Synchronization and random long time dynamics for mean-field plane rotators

We consider the natural Langevin dynamics which is reversible with respect to the mean-field plane rotator (or classical spin XY) measure. It is well known that this model exhibits a phase transition at a critical value of the interaction strength parameter \(K\) , in the limit of the number \(N\) of rotators going to infinity. A Fokker–Planck PDE captures the evolution of the empirical measure of the system as \(N \rightarrow \infty \) , at least for finite times and when the empirical measure of the system at time zero satisfies a law of large numbers. The phase transition is reflected in the fact that the PDE for \(K\) above the critical value has several stationary solutions, notably a stable manifold—in fact, a circle—of stationary solutions that are equivalent up to rotations. These stationary solutions are actually unimodal densities parametrized by the position of their maximum (the synchronization phase or center). We characterize the dynamics on times of order \(N\) and we show substantial deviations from the behavior of the solutions of the PDE. In fact, if the empirical measure at time zero converges as \(N \rightarrow \infty \) to a probability measure (which is away from a thin set that we characterize) and if time is speeded up by \(N\) , the empirical measure reaches almost instantaneously a small neighborhood of the stable manifold, to which it then sticks and on which a non-trivial random dynamics takes place. In fact the synchronization center performs a Brownian motion with a diffusion coefficient that we compute. Our approach therefore provides, for one of the basic statistical mechanics systems with continuum symmetry, a detailed characterization of the macroscopic deviations from the large scale limit—or law of large numbers—due to finite size effects. But the interest for this model goes beyond statistical mechanics, since it plays a central role in a variety of scientific domains in which one aims at understanding synchronization phenomena.     

Entropic Repulsion for the Free Field:¶Pathwise Characterization in d≥ 3

We study concentration properties of the lattice free field , i.e. the centered Gaussian field with covariance given by the Green function of the (discrete) Laplacian, when constrained to be positive in a region of volume O(N ^d ) (hard–wall condition). It has been shown in [3] that, as N→∞, the conditioned field is pushed to infinity: more precisely the typical value of the ϕ-variable to leading order is , and the exact value of c was found. It was moreover conjectured that the conditioned field, once this diverging height is subtracted, converges weakly to the lattice free field. Here we prove this conjecture, along with other explicit bounds, always in the direction of clarifying the intuitive idea that the free field with hard–wall conditioning merely translates away from the hard wall. We give also a proof, alternative to the one presented in [3], of the lower bound on the probability that the free field is everywhere positive in a region of volume N ^d . Received: 26 October 1998 / Accepted: 5 April 1999     

Log-periodic Critical Amplitudes: A Perturbative Approach

Log-periodic amplitudes appear in the critical behavior of a large class of systems, in particular when a discrete scale invariance is present. Here we show how to compute these critical amplitudes perturbatively when they originate from a renormalization map which is close to a monomial. In this case, the log-periodic amplitudes of the subdominant corrections to the leading critical behavior can also be calculated.     

On the Synthesis and Characterization of Pd (CO) (PPh3)3

Pd (CO) (PPh₃)₃ could be isolated from the reaction mixture arising from cyclohexene hydrocarboxylation by PdCl₂ (PPh₃)₂ as the catalyst precursor; furthermore, it has also been prepared through direct reaction of Pd (PPh₃)₄ with CO in benzene. For this complex, ³¹P- and ¹³C-NMR. spectra suggest a rapid dissociation of PPh₃ at room temperature and a tetrahedral structure at – 70° in solution.     

Stereochemical investigation of the reaction of cyclopentadienyl(diphosphine)methoxycarbeneruthenium(II) complexes with LiAlH4

[(η-C₅H₅)Ru{Ph₂PCHRCHR′PPh₂}({C(OCH₃)CH₂C₆H₅})]PF₆ (where R, R′ = H or CH₃) reacts with LiAlH₄ in THF at ?80° C to give the corresponding 2-phenylethyl complexes, which have an antiperiplanar conformation around the H₂CCH₂ bond in solution; the reaction takes place with retention of configuration at the ruthenium atom.     

A Geometrical Proof of Shiota's Theorem on a Conjecture of S. P. Novikov

We give a new proof of Shiota's theorem on Novikov's consecture, which states that the K.P. equation characters Jacobians among all indecomposable principally polarized abelian varieties.     

Pulsatile pipe flow of pseudoplastic fluids

This note is concerned with a laminar pipe flow of a non-Newtonian fluid under the action of a small pulsating pressure gradient superposed to a steady one. The constitutive law describing the rheological behaviour of the fluid is the so-called power law (Ostwald–de Waele). An approximated analytical solution is found for the velocity, as power series of the amplitude of the periodic disturbance. The analytic solution is compared with a direct numerical solution and the perfect accord of the values obtained is underscored.