In the first part of the paper, we introduce the concept of observable quantities associated with a macroinstrument measuring the density and temperature and with a microinstrument determining the radius of a molecule and its free path length, and also the relationship between these observable quantities. The concept of the number of degrees of freedom, which relates the observable quantities listed above, is generalized to the case of low temperatures. An analogy between the creation and annihilation operators for pairs (dimers) and the creation and annihilation operators for particles (molecules) is carried out. A generalization of the concept of a Bose condensate is introduced for classical molecules as an analog of an ideal liquid (without attraction). The negative pressure in the liquid is treated as holes (of exciton type) in the density of the Bose condensate. The phase transition gas-liquid is calculated for an ideal gas (without attraction). A comparison with experimental data is carried out.
In the other part of the paper, we introduce the concept of new observable quantity, namely, of a pair (a dimer), as a result of attraction between the nearest neighbors. We treat in a new way the concepts of Boyle temperature T_B (as the temperature above which the dimers disappear) and of the critical temperature T_c (below which the trimers and clusters are formed). The equation for the Zeno line is interpreted as the relation describing the dependence of the temperature on the density at which the dimers disappear. We calculate the maximal density of the liquid and also the maximal density of the holes. The law of corresponding states is derived as a result of an observation by a macrodevice which cannot distinguish between molecules of distinct gases, and a comparison of theoretical and experimental data is carried out.
We consider 3-dimensonal Schr¨odinger operators with complex potential. We obtain new trace formulas with new terms, associated with singular measure. In order to prove these results, we study analytic properties of a modified Fredholm determinant as a function from Hardy spaces in the upper half-plane. In fact, we reformulate spectral theory problems as problems of analytic functions from Hardy spaces.
We show that a distribution of the type of the Bose-Einstein distribution describes the van der Waals gas, while the Fermi-Dirac distribution describes the van der Waals liquid. We present the construction of the binodal, the melting curve, and the liquid-to-amorphous-solid transition under negative pressure. The notion of correlation sphere and the two-scale picture on the Hougen-Watson diagram are used. © 2015, Pleiades Publishing, Ltd.
A fluid flow along a plate with small irregularities on the surface is considered for large Reynolds numbers. The boundary layer has a double-deck structure, i.e., both a thin boundary layer and the classical Prandtl boundary layer are present. It is proved that the solution of the boundary-value problem thus obtained exists and is unique in the Prandtl boundary layer, and the stability of the solution is investigated at large times. The results of numerical modeling are given. Supported by the Basic Research Program of the National Research University “Higher School of Economics.” © 2015, Pleiades Publishing, Ltd.