Nonequilibrium statistical mechanics. I. Mathematical formalism. Theorem of N. S. Krylov

Based on an algebraic formulation of quantum mechanics we introduce concepts playing a fundamental role in the constructionof the statistical mechanics of systems having direct classical analogs. We give the definition of a macrostate, mixing in the quantum version, and also demonstrate the existence of an upper bound to the relaxation time for an isolated system. It is shown that the theory constructed here contains both quantum and classical mechanics as limiting cases, but these two cases are not reducible to each other. The existence of a transition range not describable by the Schrodinger equation is noted.Translated from Izvestiya Vysshykh Uchebnykh Zavedenii, Fizika, No. 11, pp. 42–45, November, 1981.     

In memoriam: Prof. Dr. dr.h.c. Rolf Hosemann

Dr. h.c. Rolf Hosemann, the “father of the ideal paracrystal,” passed away in Berlin on September 28, 1994, in his 83rd year. Rolf Hosemann was born in 1912 at Rostock on the Baltic Sea. He studied physics at Marburg and Freiburg, in Germany. In 1939 he obtained qualifications as a university teacher upon a “Habilitationsschrift” about the small-angle x-ray scattering of cellulose. In 1950 he laid the foundation to his theory of ideal paracrystals making wide use of convolution integrals as introduced into x-ray crystallography by P. P. Ewald. M. V. Laue was so much impressed by Hosemann's papers that he asked him to join the Fritz-Haber-Institut at Berlin-Dahlem as a staff member. It was there that Hosemann compiled his extensive work of more than 370 papers. Hosemann taught a large number of students. More than ten among them became university professors.     

h.c.p.-f.c.c. phase transition in solid parahydrogen

The difference of free energies for the h.c.p. and f.c.c. parahydrogen is calculated as a function of molar volume. It is shown that the observed transition in solid parahydrogen¹ can be explained by differences in the structure of the cubic and quartic anharmonic terms.     

О ДВИЖЕНИИ ЧАСТИЦ В НЕРЕЗОНАНСНЫХ В. Ч. И МАГНИТОСТАТИЧЕСКИХ ПОЛЯХ

The smoothed motion of a particle along the lines of force of a magnetic field in the presence of a high-frequency field is analyzed. Small gradients of all the fields and non-resonant frequency of the h.f. field are assumed. A certain conservativeness of motion is derived which permits further development of the kinetic theory. A qualitative evaluation of the possibility of trapping a rapid cluster of plasma by means of a h.f. field is made.     

Analysis of R. P. Smith's and D. C. Parris-R. B. Mclellan's carbon diffusivity data

A phenomenological model is proposed to account for the variation of carbon diffusivityD(N₁) with composition in austenite. This model is based on Parris-McLellan's statement that the increase ofD(N₁) with carbon concentrationN ₁ is due to the increase in chemical driving force. The gradient of the carbon activity in Wagner's and in Zupp-Stevenson's interpretation is presumed to be the intrinsic driving force. The predictions of the model are compared with carbonD(N₁) values both with [1] and without [5] a chemical concentration gradient. TheD(0) values and the carbon-carbon interaction coefficients ₁ are calculated by the use of the least squares method from the experimental data. The application of Zupp-Stevenson's definition of the activity coefficient ₁ results in a better fit of the theory to the experimental data.     

Orbital pair theory of N.M.R. shielding

A simple correction accounting for the effect of truncating the atomic orbital basis on the calculation of N.M.R. shielding is proposed. This correction is obtained from the expression for the magnetic shielding as a sum of orbital pair contributions: a spurious term that vanishes only if a complete basis is used appears in this expression. It is demonstrated both theoretically and numerically (using the example of PH₃) that removing this term from the results obtained with a truncated basis improves significantly the agreement with the experimental values. As a by-product the gauge dependence of the numerical results is much reduced. The additional computing time is negligible.     

75As N.Q.R. and 19F N.M.R. study of solid AsF3

The temperature dependence of the ⁷⁵As pure quadrupole resonance frequency in AsF₃ has been studied from 77 K to a temperature very close to the melting point. It is found that the effective electric field gradient at the As site does not shift much from the gas-phase value and that the temperature dependence of the ⁷⁵As N.Q.R. frequency seems to follow the Bayer-Kushida model. The occurrence of a line-width transition in the ¹⁹F N.M.R. spectrum between 200 and 150 K demonstrates the presence of low-frequency, large-amplitude molecular motions which are too slow to average out the ⁷⁵As quadrupole coupling.     

M.H.D. Generators with Superheated Steam

This paper considers a new kind of M.H.D. generator operating at extremely low temperature through the use of a working fluid which is either water vapor or metallic vapors. In the first case conductivity is achieved by seeding with electrolytes. A high thermodynamical efficiency is obtained by combination of an adiabatic expansion with an isothermal compression.     

Study of the 65Cu(n,ɣ)66Cu reaction: M. G. Delfini,J. Kopecky,J. B. M. De Haas,H. I. Liou,R.E. Chrien and P. M. Endt,Nuclear Physics A404 (1983) 250–268

The structure of the “condensed” phase of a finite nucleus is investigated by a HF calculation with a HO-type potential, one-pion-exchange potential and one-rho-meson-exchange potential. The pion field expectation value is also calculated from this nuclear source wave function and compared with the pion field of the normal phase. The time-dependent Schroedinger equation is solved for a certain approximated model of a heavy-ion reaction to see the dynamical growth of the pion field expectation value and its behaviour on such a time scale. The growth of the pion field itself is also investigated to see whether it is a normal growth or that of the condensed phase.     

N.M.R. spectra of oriented molecules

The phosphorescence spectra of anthracene-h₁₀ and -d₁₀ have been studied in a number of glasses and crystalline matrices. The line widths are very matrix dependent and they range from about 150 cm^-1 to 1 cm^-1 in n-heptane. There is a correlation between the line width and the wavenumber position of the origin. A study of the relative intensities of fluorescence and phosphorescence from various sites in Shpolskii matrices has shown that the intersystem crossing rate is site dependent. It is postulated that this involves the relative energies of the singlet state and a triplet state, nearly degenerate with it, which are site dependent.     

Site factors in N.M.R. solvent effects

An algorithm was developed enabling implementation of a Nosé—Hoover thermostat within the framework of grand canonical molecular dynamics [Lynch, C. G. and Pettitt, B. M., 1997, J. chem. Phys., 107, 8594]. The proposed algorithm could readily be extended to mixtures of molecular species with different chemical potentials as shown in the paper. This algorithm was first applied to simulate a μVT ensemble of TIP4P water molecules at 298 K by means of a system comprising a number of full particles and a single scaled (fractional) particle, with the scaling factor considered as a dynamic variable in its own right and chemical potential a pre-set parameter. Our finding showed that the scheme with a single fractional particle tended to freeze in metastable states as well as failed to reproduce either the real-life (?24.05 kJmol^?1) or the model-specific chemical potential of water (?23.0kJ mol^?1). In order to overcome the inadequacy of a single fractional particle as a chemical potential ‘probe’ the treatment of Pettitt and co-workers was extended to introduce multiple fractional particles. The extended scheme (with 4 fractional particles) was able to reproduce the actual density of water for the driving chemical potential of -24.0k mo^?1. The actual behaviour of the density as a function of the chemical potential also agreed quite well with both the results of thermo-dynamic integration and the findings of Pettitt and co-workers.     

On the mathematical structure of the B.C.S.-model. II

It is shown for the degenerate B.C.S.-model how in the limit of an infinite system the exact thermal Greens-functions approach a gauge invariant average of the one's calculated with the Bogoliubov-Haag method.Work performed as consultant to General Atomic Europe.     

Relative stability of f.c.c. and b.c.c. structures for model systems at high temperatures

Summary Free-energy, entropy and volume differences between face-centered and body-centered cubic structures have been evaluated for model rare gas and alkali metal crystals by using the method of overlapping distributions. Stable phases are predicted in agreement with the behaviour of real materials in the regions of validity of classical mechanics and in agreement with the results of previous dynamical-simulation studies of crystal nucleation from the melt and of polymorphic transformations. The existence of a stable b.c.c. phase at high pressure and temperatures is predicted in this way for Lennard-Jones solids, while no high-pressure f.c.c. phase is expected for model Rb and Cs systems. We also show the possibility of making calculations of free-energy barriers to displacive crystalline transformations along a prescribed trajectory in configuration space. Work supported by the U.S. Department of Energy, Istituto per la Ricerca Scientifica e Tecnologica, Trento, and Gruppo Nazionale di Struttura della Materia del C.N.R., Italy.     

Deuterium isotopic effect in 13C-N.M.R. spectra

We obtain the grand potential and the associated distribution functions for a system of hard rods on a line of finite length. These functions are shown to be related to those of a semi-infinite system bounded by one hard wall. It follows that the contact values of the density and distribution functions at the walls of the finite system are related to those at the wall of the semi-infinite system, and that the force on the walls is the density at contact. We obtain also the density and distribution functions for a system with repeating boundary conditions, which are also expressible in terms of those of the finite and semi-infinite systems.     

N.M.R. Spectral Studies of Some Quinolone Derivatives. Part VI.1 N-Methyl-2-Quinolone Derivatives

The chemical shifts of the N-CH₃ protons for a series of 1-methyl-2-quinolone derivatives has been studied. Compounds with an 8-methyl substituent exhibit a slight downfield shift.     

N.M.R. relaxation study of crystalline quinuclidine

Transient signals in the fluorescence intensity induced by microwaves can be used to acquire quantitative information on the molecular rates of populating and depopulating of the lowest triplet state of organic molecules. This method seems especially promising when the molecule emits no detectable phosphorescence.

Part of this paper concerns the introduction of an appropriate mathematical model for describing the fluorescence transients. The model is formulated in conjunction with a specific series of experiments from which one may evaluate the molecular quantities in a systematic way. The kinetics of the lowest triplet state of free base porphin (H₂P) are determined explicitly by this method.

In the discussion the results for H₂P are compared with those for Zn porphin. A satisfactory qualitative explanation based on the theory of radiationless transitions can be given for the great difference in behaviour of these two porphyrin molecules.

Short attention is given to effects due to the photochemical shift of the two H atoms in the centre of H₂P, which is proven to occur even at a temperature as low as 1·3 K.     

E.S.R. spectroscopy of non-cyclic enones

The π-bond model is applied to the calculation of the proton chemical shifts of a range of benzenoid hydrocarbons. The simple calculations give results which compare well with experiment and with calculations based on the ring current model. Least squares fitting of the predicted chemical shifts to the observed values can give an indication of the occurrence of local aromatic groups in larger molecules. This procedure also yields an empirically adjusted estimate of the anisotropy in the molar magnetic susceptibility from the experimental proton chemical shifts. A simple criterion for aromaticity is also given. The overall results suggest that there is no need to associate a special ring current attribute to π conjugation in aromatic molecules.     

Theoretical models for the interpretation of E.S.C.A. spectra

Three features of E.S.C.A. spectra are discussed: the chemical shift, multiplet splitting, and satellite bands. In the chemical shift effect a perturbation theory treatment shows that the electronic relaxation energy in the hole state of the molecules can be attributed to a flow of electrons on to the ionized atom. Thus the relative abilities of surrounding atoms to “feed” electrons to the ionized atom determines the relative importance of the relaxation energy contribution to the chemical shift. This explains why neutral atom charges don't always determine the direction of the chemical shift. In multiplet splitting the same perturbation treatment shows that the sign and magnitude of the relaxation energy contribution to the multiplet splitting is determined by the direction and magnitude of flow of majority electron spin density in going to the relaxed hole state. Unrestricted Hartree-Fock calculations on the series MF ₂, with M first row transition metals, are compared with E.S.C.A. experiments on the corresponding octahedral complexes to show that complex ion formation has only a very small effect on core level multiplet splitting. Relaxed hole state calculations on a series of first row atom (C,N,O,F) containing radicals leads to the conclusion that in these cases correlation energy effects are larger for the high spin multiplets than for the low spin multiplets. The assignments of satellite structure in transition metal complexes is reviewed and multiconfiguration self-consistent field results are presented and discussed for the satellite structure     

Modified independent modal space control of m.d.o.f. systems

The performance of the independent modal space control (IMSC) algorithm for structural vibration control is examined in this paper. Both the theoretical analysis and numerical simulation show that, for a multi-degree-of-freedom system, the modal control forces may increase the contributions of the vibration of higher modes (uncontrolled modes) to the system response if the IMSC algorithm is used to design a structural control system. Therefore, the responses of the controlled structure may be underestimated if the effects of control forces on the higher modes are not considered in the response analysis. A new control algorithm—modified independent modal space control (MIMSC) algorithm is proposed in this paper for eliminating the effect of modal control force on the uncontrolled modes. Numerical example shows that the structural responses can be effectively reduced when control system design is carried out based on the proposed algorithm. By comparing the simulated results obtained by the IMSC and MIMSC algorithms, it is found that, in order to achieve the same control objective, the proposed algorithm is more effective than IMSC since the modal control forces do not have any effect on the uncontrolled modes. In order to verify the effectiveness of the proposed algorithm, a practical example—active control design of UCLA Math-Science Building is presented and discussed.