Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

ISRN Condensed Matter Physics

Free Energy of Three-Dimensional Uniaxial Magnet in the Higher Non-Gaussian Approximation and in the Presence of an External Field

Mykhailo P. Kozlovskii, Ihor V. Pylyuk

Published Dec 27, 2011• 1 citation
Publisher

Abstract

A three-dimensional Ising-like system in a homogeneous external field is studied on the basis of the higher non-Gaussian measure density (the model). The presented solutions of recurrence relations for the coefficients of the effective measure densities and the generalized point of exit of the system from the critical regime are used for calculating the free energy of the system at temperatures ( is the phase transition temperature in the absence of an external field). A calculation technique is based on the first principles of statistical physics and is naturally realized without any general assumptions and without any adjustable parameters. The obtained expression for the free energy does not involve series expansions in the scaling variable and is valid near the critical point not only in the regions of the so-called weak and strong external fields, but also in the crossover region between these fields, where power series in the scaling variable are not effective.

DOI
10.5402/2011/260750

Related papers

Calculation of free energy of three-dimensional uniaxial magnet in external field based on the higher non-Gaussian distribution

Ihor V. Pylyuk, Mykhailo P. Kozlovskii, Yurij V. Holovatch +3 more

The analytical calculation of the free energy of a three‐dimensional Ising‐like system in a homogeneous external field is performed in the higher non‐Gaussian approximation (the ρ6 model) at temperatures above the critical value of Tc (⁠Tc is the phase‐transition temperature in the absence of an external field). The free energy of the system is found by separating the contributions from the short‐ and long‐wave spin‐density oscillation modes taking into account the generalized point of exit of the system from the critical regime as a function of both the temperature and field variables. A calculation technique is based on the first principles of statistical physics and is naturally realized without any general assumptions and without any adjustable parameters. The obtained expression for the free energy does not involve series expansions in the scaling variable and is valid near the critical point not only in the regions of the so‐called weak and strong external fields but also in the crossover region between these fields. In this region, the temperature and field effects on the system are equivalent, the scaling variable is of the order of unity and power series are not efficient. The free energy contains the leading terms and terms determining the temperature and field confluent corrections.

AIP conference proceedings•Jan 1, 2009•0 citations

Method of Calculating the Free Energy of Three-Dimensional Ising-Like System in an External Field with the Use of the $ρ^6$ Model

Ihor V. Pylyuk, Mykhailo P. Kozlovskii

The microscopic approach to calculating the free energy of a three-dimensional Ising-like system in a homogeneous external field is developed in the higher non-Gaussian approximation (the $ρ^6$ model) at temperatures above the critical value of $T_c$ ($T_c$ is the phase-transition temperature in the absence of an external field). The free energy of the system is found by separating the contributions from the short- and long-wave spin-density oscillation modes taking into account both the temperature and field fluctuations of the order parameter. Our analytical calculations do not involve power series in the scaling variable and are valid in the whole field-temperature plane near the critical point including the region in the vicinity of the limiting field $\tilde h_c$, which divides external fields into the weak and strong ones (i.e., the crossover region). In this region, the temperature and field effects on the system are equivalent, the scaling variable is of the order of unity, and power series are not efficient. The obtained expression for the free energy contains the leading terms and terms determining the temperature and field confluent corrections.

arXiv:0812.1955•Dec 10, 2008•0 citations

Three-dimensional Ising-like system in an external field: Microscopic calculation of the free energy in the higher non-Gaussian approximation

Ihor V. Pylyuk

An analytic method for deriving the free energy of a three-dimensional Ising-like system near the critical point in a homogeneous external field is developed in the the ρ6 model approximation. The mathematical description proposed for temperatures T>Tc (Tc is the phase-transition temperature in the absence of an external field) is valid for fields near , where the scaling variable is of the order of unity and power series in this variable are not effective. At the limiting field , the temperature and field effects on the system in the vicinity of the critical point are equivalent. The total free energy is obtained as a function of temperature, field and microscopic parameters of the system without using series expansions in the scaling variable. In addition to leading terms, the expression for the free energy includes the terms determining the temperature and field confluent corrections.

arXiv:0711.3156•Phase Transitions•Jan 1, 2007•2 citations

Microscopic description of the critical behavior of three-dimensional Ising-like systems in an external field

Mykhailo P. Kozlovskii, Ihor V. Pylyuk, O. O. Prytula

We study the behavior of a three-dimensional Ising-like system in an external field near the critical point by using the non-Gaussian spin-density fluctuations, namely, the quartic measure density with the even and odd powers of the variable (the asymmetric ${\ensuremath{\rho}}^{4}$ model). The basic idea of the analytic method for deriving complete expressions of the thermodynamic characteristics (including the scaling functions) is described. The proposed method allows us to perform the calculations on the microscopic level without any adjustable parameters. Explicit expressions for the total free energy, order parameter, susceptibility, entropy, and specific heat of the system are obtained as functions of the temperature and field. The regions of the so-called weak and strong fields are considered for temperatures above and below ${T}_{c}$ (${T}_{c}$ is the phase-transition temperature in the absence of an external field). The average spin moment and susceptibility, depending on the field variation and the proximity to ${T}_{c}$, are investigated. It is confirmed that the temperature and field fluctuations for the order parameter play the leading roles in the weak and strong fields, respectively.

Physical Review B•May 5, 2006•12 citations

Analytical Description of the Critical Behavior of a Three-Dimensional Uniaxial Magnet in an External Field by Singling out a Reference System

Mykhailo P. Kozlovskii, Ihor V. Pylyuk

The critical behavior of systems belonging to the universality class of the three-dimensional Ising model has been studied theoretically. A three-dimensional Ising-like system with exponentially decreasing interaction potential and in the presence of a homogeneous external field was considered in the framework of the collective variables method. A specific feature in the calculation of the partition function and the free energy of a uniaxial magnet consists in singling out a reference system. The role of the latter is played by the molecular-field Hamiltonian. A method to describe the critical behavior with the use of a singled out reference system is developed on the basis of a non-Gaussian (quartic) distribution of order-parameter fluctuations (the p4 model).

Ukrainian Journal of Physics•Apr 1, 2015•0 citations

Critical behaviour of a three-dimensional one-component magnet in strong and weak external fields at T>Tc

Mykhailo P. Kozlovskii, Ihor V. Pylyuk, O. O. Prytula

The influence of a homogeneous external field on a three-dimensional uniaxial magnet behaviour near the critical point is investigated on the microscopic level by the nonperturbative collective variable (CV) method using the ρ4 model. The research is carried out in the high-temperature region. The critical exponents related with external field are calculated. The analytic explicit expressions for the free energy, average spin moment and susceptibility are obtained without using any phenomenological assumptions and adjusting parameters. The field dependences of the order parameter are plotted for the field region including the neighbourhood of the pseudocritical line. It is established that long-wave fluctuations of the order parameter are responsible for main features of the crossover between the temperature-dependence and field-dependence critical behaviour of the system. The maximum locations on the temperature scale for the long-wave part of the susceptibility are determined for different values of the field.

•Jan 1, 2006•0 citations
A three-dimensional Ising-like system in a homogeneous external field is studied on the basis of the higher non-Gaussian measure density (the model). The presented solutions of recurrence relations for the coefficients of the effective measure densities and the generalized point of exit of the system from the critical regime are used for calculating the free energy of the system at temperatures ( is the phase transition temperature in the absence of an external field). A calculation technique is based on the first principles of statistical physics and is naturally realized without any general assumptions and without any adjustable parameters. The obtained expression for the free energy does not involve series expansions in the scaling variable and is valid near the critical point not only in the regions of the so-called weak and strong external fields, but also in the crossover region between these fields, where power series in the scaling variable are not effective.
The analytical calculation of the free energy of a three‐dimensional Ising‐like system in a homogeneous external field is performed in the higher non‐Gaussian approximation (the ρ6 model) at temperatures above the critical value of Tc (⁠Tc is the phase‐transition temperature in the absence of an external field). The free energy of the system is found by separating the contributions from the short‐ and long‐wave spin‐density oscillation modes taking into account the generalized point of exit of the system from the critical regime as a function of both the temperature and field variables. A calculation technique is based on the first principles of statistical physics and is naturally realized without any general assumptions and without any adjustable parameters. The obtained expression for the free energy does not involve series expansions in the scaling variable and is valid near the critical point not only in the regions of the so‐called weak and strong external fields but also in the crossover region between these fields. In this region, the temperature and field effects on the system are equivalent, the scaling variable is of the order of unity and power series are not efficient. The free energy contains the leading terms and terms determining the temperature and field confluent corrections.
Method of Calculating the Free Energy of Three-Dimensional Ising-Like System in an External Field with the Use of the ρ6ρ^6ρ6 Model
The microscopic approach to calculating the free energy of a three-dimensional Ising-like system in a homogeneous external field is developed in the higher non-Gaussian approximation (the ρ6ρ^6ρ6 model) at temperatures above the critical value of TcT_cTc​ (TcT_cTc​ is the phase-transition temperature in the absence of an external field). The free energy of the system is found by separating the contributions from the short- and long-wave spin-density oscillation modes taking into account both the temperature and field fluctuations of the order parameter. Our analytical calculations do not involve power series in the scaling variable and are valid in the whole field-temperature plane near the critical point including the region in the vicinity of the limiting field h~c\tilde h_ch~c​, which divides external fields into the weak and strong ones (i.e., the crossover region). In this region, the temperature and field effects on the system are equivalent, the scaling variable is of the order of unity, and power series are not efficient. The obtained expression for the free energy contains the leading terms and terms determining the temperature and field confluent corrections.
An analytic method for deriving the free energy of a three-dimensional Ising-like system near the critical point in a homogeneous external field is developed in the the ρ6 model approximation. The mathematical description proposed for temperatures T>Tc (Tc is the phase-transition temperature in the absence of an external field) is valid for fields near , where the scaling variable is of the order of unity and power series in this variable are not effective. At the limiting field , the temperature and field effects on the system in the vicinity of the critical point are equivalent. The total free energy is obtained as a function of temperature, field and microscopic parameters of the system without using series expansions in the scaling variable. In addition to leading terms, the expression for the free energy includes the terms determining the temperature and field confluent corrections.
We study the behavior of a three-dimensional Ising-like system in an external field near the critical point by using the non-Gaussian spin-density fluctuations, namely, the quartic measure density with the even and odd powers of the variable (the asymmetric ${\ensuremath{\rho}}^{4}$ model). The basic idea of the analytic method for deriving complete expressions of the thermodynamic characteristics (including the scaling functions) is described. The proposed method allows us to perform the calculations on the microscopic level without any adjustable parameters. Explicit expressions for the total free energy, order parameter, susceptibility, entropy, and specific heat of the system are obtained as functions of the temperature and field. The regions of the so-called weak and strong fields are considered for temperatures above and below Tc{T}_{c}Tc​ (Tc{T}_{c}Tc​ is the phase-transition temperature in the absence of an external field). The average spin moment and susceptibility, depending on the field variation and the proximity to Tc{T}_{c}Tc​, are investigated. It is confirmed that the temperature and field fluctuations for the order parameter play the leading roles in the weak and strong fields, respectively.
The critical behavior of systems belonging to the universality class of the three-dimensional Ising model has been studied theoretically. A three-dimensional Ising-like system with exponentially decreasing interaction potential and in the presence of a homogeneous external field was considered in the framework of the collective variables method. A specific feature in the calculation of the partition function and the free energy of a uniaxial magnet consists in singling out a reference system. The role of the latter is played by the molecular-field Hamiltonian. A method to describe the critical behavior with the use of a singled out reference system is developed on the basis of a non-Gaussian (quartic) distribution of order-parameter fluctuations (the p4 model).
The influence of a homogeneous external field on a three-dimensional uniaxial magnet behaviour near the critical point is investigated on the microscopic level by the nonperturbative collective variable (CV) method using the ρ4 model. The research is carried out in the high-temperature region. The critical exponents related with external field are calculated. The analytic explicit expressions for the free energy, average spin moment and susceptibility are obtained without using any phenomenological assumptions and adjusting parameters. The field dependences of the order parameter are plotted for the field region including the neighbourhood of the pseudocritical line. It is established that long-wave fluctuations of the order parameter are responsible for main features of the crossover between the temperature-dependence and field-dependence critical behaviour of the system. The maximum locations on the temperature scale for the long-wave part of the susceptibility are determined for different values of the field.