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.     

Ground state of a spin-phonon system. I. Variational estimates

A study is made of the ground-state energy of a spin-one-half particle in a fieldB and interacting with a phonon bath. The infrared-sensitive case of acoustic phonons with point coupling in three dimensions is characterized by two parameters, a coupling constant andB. Units are used where the high-momentum phonon cutoff is unity. There is a curve (B) separating a symmetry-breaking region with a long-range phonon field from a normal region. Two simple, well-known, approximations are compared. The source theory yields discontinuities in the first derivatives of the energy with respect toB and whenB>e ^–1 and an infinite-order transition whenB<e ^–1, but is trivial in the large- region. The classical theory yields discontinuities in the second derivatives but is trivial in the small- region. An improved variationally fixed ground-state wave function is analyzed. It gives a new (B) curve with an infinite-order transition with continuous energy derivatives whenB<e/(e ²–1/4) and with discontinuous derivatives whenB is larger than this value. It is nontrivial in the entire (B) plane. The crossover to classical behavior occurs near =1/2 forB1. But the wave function does not describe quantum fluctuations in the large- phase. A second way of combining source and classical effects is described. It yields a second-order transition (near =1/2 forB1) everywhere. These theories are special cases of a symmetry-breaking transformation together with a one-mode treatment of quantum fluctuations. The transition is viewed in terms of a single mode with a variable length, coupled dynamically to the spin.     

Mechanism of breakdown development in a gas in a subthreshold HF field

The development of a breakdown wave in subthreshold HF fields is considered for the condition that the field energy density exceed the plasma energy density E²/4 > nT. It is shown that the breakdown wave propagation mechanism is controlled by HF diffusion with a coefficient _iE²/4en. A model of breakdown development along the direction of the linear field polarization is constructed on the basis of this mechanism.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 55–60, August, 1989.     

Conducting electrically active films obtained by the electrochemical oxidation of aniline in molten electrolytes and concentrated hydrochloric acid

The feasibility of obtaining electrically conducting and active films by the electropolymerization of aniline in butylpyridinium tetrachloroaluminate or hexachlorodialuminate as the medium has been demonstrated. Identical voltamperometric characteristics with a potential peak of 0.5 V were found in these films and the compound formed upon the electrochemical polymerization of aniline in 20% hydrochloric acid.Translated from Izvestiya Akademii Nauk SSSR, Seriya Khimicheskaya, No. 1, pp. 212–214, January, 1991.     

XX International Chugaev Conference on Coordination Chemistry

Russian Journal of Coordination Chemistry -     

Alkylation ofβ-diketones of the hexahydroindan series and their enamines

Conclusions The reactions of hexahydro-1,3-indandlone with ammonia and some amines were studied, and the structures of the products formed were proved. In the alkylation of-diketones of the hexahydroindan series and their enamines under the usual conditions C-alkylation products are not formed.Translated from Izvestiya Akademii Nauk SSSR, Seriya Khimicheskaya, No. 7, pp. 1214–1218, July, 1966.     

Structural mining: self-consistent design on flexible protein-peptide docking and transferable binding affinity potential

A flexible protein-peptide docking method has been designed to consider not only ligand flexibility but also the flexibility of the protein. The method is based on a Monte Carlo annealing process. Simulations with a distance root-mean-square (dRMS) virtual energy function revealed that the flexibility of protein side chains was as important as ligand flexibility for successful protein-peptide docking. On the basis of mean field theory, a transferable potential was designed to evaluate distance-dependent protein-ligand interactions and atomic solvation energies. The potential parameters were developed using a self-consistent process based on only 10 known complex structures. The effectiveness of each intermediate potential was judged on the basis of a Z score, approximating the gap between the energy of the native complex and the average energy of a decoy set. The Z score was determined using experimentally determined native structures and decoys generated by docking with the intermediate potentials. Using 6600 generated decoys and the Z score optimization criterion proposed in this work, the developed potential yielded an acceptable correlation of R(2) = 0.77, with binding free energies determined for known MHC I complexes (Class I Major Histocompatibility protein HLA-A(*)0201) which were not present in the training set. Test docking on 25 complexes further revealed a significant correlation between energy and dRMS, important for identifying native-like conformations. The near-native structures always belonged to one of the conformational classes with lower predicted binding energy. The lowest energy docked conformations are generally associated with near-native conformations, less than 3.0 Angstrom dRMS (and in many cases less than 1.0 Angstrom) from the experimentally determined structures.     

The separation of palladium and platinum by ion flotation

Differences in the ion flotation properties of palladium(II) and platinum(IV) chloro complexes in aqueous solutions are used to achieve separations of these metals. The anionic chloro complex PtCl^2-₆ is floated selectively with cationic surfactants of the type, RNR'₃Br, from solutions of PdCl^2-₄ and various concentrations of hydrochloric acid. The palladium(II) does not float from solutions of ? 3.0 M HCl and the platinum(IV) floated from these solutions can be recovered free of palladium. However, the separation is incomplete as much of the platinum(IV) is also unfloated from these solutions. Quantitative separations are obtained by conversion of the palladium(II) to the cationic ammine, Pd(NH₃)₄²⁺ with aqueous ammonia prior to flotation. The anionic chloro complex of platinum(IV) is unaffected by the presence of ammonia and is floated quantitatively with the surfactant n-hexadecyltri-n-propylammonium bromide from 0.01 M ammonia solutions.