonequilibrium structure factor of fluids below the Rayleigh-Bénard instability

We consider a horizontal fluid layer between two rigid boundaries, maintained in a stationary thermal nonequilibrium state below the convective Rayleigh-Bénard instability. We derive an explicit expression for the nonequilibrium structure factor in a first-order Galerkin approximation valid for negative and positive Rayleigh numbers R up to the critical Rayleigh number R(c) associated with the appearance of convection. The results obtained for rigid boundaries by the Galerkin-approximation method are compared with exact results previously derived for the case of free boundaries. The nonequilibrium structure factor exhibits a maximum as a function of the wave number q of the fluctuations. This maximum is associated with a crossover from a q(-4) dependence for larger q to a q(2) dependence for small q. This maximum is present at both negative and positive R, becomes pronounced at positive R and diverges as R approaches the critical value R(c).

Hydrodynamic interactions in ordering process of two-dimensional quenched block copolymers

The hydrodynamic coarsening of microphase separation in two-dimensional diblock copolymers is studied using numerical simulations. Results for symmetric and asymmetric block copolymers are compared. In contrast to the formation of the hexagonal phase where hydrodynamic flow appears not to be effective in enhancing domain coarsening, the late-time evolution of the lamellar phase proceeds faster, thus leading to a different power-law scaling with the addition of coupling of the velocity field to the order parameter.

Temporal and spatial properties of fluctuations below a supercritical primary bifurcation to traveling oblique-roll electroconvection

We present measurements of thermally-induced oblique-roll traveling-wave (TW) fluctuations below the supercritical primary bifurcation to electroconvection (EC) in the nematic liquid crystal 4-ethyl-2-fluoro-4'-[2-(trans-4-pentylcyclohexyl)ethyl]-biphenyl (I52). First we analyze time sequences of one-dimensional shadowgraph images taken parallel to the director to obtain the TW frequency omega and the fluctuation lifetime tau. Within our resolution we find that omega is independent of epsilon [triple bond] V/V(c)-1 (V is the applied voltage amplitude and V(c) its value at the onset of convection). Contrary to linear theory, the relaxation rate 1/tau remains finite at the bifurcation. Next we present the analysis of temporally uncorrelated two-dimensional shadowgraph images of the fluctuations for several values of the electrical conductivity sigma. We fitted an anisotropic two-dimensional Lorentzian function, corresponding to oblique-roll EC, to the time-averaged structure factors S(k) derived from the images. This yielded information about the components of the mean wave vector k(0) and about the correlation length xi as a function of sigma and epsilon. The angle of obliqueness theta of the roll patterns was independent of sigma but decreased anomalously as epsilon approached zero. The modulus k(0) of k(0) depended on sigma. It also showed an anomalous reduction close to onset. The anomalous epsilon dependence of k(0) and theta disagrees with linear theory, which predicts a smooth, essentially linear dependence on epsilon, and presumably is caused by nonlinear interactions between the fluctuations.

Influence of inlet and bulk noise on Rayleigh-Bénard convection with lateral flow

Spatiotemporal properties of convective fluctuations and of their correlations are investigated theoretically in the vicinity of the threshold for onset of convection in the presence of a lateral through-flow using the full linearized equations of fluctuating hydrodynamics. The effect of external forcing by inlet boundary conditions on the downstream evolution of convective fields is separated from the effect of internal bulk thermal forcing with the use of spatial Laplace transformations. They show how the spatial variation of fluctuations and of their correlations are governed by the six spatial characteristic exponents of the field equations.

Renormalization-group theoretical reduction of the Swift-Hohenberg model

The Swift-Hohenberg model of the cellular pattern formation is exploited with a proto renormalization-group (RG) scheme. The method dispenses with the explicit perturbation solutions which are required in the standard RG approach. The RG equations obtained are the well-known reductive perturbation results such as a rotationally covariant amplitude equation and the nonlinear phase equations.

Kinetics of ordering in fluctuation-driven first-order transitions: simulation and theory

Many systems involving competing interactions or interactions that compete with constraints are well described by a model first introduced by Brazovskii [Zh. Eksp. Teor. Fiz. 68, 175 (1975) [Sov. Phys. JETP 41, 85 (1975)]]. The hallmark of this model is that the fluctuation spectrum is isotropic and has a maximum at a nonzero wave vector represented by the surface of a d-dimensional hypersphere. It was shown by Brazovskii that the fluctuations change the free energy structure from a straight phi(4) to a straight phi(6) form with the disordered state metastable for all quench depths. The transition from the disordered phase to the periodic lamellar structure changes from second order to first order and suggests that the dynamics is governed by nucleation. Using numerical simulations we have confirmed that the equilibrium free energy function is indeed of a straight phi(6) form. A study of the dynamics, however, shows that, following a deep quench, the dynamics is described by unstable growth rather than nucleation. A dynamical calculation, based on a generalization of the Brazovskii calculations, shows that the disordered state can remain unstable for a long time following the quench.

Deviations from linear theory for fluctuations below the supercritical primary bifurcation to electroconvection

Over two decades ago it was predicted that nonlinear interactions between thermally driven fluctuations in dissipative nonlinear nonequilibrium systems lead to deviations from mean-field theory. Here we report experimental observations of such deviations as a supercritical primary bifurcation is approached. We measured the mean-square director-angle fluctuations <straight theta(2)> below the bifurcation to electroconvection of two different nematic liquid crystals. For epsilon(mf) identical withV2/V(2)(c,mf)-1 less, similar-0.1 ( V is the applied voltage) we find <straight theta(2)> approximately |epsilon(mf)|(-gamma) with gamma given by linear theory (LT). Closer to the bifurcation there are deviations from LT with a smaller gamma and with V(2)(c)>V(2)(c,mf).

Bifurcation to worms in electroconvection

The primary bifurcation to electroconvection of the liquid crystal 4-ethyl-2-fluoro-4'-[2-(trans-4-pentylcyclohexyl)ethyl]-biphenyl (I52) with planar alignment leads to localized structures of convection rolls known as "worms" when the conductivity of the fluid is relatively small. Worms coexist with the conduction state. They have a unique small width in the direction perpendicular to the director and a varying, usually much greater, length parallel to the director. Previous experiments had not determined whether the bifurcation to worms is supercritical or subcritical. We estimated the voltage V(c) corresponding to the stability limit of the conduction state by measuring the mean-square amplitude of the thermally induced fluctuations below onset and extrapolating to V(c). We found that worms appear already well below V(c). Thus the bifurcation is subcritical. Measurements of the lifetime of the conduction state below V(c) gave information about the voltage V(s) corresponding to the saddle node below which no worms form. We measured V(c) and epsilon(s)=V(s)2/V(c)2-1 as a function of the conductance sigma for a cell of thickness 24 microm and found for our sample that epsilon(s) approaches zero from negative values near sigma approximately 1.2 x 10(-8) Omega(-1) m(-1) as sigma increases. For larger sigma we found the bifurcation to be supercritical. We have been unable to determine so far whether the experimentally observed transition with decreasing sigma from a supercritical to a subcritical bifurcation occurs via a tricritical bifurcation, or whether the worm saddle node is disconnected from the primary supercritical bifurcation line as suggested by theory.