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

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

Phase transition in a cell fluid model

We propose a method of describing a phase transition in a cell fluid model with pair interaction potential that includes repulsive and attractive parts. An exact representation of the grand partition function of this model is obtained in the collective variables set. The behavior of the system at temperatures below and above the critical one is explored in the approximation of a mean-field type. An explicit analytic form of the equation of state which is applicable in a wide range of temperatures is derived, taking into account an equation between chemical potential and density. The coexistence curve, the surface of the equation of state and the phase diagram of the cell Morse fluid are plotted.Comment: 18 pages, 12 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

Free energy of 3D Ising-like system near the phase transition point`

A generalized representation for the scaling form of free energy of the system near the phase transition point is proposed. Explicit expressions for coefficients as functions of the reduced temperature and external field in the case of T>Tc are obtained at the microscopic level

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