To calculate thermodynamic properties of the anisotropic Ising model, a double chain approximation (DCA), based on Bogolyubov's variational procedure, is presented. The critical temperatures are compared to those obtained by the usual mean field approximation (MFA), the linear chain approximation (LCA), and the exact or series results, when existing. A fitting of the spontaneous polarization and the dielectric constant data for the quasi-one-dimensional ferroelectric crystal PMDP (PbDPO4) is also made.



Zur Berechnung der thermodynamischen Eigenschaften eines anisotropen Ising-Modells wird eine auf dem Bogolyubov- Variationsverfahren beruhende Doppelkettennaherung (DCA) angegeben. Die kritischen Temperaturen werden mit denen verglichen, die mit der ublichen Naherung des mittleren Feldes (MFA) und der linearen Kettennaherung (LCA) erhalten werden und wo vorhanden, mit den exakten oder Serienergebnissen. Eine Anpassung der spontanen Polarisierung und der Dielektrizitatskonstante fur den quasi-eindimensionalen ferroelektrischen Kristall PMDP (PbDPO4) wird ebenfalls durchgefuhrt.

A Double Chain Approximation to the Ising Model

A derivation of the relativistic Hartree-Fock-Approximation for finite temperatures is given by utilizing a thermodynamic extremum principle for the construction of an approximate (thermal HF-) hamiltonian of single-particle structure. By use of the Wigner transform one can obtain the so-called quasi-classical expansion of the theory; i.e. the thermal Thomas-Fermi theory plus quantum corrections. The details are illustrated for the Walecka Lagrangian density.

The relativistic Hartree-Fock-Approximation for finite temperatures and its semiclassical expansion

One-Dimensional Ising Ferromagnet with Phase Transition†

To calculate thermodynamic functions in statistical mechanics the authors propose a new variational approach which has as particular cases the approaches of Bogoliubov, Oguchi and Tsallis-da Silva (1982), but introduces some improvements, mainly in the short-range order effects above the critical temperature. It requires a choice of a trial Hamiltonian that defines an approximate ensemble through which the thermodynamic functions may be calculated. To check the efficiency of their approach they apply it to the ferromagnetic Ising model by using the simplest possible trial Hamiltonian.

A variational approach for the free energy

A New Effective Ensemble Approach for the Free Energy

It is shown that a large class of approximations in statistical thermodynamics that are based on the free-energy variational principle for the density operator yield expressions for the macroscopic quantities of the system that are consistent from both the statisticalmechanical and thermodynamical points of view. This corrects some erroneous statements in the literature. In an exact treatment of the statistical mechanics of a system in thermal equilibrium there is complete consistency between the definitions of the macroscopic quantities from the statisticalmechanical point of view and the corresponding ones from the standpoint of phenomenological thermodynamics. This great achievement of statistical mechanics can easily be established within the formulation of statistical therraodynamics„because the exact Hamiltonian of the system is a purely mechanical quantity independent of any thermodynamic parameters, such as temperature, etc. In various unavoidable approximation procedures for the thermal-equilibrium properties of systems, ho~ever, the Hamiltonian is approximated by an effective one that depends on such thermodynamic parameters. In such a case the statistical-thermodynamic consistency mentioned above is brought into question. In fact, it has been stated' that in the weil-known thermal Hartree-Fock approximation" for a system of interacting fermions this consistency is sacrificed. An approximation scheme in statistical thermodynamics that yields results that violate the consistency between the statistical and thermodynamical definitions of various observable quantities is clearly undesirable (at the very least one would have to give a reason for choosing one rather than the other definition}. In the following we shall show that there is a large class of approximations in statistical thermodynamics that are free from this serious drawback. According to the laws of thermodynamics' the complete macroscopic description of a system in thermodynamic equilibrium under certain constraints is provided by the fundamental relation appropriate to the constraints on the system, i.e., by a function of the appropriate independent variables. For example, for a simple, singlecomponent system of volume V in contact with a heat reservoir at temperature T and a particle reservoir at chemical potential p, , the fundamental relation is

Nilton P. Silva

Papers

Consistency of variational approximations in statistical thermodynamics