Gibbs free energy and Helmholtz free energy for a three-dimensional Ising-like model

The critical behavior of a 3D Ising-like system is studied at the microscopic level of consideration. The free energy of ordering is calculated analytically as an explicit function of temperature, an external field and the initial parameters of the model. Within a unified approach, both Gibbs and Helmholtz free energies are obtained and the dependencies of them on the external field and the order parameter, respectively, are presented graphically. The regions of stability, metastability, and unstability are established on the order parameter-temperature plane. The way of implementation of the well-known Maxwell construction is proposed at microscopic level.Comment: 10 pages, 4 figure

A non-classical van der Waals loop: Collective variables method

The equation of state is investigated for an Ising-like model in the framework of collective variables method. The peculiar feature of the theory is that a non-classical van der Waals loop is extracted. The results are compared with the ones of a trigonometric parametric model in terms of normalized magnetization, \tilde{M}, and field, \tilde{H}.Comment: 9 pages, 2 figure

The equation of state of a cell fluid model in the supercritical region

The analytic method for deriving the equation of state of a cell fluid model in the region above the critical temperature ($T \geqslant T_\text{c}$) is elaborated using the renormalization group transformation in the collective variables set. Mathematical description with allowance for non-Gaussian fluctuations of the order parameter is performed in the vicinity of the critical point on the basis of the $\rho^4$ model. The proposed method of calculation of the grand partition function allows one to obtain the equation for the critical temperature of the fluid model in addition to universal quantities such as critical exponents of the correlation length. The isothermal compressibility is plotted as a function of density. The line of extrema of the compressibility in the supercritical region is also represented.Comment: 26 pages, 6 figures, 1 tabl

The Procedure of Termination of the Criminal Case (Criminal Prosecution) on Rehabilitation Bases in the Russian Federation and Mongolia

The article describes the features of the modern criminal procedure of Mongolia, which consist in the absence of the stage of initiation of criminal proceedings, confluence of a decision of initiation of criminal proceedings and decision about prosecution as an accused, in existence of uniform termination procedure of investigation of the criminal cases subject to termination or referral to court. Comparative legal analysis of rehabilitating grounds for termination of criminal case (criminal prosecution) is carried out. The content of criminal procedural activities at the end of the investigation is considered. The procedure for appealing against decisions on termination of the criminal case is researched. It is characterized by a lack of judicial control in pre-trial proceedings in Mongolia. The procedure and law enforcement practice of realization of the right to rehabilitation is analyzed. The conclusion that departmental regulation hinders its implementation is made. The Authors state that the procedure for termination of the criminal case provided by the Criminal procedure code of Mongolia contains fairly extended guarantees of establishing the truth. The normative regulation completely excludes multiple transfer of the criminal case from the police to prosecutor in the event of discrepancy in the assessment of the qualification of the crime and the completeness of the investigation. The absence of a procedure for the termination of criminal prosecution due to non-involvement is a serious drawback of Mongolian legislation. In general, the mechanism of termination of the criminal case in the criminal procedure legislation of Mongolia is conceptually more consistent and logical, but in terms of elaboration of procedural mechanisms is much inferior to Russian legislation

Analytical calculation of the critical temperature and estimation of the critical region size for a fluid model

An analytical procedure for calculating the critical temperature and estimating the size of critical region for a cell fluid model is developed. Our numerical calculations are illustrated by the case of the Morse potential parameters characterizing the alkali metals (sodium and potassium). The critical temperatures found for liquid sodium and potassium as solutions of the resulting quadratic equation agree with experimental data. The expression for the relative temperature determining the critical region size is obtained proceeding from the condition for the critical regime existence. In the cases of sodium and potassium, the value of this temperature is of the order of a few hundredths.Comment: 15 pages, 2 table

Morse fluids in the immediate vicinity of the critical point: Calculation of thermodynamic coefficients

The previously proposed approach for the microscopic description of the critical behavior of Morse liquids based on the cell fluid model is applied to the case where the parameters of the Morse interaction potential correspond to alkali metals (sodium and potassium). The critical temperatures, densities, and pressures obtained for sodium and potassium agree with the experimental results. The thermodynamic coefficients (isothermal compressibility, density fluctuations, and thermal expansion) of sodium are investigated in the supercritical temperature region. Numerical calculations of thermodynamic coefficients are performed close to the critical point, where carrying out theoretical and experimental research is challenging. The change in compressibility with increasing density at various temperature values is traced. The behavior of density fluctuations approaching the critical point is shown for different temperatures. The variation in the magnitude of the thermal expansion with increasing temperature for different pressure values is illustrated.Comment: 14 pages, 6 figures, 1 tabl

Ab initio study of the vapour-liquid critical point of a symmetrical binary fluid mixture

A microscopic approach to the investigation of the behaviour of a symmetrical binary fluid mixture in the vicinity of the vapour-liquid critical point is proposed. It is shown that the problem can be reduced to the calculation of the partition function of a 3D Ising model in an external field. For a square-well symmetrical binary mixture we calculate the parameters of the critical point as functions of the microscopic parameter r measuring the relative strength of interactions between the particles of dissimilar and similar species. The calculations are performed at intermediate ($\lambda=1.5$) and moderately long ($\lambda=2.0$) intermolecular potential ranges. The obtained results agree well with the ones of computer simulations.Comment: 14 pages, Latex2e, 5 eps-figures included, submitted to J.Phys:Cond.Ma