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

Free Energy and Equation of State of Ising-like Magnet Near the Critical Point

The description of a three-dimensional Ising-like magnet in the presence of an external field in the vicinity of the critical point by the collective variables method is proposed. Using the renormalization group transformations, the scaling region size is defined as a function of temperature and field. The obtained expressions for the free energy, equation of state and susceptibility allow one to analyse their dependence on microscopic parameters of the system. The critical exponents of the correlation length and order parameter are calculated as well. The results agree qualitatively with ones obtained within the framework of the parametric representation of the equation of state and Monte-Carlo simulations. The calculations do not involve any parametrization, phenomenological assumptions and adjustable parameters. The approach can be extended to models with a multicomponent order parameter.Comment: 9 pages 2 figures in journal Nuclear physics B (in press but with ref. v.753, pages 242-251

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

3D Ising system in an external field. Recurrence relations for the asymmetric ρ⁶ model

The 3D one-component spin system in an external magnetic field is studied using the collective variables method. The integration of the partition function of the system over the phase space layers is performed in the approximation of the sextic measure density including the even and the odd powers of the variable (the asymmetric ρ⁶ model). The general recurrence relations between the coefficients of the effective measure densities are obtained. The new functions appearing in these recurrence relations are given in the form of a convergent series.Тривимірна однокомпонентна спінова система в зовнішньому магнітному полі досліджується з допомогою методу колективних змінних. Інтегрування статистичної суми системи по шарах фазового простору здійснюється в наближенні густини міри, яка містить в собі парні та непарні степені змінної до шостого включно (асиметрична модель ρ⁶). Одержано загальні рекурентні співвідношення між коефіцієнтами ефективних густин мір. Нові функції, що появляються в цих рекурентних співвідношеннях, подані у вигляді збіжних рядів

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