On the domain of fractional Laplacians and related generators of Feller processes

In this paper we study the domain of the generator of stable processes, stable-like processes and more general pseudo- and integro-differential operators which naturally arise both in analysis and as infinitesimal generators of Lévy- and Lévy-type (Feller) processes. In particular we obtain conditions on the symbol of the operator ensuring that certain (variable order) Hölder and Hölder–Zygmund spaces are in the domain. We use tools from probability theory to investigate the small-time asymptotics of the generalized moments of a Lévy or Lévy-type process ${(X_t)}_{t\ge 0}$,

$\underset{t\to 0}{\lim }?\frac{1}{t}({\mathbb{E}}^{x}f(X_t)?f(x)),x\in {\mathbb{R}}^d,$

for functions f which are not necessarily bounded or differentiable. The pointwise limit exists for fixed $x\in {\mathbb{R}}^d$ if f satisfies a Hölder condition at x. Moreover, we give sufficient conditions which ensure that the limit exists uniformly in the space of continuous functions vanishing at infinity. As an application we prove that the domain of the generator of ${(X_t)}_{t\ge 0}$ contains certain Hölder spaces of variable order. Our results apply, in particular, to stable-like processes, relativistic stable-like processes, solutions of Lévy-driven SDEs and Lévy processes.     

Dirichlet operators and the positive maximum principle

Let (A, D(A)) denote the infinitesimal generator of some strongly continuous sub-Markovian contraction semigroup onL ^p (m), p1 andm not necessarily -finite. We show under mild regularity conditions thatA is a Dirichlet operator in all spacesL ^q (m), qp. It turns out that, in the limitq,A satisfies the positive maximum principle. If the test functionsC _c D(A), then the positive maximum principle implies thatA is a pseudo-differential operator associated with a negative definite symbol, i.e., a Lévy-type operator. Conversely, we provide sufficient criteria for an operator (A, D(A)) onL ^p(m) satisfying the positive maximum principle to be a Dirichlet operator. If, in particular,A onL ² (m) is a symmetric integro-differential operator associated with a negative definite symbol, thenA extends to a generator of a regular (symmetric) Dirichlet form onL ² (m) with explicitly given Beurling-Deny formula.     

More efficient time integration for Fourier pseudospectral DNS of incompressible turbulence

Time integration of Fourier pseudospectral DNS is usually performed using the classical fourth-order accurate Runge-Kutta method or other second- or third-order methods, with a fixed step size. We investigate the use of higher-order Runge-Kutta pairs and automatic step size control based on local error estimation. We find that the fifth-order accurate Runge-Kutta pair of Bogacki and Shampine gives much greater accuracy at a significantly reduced computational cost. Specifically, we demonstrate speedups of 2× to 10× for the same accuracy. Numerical tests (including the Taylor-Green vortex, Rayleigh-Taylor instability, and homogeneous isotropic turbulence) confirm the reliability and efficiency of the method. We also show that adaptive time stepping provides a significant computational advantage for some problems (like the development of a Rayleigh-Taylor instability) without compromising accuracy.     

Helfrich repulsion and dynamical phase separation of multicomponent lipid bilayers

Thermal fluctuations of surfactant bilayers in an aqueous solution produce an effective, long-range repulsion that can lead to a continuous unbinding transition. We report on an optical interferometry study of the thermal fluctuations of multicomponent bilayers close to the unbinding transition. We find that, in contrast to the case of single-component bilayers, the thermal fluctuation spectrum of multicomponent bilayers does not agree with a continuous unbinding transition but instead indicates the proximity of an unbinding tricritical point.     

Homogeneity of a supersaturated solid solution

Extended x-ray absorption fine structures, small-angle x-ray scattering, and atomistic model calculations have been employed to probe the homogeneity of the fcc solution created in Ag-Cu, a classical system demonstrating the extension of solubility across the entire miscibility gap through rapid quenching. Our results reveal that in many cases the supersaturated solutions formed have decomposition features on the scale of 1 nm. Conventional diffraction methods are inadequate in determining the level of supersaturation or the uniformity of such solid solution alloys.     

Dynamical precursor of nematic order in a dense fluid of hard ellipsoids of revolution

We investigate hard ellipsoids of revolution in a parameter regime where no long range nematic order is present but already finite-size domains are formed which show orientational order. Domain formation leads to a substantial slowing down of a collective rotational mode which separates well from the usual microscopic frequency regime. A dynamic coupling of this particular mode into all other modes provides a general mechanism which explains an excess peak in spectra of molecular fluids. Using molecular dynamics simulation on up to 4096 particles and on solving the molecular mode coupling equation we investigate dynamic properties of the peak and prove its orientational origin. Received 19 September 2001 and Received in final form 19 March 2002     

Self-poisoning of crystal nuclei in hard-rod liquids

We report a Monte Carlo study of the pathway for crystal nucleation in a fluid of hard, colloidal rods. In the earliest stages of nucleation, a lamellar crystallite forms. Subsequent thickening of this lamella is hampered by the fact that the top and bottom surfaces of this crystallite are preferentially covered by rods that align parallel to the surface. As a consequence, subsequent growth of individual crystals is stunted. Experimental evidence for such stunted crystal growth has recently been reported by Maeda and Maeda in experiments on suspensions of colloidal rods [Phys. Rev. Lett. 90, 018303 (2003)]]. The simulations suggest that, in experiments, the growth of multilayer colloidal crystals can be selectively enhanced by the application of an external aligning field.