Distribution Functions in Quantum Mechanics and Wigner Functions

We formulate and solve the problem of finding a distribution function F(r,p,t) such that calculating statistical averages leads to the same local values of the number of particles, the momentum, and the energy as those in quantum mechanics. The method is based on the quantum mechanical definition of the probability density not limited by the number of particles in the system. The obtained distribution function coincides with the Wigner function only for spatially homogeneous systems. We obtain the chain of Bogoliubov equations, the Liouville equation for quantum distribution functions with an arbitrary number of particles in the system, the quantum kinetic equation with a self-consistent electromagnetic field, and the general expression for the dielectric permittivity tensor of the electron component of the plasma. In addition to the known physical effects that determine the dispersion of longitudinal and transverse waves in plasma, the latter tensor contains a contribution from the exchange Coulomb correlations significant for dense systems.     

Gaussian Approximations in the Stochastic Theory of Spontaneous Ignition of Coal Particles

Stochastic models of the process of spontaneous ignition of coal particles are distinguished by a pronounced nonlinearity due to the Arrhenius approximation for the constant of the reaction rate and by a complex structure of the stochastic component (noise). We present models describing the dynamics of the distribution of this quantity. The models obtained consist of nonlinear ordinary differential equations, which are solved by numerical methods with the use of computer technologies.     

Different modes of alkaline hydrolysis of fervenulin and isofervenulin

Unlike rheumycin and fervenulin, isofervenulin and 3-methylisofervenulin are hydrolyzed by aqueous alkalies at the N₍₅₎-C₍₆₎ bond. On acidification of the reaction mixture, the N-carboxy-N-methylcarbamoyltriazines formed are reconverted into the starting isofervenulins, and on basification (pH>10) into methylaminotriazinecarboxamides. A by-product of the alkaline hydrolysis of isofervenulin is a product of the contraction of the uracil ring, namely imidazotriazinone-7a-carboxylic acid.Translated from Khimiya Geterotsiklicheskikh Soedinenii, No. 11, pp. 1555–1559, November, 1987.     

Heterocyclic analogs of pleiadiene. 50. Sulfonation of perimidines,perimidones, and 2,3-dihydroperimidines

The action of concentrated sulfuric acid on perimidine and 2-substituted perimidines, 1,3-dimethylperimidinium perchlorate, and the 1,3-dimethyl derivatives of perimidone and 2,3-dihydroperimidine gave the 6(7)-sulfonic acids of these compounds.See [1] for communication 49.Translated from Khimiya Geterotsiklicheskikh Soedinenii, No. 5, pp. 691–696, May, 1980.     

Determination of thermodynamic functions of sorption for cyclohexyl-, piperidyl-, morpholyl-, and thiomorpholyl-n-alkanes by capillary gas-liquid chromatography

The values of partial molar free energy (G), enthalpy (H), and entropy (S) of sorption in the homologous series ofN-alkylpiperidines,N-alkylmorpholines,N-alkyl thiomorpholines, and alkylcyclohexanes were determined. It was found that the free energy of sorption is determined to a greater extent by the enthalpy term than by the entropy one. The free energy of sorption of the first homolog decreases when then-alkyl chain is attached directly to the carbon atom of the cycle and increases in the case ofN-alkylsubstituted heterocycles. The influence of the heteroatom nature on intermolecular interactions of homologs with the nonpolar stationary phase was quantitatively estimated on the basis of thermodynamic data.Dead time necessary for calculation of the retention factor was determined by the retention of methane injected into the column simultaneously with the sample.Translated from Izvestiya Akademii Nauk. Seriya Khimicheskaya, No. 8, pp. 2030–2032, August, 1996.     

Ring contraction of thia(seleno)diazine rings into pyrazoles in the reaction of 3-aryl-10-methyl-2H-1,3,4-thia(seleno)diazino[3, 2-a]benzimidazolium salts with bases

Reaction of 3-aryl-10-methyl-2H-1,3,4-thiadiazino[3,2-a]benzimidazolium salts with triethylamine led to ring contraction of the thiadiazine ring to a pyrazole ring with the formation of a mixture of derivatives of 3-mercaptopyrazolobenzimidazole and di(pyrazolo[1,5-a]benzimidazolyl-3) disulfide. The disulfides were formed exclusively when these salts react with ethanolic alkali. The reaction with potassium carbonate in acetic anhydridegave3-acetylthiopyrazolo[1,5- a]benzimidazoles. 2-Aryl-4- alkylpyrazolo[1, 5- a]benzimidazoles were formed by heating thiadiazino[3,2-a]benzimidazolium salts in formamide and by treating selenodiazino[3,2-a]benzimidazolium salts with potassium carbonate in acetic anhydride.Translated from Khimiya Geterotsiklicheskikh Soedinenii, No. 12, pp. 1698–1705, 1992.