Statistical mechanics of the nonlinear Schrödinger equation

We investigate the statistical mechanics of a complex fieldø whose dynamics is governed by the nonlinear Schrödinger equation. Such fields describe, in suitable idealizations, Langmuir waves in a plasma, a propagating laser field in a nonlinear medium, and other phenomena. Their Hamiltonian $$H(\phi ) = \int_\Omega {[\frac{1}{2}|\nabla \phi |^2 - (1/p) |\phi |^p ] dx}$$ is unbounded below and the system will, under certain conditions, develop (self-focusing) singularities in a finite time. We show that, whenΩ is the circle and theL ² norm of the field (which is conserved by the dynamics) is bounded byN, the Gibbs measureυ obtained is absolutely continuous with respect to Wiener measure and normalizable if and only ifp andN are such that classical solutions exist for all time—no collapse of the solitons. This measure is essentially the same as that of a one-dimensional version of the more realisitc Zakharov model of coupled Langmuir and ion acoustic waves in a plasma. We also obtain some properties of the Gibbs state, by both analytic and numerical methods, asN and the temperature are varied.     

Local mean field models of uniform to nonuniform density fluid-crystal transitions

We investigate the existence of nontranslation invariant (periodic) density profiles, for systems interacting via translation invariant long-range potentials, as minimizers of local mean field free energy functionals. The existence of a second-order transition from a uniform to a nonuniform density at a specified temperature is proven for a class of model systems.     

Self-diffusion in a non-uniform one dimensional system of point particles with collisions

Summary We generalize the results of Spitzer, Jepsen and others [1–4] on the motion of a tagged particle in a uniform one dimensional system of point particles undergoing elastic collisions to the case where there is also an external potential U(x). When U(x) is periodic or random (bounded and statistically translation invariant) then the scaled trajectory of a tagged particle converges, as A , to a Brownian motion W _D (t) with diffusion constant , where is the average density, is the mean absolute velocity and ^–1 the temperature of the system. When U(x) is itself changing on a macroscopic scale, i.e. , then the limiting process is a spatially dependent diffusion. The stochastic differential equation describing this process is now non-linear, and is particularly simple in Stratonovich form. This lends weight to the belief that heuristics are best done in that form.Dedicated to Frank Spitzer on the occasion of his 60th birthdayWork supported in part by NSF Grants No. PHY 8201708 and No. DMR 81-14726Heisenberg-fellowAlso Department of Physics     

Percolation in strongly correlated systems: The massless Gaussian field

We derive a number of new results for correlated nearest neighbor site percolation onZ ^d. We show in particular that in three dimensions the strongly correlated massless harmonic crystal, i.e., the Gaussian random field with mean zero and covariance –, has a nontrivial percolation behavior: sites on whichS _x h percolate if and only ifh _c . with0 _c < . This provides the first rigorous example of a percolation transition in a system with infinite susceptibility.     

Resolving Component Overlap in Multiwavelength Kinetic Spectroscopy: Application to Raman Scattering from Intermediates in Triplet-State Photoreactions

Abstract— Understanding of reaction mechanisms in photochemistry is advanced both by kinetic measurements to determine rates of reaction and by spectroscopic studies of the structures of precursors and photoproducts. When kinetics and spectroscopy are combined in a time-resolved, multiwavelength spectroscopic measurement on a reacting system, spectra and structures of intermediates can also be determined. In this paper, the application of multidimensional least-squares and factor analysis techniques for resolving overlapped spectra from intermediates in photochemical kinetics are discussed. The methods are employed specifically to resolve Raman spectra of intermediates in triplet-state photoreactions. By varying excitation intensity, spectra of excited-triplet states are resolved from ground states and solvent. By varying the concentration of a triplet quencher in the sample, the Raman spectrum of a sensitizer excited state (benzophe-none) can be resolved from those of acceptor-triplet states or radical photoproducts. Applications of these concepts in other areas of photochemical kinetics are addressed.     

Composite alignment media for the measurement of independent sets of NMR residual dipolar couplings

The measurement of independent sets of NMR residual dipolar couplings (RDCs) in multiple alignment media can provide a detailed view of biomolecular structure and dynamics, yet remains experimentally challenging. It is demonstrated here that independent sets of RDCs can be measured for ubiquitin using just a single alignment medium composed of aligned bacteriophage Pf1 particles embedded in a strained polyacrylamide gel matrix. Using this composite medium, molecular alignment can be modulated by varying the angle between the directors of ordering for the Pf1 and strained gel matrix, or by varying the ionic strength or concentration of the Pf1 particles. This approach offers significant advantages in that greater experimental control can be exercised over the acquisition of multi-alignment RDC data while a homogeneous chemical environment is maintained across all of the measured RDC data.     

A theoretical study of the geometry of the first triplet of acetaldehyde and of its fragmentation into free radicals

The energy profiles for the decomposition of the first triplet of acetaldehyde via the reactions: CH₃CHO^ν → CH₃CO° + H°, CH₃CHO^* → CH^.₃ + HCO^., have been computed using a non-empirical LCAO MO SCF procedure. Both reactions were found to have energy barriers of similar height (245 KJ mole ^?1) and at similar extensions of the respective rupturing bonds (0.71–0.76 Å). The energy profile for internal rotation about the C-C bond was also calculated and showed a small barrier (3.5 kJ mole^?1).     

Mono- and di-substituted urea derivatives of cyclodiphosphazane: [ClP(μ-NBu)2PN(Me)CON(H)Me] and [Me(H)NCON(Me)P(μ-NBu)]2

The reaction of cis-[ClP(μ-N^tBu)]₂ with N,N′-dimethylurea leads to the formation of both mono- and di-substituted derivatives [ClP(μ-N^tBu)₂P(NMeCON(H)Me)] and [(μ-N^tBu)P(NMeCON(H)Me)]₂, instead of monomeric, dimeric or higher oligomeric macrocycles. The structure of [ClP(μ-N^tBu)₂P(NMeCON(H)Me)] shows rare non-bonded P?Cl and intermolecular hydrogen bonding interactions leading to a 2D-sheet like structure.     

Silver and palladium complexes of some aromatic azo and azoxy compounds

The complexing ability of derivatives of azobenzene (I) is well known (I) and responsible for the production of a wide variety of dyestuffs and analytical chemicals. While the azo group generally participates in the coordination, the determination of the degree of its interaction is complicated by other functional groups which are also coordinated with the transition metal. In a previous publication (2), we reported the preparation of the silver and palladium complexes of benzo[c]cinnoline (II) and proposed that these results might be used to explain the electron donor properties of azobenzene. We are now reporting the preparation of some additional complexes with ligands containing the azo group.