Cube–Root Boundary Fluctuations¶for Droplets in Random Cluster Models

For a family of bond percolation models on ℤ² that includes the Fortuin–Kasteleyn random cluster model, we consider properties of the “droplet” that results, in the percolating regime, from conditioning on the existence of an open dual circuit surrounding the origin and enclosing at least (or exactly) a given large area A. This droplet is a close surrogate for the one obtained by Dobrushin, Kotecky and Shlosman by conditioning the Ising model; it approximates an area-A Wulff shape. The local part of the deviation from the Wulff shape (the “local roughness”) is the inward deviation of the droplet boundary from the boundary of its own convex hull; the remaining part of the deviation, that of the convex hull of the droplet from the Wulff shape, is inherently long-range. We show that the local roughness is described by at most the exponent 1/3 predicted by nonrigorous theory; this same prediction has been made for a wide class of interfaces in two dimensions. Specifically, the average of the local roughness over the droplet surface is shown to be O(l ^1/3(log l)^2/3) in probability, where is the linear scale of the droplet. We also bound the maximum of the local roughness over the droplet surface and bound the long-range part of the deviation from a Wulff shape, and we establish the absense of “bottlenecks”, which are a form of self-approach by the droplet boundary, down to scale log l. Finally, if we condition instead on the event that the total area of all large droplets inside a finite box exceeds A, we show that with probability near 1 for large A, only a single large droplet is present. Received: 20 January 2000 / Accepted: 7 August 2001     

Cluster description of water-in-oil microemulsions near the critical and percolation points

The three-component ionic microemulsion system consisting of AOT/water/decane shows an unusual phase behavior in the vicinity of room temperature. The phase diagram in the temperature-volume fraction (of the dispersed phase) plane exhibits a lower consolute critical point at about 40 degrees centigrades and 10% volume fraction. A percolation line, starting from the vicinity of the critical point, cuts across the plane, extending to high volume fraction side at progressively lower temperatures. In this paper we review the evidence that allows to interpret the phase behavior of our system in terms of interacting spherical droplets. We also investigate the dynamics of droplets, below and approaching the critical point by dynamic light scattering. The first cumulant and time evolution of the droplet density correlation function can be quantitatively calculated by assuming the existence of polydispersed fractal clusters formed by the microemulsion droplets due to attraction. The relaxation phenomena observed in an extensive set of measurements of electrical conductivity and permittivity close to percolation is also reviewed and interpreted through the same cluster-forming mechanism, which reproduces the most relevant features of the frequency-dependent complex dielectric constant of this system. Paper presented at the I International Conference on Scaling Concepts and Complex Fluids, Copanello, Italy, July 4–8, 1994.     

Droplet combustion in standing sound waves

Interaction between droplet combustion and acoustic oscillation is clarified. As the simplest model, an isolated fuel droplet is combusted in a standing sound wave. Apart from the conventional idea that oscillatory component of flow influences heat and mass transfer and promotes combustion, a new model that a secondary flow dominates combustion promotion is examined. The secondary flow, found by the authors in the previous work, is driven by acoustic radiation force due to Reynolds normal stress, and named as thermo-acoustic streaming. Since the force is described by the same equation as buoyancy, i.e., F = ΔρVg, the nature of the streaming is thought to be the same as natural convection. The flow patterns of the streaming are analyzed and its influence on burning rate of a droplet is predicted. Experimental investigation was mainly done with burning droplets located in the middle of node and anti-node of standing sound waves. This location realizes the strongest streaming. By varying sound pressure level, ambient pressure, and acoustic frequency, the strength of the streaming was controlled. Flame configuration including soot and burning rate were examined. Microgravity conditions were employed to clarify the influence of acoustic field through the streaming, since it is similar to and must be distinguished from natural convection. Experiments using microgravity conditions confirmed the new combustion promotion model and the way to quantify it. By introducing a new non-dimensional number Gr_a, that is the ratio of acoustic radiation force to viscosity, burning rate constants for various ambient and sound conditions are rearranged. As a result, it was found that the excess burning rate (k/k₀ − 1) is proportional to or , for weak sound and for strong sound, respectively.     

Small-angle light scattering studies of dense AOT-water-decane microemulsions

Summary We have performed extensive studies of a three-component microemulsion system composed of AOT-water-decane (AOT=sodium-bis-ethylhexyl-sulfosuccinate is an ionic surfactant) using small-angle light scattering (SALS). The small-angle scattering intensities are measured in the angular interval 0.001–0.1 radians, corresponding to a Bragg wave number range of 0.14 μm⁻¹<Q<<1.4 μm⁻¹. The measurements were made by changing temperature and volume fraction ϕ of the dispersed phase (water + AOT) in the range 0.05<ϕ<0.75. All samples have a fixed water-to-AOT molar ratio,w=[water]/[AOT]=40.8, in order to keep the same average droplet size in the stable one-phase region. With the SALS technique, we have been able to observe all the phase boundaries of a very complex phase diagram with a percolation line and many structural organizations within it. We observe at the percolation transition threshold, a scaling behavior of the intensity data. This behavior is a consequence of a clustering among microemulsion droplets near the percolation threshold. In addition, we describe in detail a structural transition from a droplet microemulsion to a bicontinuous one as suggested by a recent small-angle neutron scattering experiment. The loci of this transition are located several degrees above the percolation temperatures and are coincident with the maxima previously observed in shear viscosity. From the data analysis, we show that both the percolation phenomenon and this novel structural transition are derived from a large-scale aggregation between microemulsion droplets.     

Experimental study of the effect of helium/nitrogen concentration and initial droplet diameter on nonane droplet combustion with minimal convection

In this paper, we study the influence of inert concentration and initial droplet diameter on nonane (C₉H₂₀) droplet combustion in an environment that promotes spherical droplet flames. The oxygen concentration is fixed while the inert is varied between nitrogen and helium. A range of initial droplet diameters (D_o) are examined in each ambient gas: 0.4 mm < D_o < 0.8 mm; and an oxidizing ambiance consisting of 30% oxygen (fixed) and 70% inert (fixed), with the inert in turn composed of mixtures of nitrogen and helium in concentrations of 0, 25, 50, 75, and 100% N₂. The experiments are carried out at normal atmospheric pressure in a cold ambiance (room temperature) under low gravity to minimize the influence of convection and promote spherical droplet flames. For burning within a helium inert (0% N₂), the droplet flames are entirely blue and there is no influence of initial droplet diameter on the local burning rate (K). With increasing dilution by nitrogen, droplet flames show significant yellow luminosity indicating the presence of soot and the individual burning histories show K reducing with increasing D_o. The evolution of droplet diameter D(t) is nonlinear for a given D_o in the presence of either helium or nitrogen inerts indicating that soot formation has little to do with nonlinear burning. A correlation is presented of the data in the form where the effective burning rate, K′, and ε are concentration-dependent. Correlations for these parameters are presented in the paper.     

Modulated inflation

We have studied modulated inflation that generates curvature perturbation from light-field fluctuation. As discussed in previous works, even if the fluctuation of the inflaton itself does not generate the curvature perturbation, fluctuation of a light field may induce fluctuation for the end-line of inflation and this may lead to generation of cosmological perturbation “at the end of the inflation”. Our scenario is different from this kind of modulated scenario, as clearly explained in this Letter by using δN formalism. We also explain the crucial difference from the standard multi-field inflation model. We show concrete examples of the modulated inflation scenario in which large non-gaussianity can be generated. We also discuss the running of the non-gaussianity parameter.     

Fine-scale anisotropy of the cosmic γ-ray background to identify the underlying generation process

The study of the intrinsic anisotropies of the extragalactic γ-ray background is of cosmological importance. As the first step to investigate the anisotropies, we introduce new evolution results of the active galaxies, and calculate the fine-scale anisotropy due to the possible contribution from the active galactic nuclei (AGN) model. By use of Monte Carlo simulations, we discuss the fine-scale fluctuation patterns, and propose that these patterns or anisotropies form a unique test for the AGN generation mechanism.We also suggest that the best energy range to test the AGN model is E_o 10 MeV.     

Brownian Motion with Dry Friction

A small object (Solid or droplet) is placed on a horizontally vibrating plate, imposing an acceleration γ(t) in the form of a white noise. The object experiences dry friction (due to soild/solid interaction, or to contact angle hysteresis in the case of a droplet). The object is driven by a force γ(t) – Δσ(t) where σ(t), =±1, depending on the sign of the velocity. We discuss the motion at two levels: (i) in terms of simple scaling laws, (ii) by a propagator technique. (a) When Δ is below a certain crossover value Δ*, we expect an unperturbed (Langevin) Brownian motion. (b) When Δ > Δ*, we expect a reduced diffusion coefficient proportional to Δ⁻⁴ for small Δ.     

Kinetic theory of dense fluids. IV. Entropy balance equation and irreversible thermodynamics

The viscosity coefficient obtained in a previous paper of this series is calculated as a function of density by developing the N-particle collision operator into a dynamic cluster expansion. The excess transport coefficient Δη is given in an exponential form, where η₀ is the two-body Chapman-Enskog result for the transport coefficient, n is the density, and β_l is a density-independent quantity consisting of connected cluster contributions of (l + 2) particles. Therefore, the leading term β₁ consists of connected three-body cluster contributions. The excess shear viscosity coefficient is calculated for a monatomic hard-sphere fluid by computing β_l up to the three-body contributions and the result is compared with the molecular dynamics result by Ashurst and Hoover and also with the experimental data on Ar at 75°C. In spite of the crudity of the potential model used and the approximations made the agreement is good. The result can be improved if l-body clusters (l 4) are included in the calculation. The thermal conductivity coefficient can be obtained in a similar form by using exactly the same procedure used for the viscosity coefficient.     

Depletion-induced sphere-cylinder transition in C12E5 microemulsion: a small-angle X-ray scattering study

Small-angle X-ray scattering was used to study the mixture of C₁₂E₅ (pentaethylene glycol monododecyl ether)/H₂O/n-decane microemulsion and polyethylene glycol (PEG). The size, shape and the structure factor of the microemulsion were investigated by adding the polymer (PEG) to the mixture. Attractive depletion potential was induced between the microemulsion droplets by the non-adsorb polymer. The range and strength of the attractive potential were changed by varying the molecular weight and concentration of PEG. The forward scattering, S(0), of the spherical microemulsion, declined gradually as the polymer concentration decreased. For PEG with the molecular weight of M_n = 285?315, the microemulsion morphology remained spherical, but the main peak of the structure factor moved towards a bigger q. When PEG with molecular weights of M_n = 2200 and M_n = 6000 were used, a shape transition from spherical to cylindrical was induced in line with increasing polymer concentration.     

Algorithmic Reduction of Poincaré–Dulac Normal Forms and Lie Algebraic Structure

The Poincaré–Dulac normal form of a given resonant system is in general nonunique; given a specific normal form, one would like to further reduce it to a simplest normal form. In this Letter we give an algorithm, based on the Lie algebraic structure of the set of normal forms, to obtain this. The algorithm can be applied under some condition, nongeneric but often met in applications. When applicable, it is only necessary to solve linear equations, and is more powerful than the one proposed in previous work by the same author [Lett. Math. Phys. 42 (1999), 103–114; and Ann. Inst. H. Poincaré Phys. Théor. 70 (1999), 461–514].     

Theory of collective opinion shifts: from smooth trends to abrupt swings

We unveil collective effects induced by imitation and social pressure by analyzing data from three different sources: birth rates, sales of cell phones and the drop of applause in concert halls. We interpret our results within the framework of the Random Field Ising Model, which is a threshold model for collective decisions accounting both for agent heterogeneity and social imitation. Changes of opinion can occur either abruptly or continuously, depending on the importance of herding effects. The main prediction of the model is a scaling relation between the height h of the speed of variation peak and its width w of the form h ∼w^-κ, with κ= 2/3 for well connected populations. Our three sets of data are compatible with such a prediction, with κ≈0.62 for birth rates, κ≈0.71 for cell phones and κ≈0.64 for clapping. In this last case, we in fact observe that some clapping samples end discontinuously (w=0), as predicted by the model for strong enough imitation.     

DeuteronA(q 2 ) structure function in the low and mediumq 2 range

The deuteronA(q ² ) structure function is calculated up toq ²=20 fm^–2 with the Paris potential including relativistic nucleonic and mesonic corrections. Different parametrizations (Iachello et al., Galster et al., Höhler et al., Gari and Krümpelmann (GK)) of the nucleon electromagnetic form factors (emff) are used. Unlike the data of Simon et al., those recently taken at Saclay are reasonably well reproduced by our full predictions for the first three emff considered. On the contrary, the predictions for the GK model fail to fit the Saclay data. The dependence ofA(q ² ) on the cut-off masses of the hadronic form factors is also investigated.     

Multifractal behaviour of n-simplex lattice

We study the asymptotic behaviour of resistance scaling and fluctuation of resistance that give rise to flicker noise in an n-simplex lattice. We propose a simple method to calculate the resistance scaling and give a closed-form formula to calculate the exponent, β _L, associated with resistance scaling, for any n. Using current cumulant method we calculate the exact noise exponent for n-simplex lattices.     

Manipulating nonclassical quantum statistical properties of light field by an f-deformed Bose–Einstein condensate

We consider the interaction between an f-deformed Bose–Einstein condensate and a single-mode quantized light field. By using the Gardiner’s phonon operators, we find that there exists a natural deformation in the model which modifies the Bogoliubov approximation under the condition of large but finite number of particles in condensate. This approach introduces an intrinsically deformed Bose–Einstein condensate, where the deformation parameter, well-defined by the particle number N in condensate, controls the strength of the associated nonlinearity. By introducing the deformed Gardiner’s phonon operators we modify the very dilute-gas approximation through including atomic collisions in condensate. The rate of atomic collisions κ, as a new deformation parameter in the deformed Bose–Einstein condensate, controls the nonlinearity related to the atomic collisions. We show that by controlling the nonlinearities in the f-deformed atomic condensate through the two atomic parameters N and κ, it is possible to generate and manipulate the nonclassical quantum statistical properties of radiation field, such as, the sub-Poissonian photon statistics and quadrature squeezing. Also, it is possible to control the collapses and revivals phenomena in the average number of photons by atomic parameters N and κ.     

Phase transition of directed polymer in random potentials on 4+1 dimensions

Finite-temperature-directed polymer in random potentials is described by a transfer matrix method. On 4+1 dimensions, the evidence for a finite-temperature phase transition is found at T_c≈0.18, where the free energy fluctuation grows logarithmically as a function of time t. When TT_c, the fluctuation of the free energy grows as t^ω with ω≈0.156. The phase transition of the restricted solid-on-solid model, which is closely related to the directed polymer problem through the Kardar–Parisi–Zhang equation, is also discussed.     

Quantum n-Vector Anharmonic Crystal II: Displacement Fluctuations

Based on the 1/n-expansion derived in a previous paper, the displacement fluctuations are analyzed in a quantum n-vector model of anharmonic crystal in the large n regime. It is shown that in the ferroelectric phase the n limit of the local fluctuation field has faster large-distance correlation decay than its Hartree–Fock approximation. Also, the critical exponent of the global displacement fluctuation is strictly smaller there than the Hartree–Fock exponent. In particular, the displacement fluctuations may be normal in the ferroelectric phase in spite of the Hartree–Fock prediction.     

Stochastic epidemics and rumours on finite random networks

In this paper, we investigate the stochastic spread of epidemics and rumours on networks. We focus on the general stochastic (SIR) epidemic model and a recently proposed rumour model on networks in Nekovee et al. (2007) [3], and on networks with different random structures, taking into account the structure of the underlying network at the level of the degree–degree correlation function. Using embedded Markov chain techniques and ignoring density correlations between neighbouring nodes, we derive a set of equations for the final size of the epidemic/rumour on a homogeneous network that can be solved numerically, and compare the resulting distribution with the solution of the corresponding mean-field deterministic model. The final size distribution is found to switch from unimodal to bimodal form (indicating the possibility of substantial spread of the epidemic/rumour) at a threshold value that is higher than that for the deterministic model. However, the difference between the two thresholds decreases with the network size, n, following a n^−1/3 behaviour. We then compare results (obtained by Monte Carlo simulation) for the full stochastic model on a homogeneous network, including density correlations at neighbouring nodes, with those for the approximating stochastic model and show that the latter reproduces the exact simulation results with great accuracy. Finally, further Monte Carlo simulations of the full stochastic model are used to explore the effects on the final size distribution of network size and structure (using homogeneous networks, simple random graphs and the Barabasi–Albert scale-free networks).     

Stochastic modeling of atomizing spray in a complex swirl injector using large eddy simulation

Large-eddy simulation of an atomizing spray issuing from a gas-turbine injector is performed. The filtered Navier–Stokes equations with dynamic subgrid scale model are solved on unstructured grids to compute the swirling turbulent flow through complex passages of the injector. The collocated grid, incompressible flow algorithm on arbitrary shaped unstructured grids developed by Mahesh et al. (J. Comp. Phys. 197 (2004) 215–240) is used in this work. A Lagrangian point-particle formulation with a stochastic model for droplet breakup is used for the liquid phase. Following Kolmogorov’s concept of viewing solid particle-breakup as a discrete random process, the droplet breakup is considered in the framework of uncorrelated breakup events, independent of the initial droplet size. The size and number density of the newly produced droplets is governed by the Fokker–Planck equation for the evolution of the pdf of droplet radii. The parameters of the model are obtained dynamically by relating them to the local Weber number and resolved scale turbulence properties. A hybrid particle-parcel is used to represent the large number of spray droplets. The predictive capability of the LES together with Lagrangian droplet dynamics models to capture the droplet dispersion characteristics, size distributions, and the spray evolution is examined in detail by comparing it with the spray patternation study for the gas-turbine injector. The present approach is computationally efficient and captures the global features of the fragmentary process of liquid atomization in complex configurations.     

Direct numerical simulations of autoignition in turbulent two-phase flows

Three-dimensional direct numerical simulations (DNS) were carried out to investigate the impact of evaporation of droplets on the autoignition process under decaying turbulence. The droplets were taken as point sources and were tracked in a Lagrangian manner. Three cases with the same initial equivalence ratio but different initial droplet size were simulated and the focus was to examine the influence of the droplet evaporation process on the location of autoignition. It was found that an increase in the initial droplet size results in an increase in the autoignition time, that highest reaction rates always occur at a specific mixture fraction ξ_MR, as in purely gaseous flows, and that changes in the initial droplet size did not affect the value of ξ_MR. The conditional correlation coefficient between scalar dissipation rate and reaction rates was only mildly negative, contrary to the strongly negative values for purely gaseous autoigniting flows, possibly due to the continuous generation of mixture fraction by the droplet evaporation process that randomizes both the mixture fraction and the scalar dissipation fields.