Remarks on the star-triangle relation in the Baxter-Bazhanov model

We show that the restricted star-triangle relation introduced by Bazhanov and Baxter can be obtained either from the star-triangle relation of the chiral Potts model or from the star-square relation proposed by Kashaev, Mangazeev, and Stroganov, and give a response to a guess of Bazhanov and Baxter.     

New solvable lattice models in three dimensions

In this paper we establish a remarkable connection between two seemingly unrelated topics in the area of solvable lattice models. The first is the Zamolodchikov model, which is the only nontrivial model on a three-dimen-sional lattice so far solved. The second is the chiral Potts model on the square lattice and its generalization associated with theU _q(sl(n)) algebra, which is of current interest due to its connections with high-genus algebraic curves and with representations of quantum groups at roots of unity. We show that this last sl(n)-generalized chiral Potts model can be interpreted as a model on a threedimensional simple cubic lattice consisting ofn square-lattice layers with anN- valued (N2) spin at each site. Further, in theN=2 case this three-dimen-sional model reduces (after a modification of the boundary conditions) to the Zamolodchikov model we mentioned above.     

Three-dimensional integrable models and associated tangle invariants

In this paper we show that the Boltzmann weights of the three-dimensional Baxter-Bazhanov model give representations of the braid group if some suitable spectral limits are taken. In the trigonometric case we classify all possible spectral limits which produce braid group representations. Furthermore, we prove that for some of them we get cyclotomic invariants of links and for others we obtain tangle invariants generalizing the cyclotomic ones.     

Knot invariants associated with a particularN→∞ continuous limit of the Baxter-Bazhanov model

First we briefly recall the definition of the three-dimensional Baxter-Bazhanov lattice model. The spins of this model are elements ofZ _N and theR-matrix is associated to the algebraU _q sl(n) ifq is a primitiveNth root of unity. Then we construct a particularN limit of the model, in which it is meaningful to interpret the spins as elements ofR and which gives the free Gaussian boson model. Finally, we study special limits of the rapidity variables in which we obtain braid group representations and we show that forn odd the associated knot invariants are given by the inverse of products of Alexander polynomials, evaluated at certain roots of unity.     

Chiral Potts model as a descendant of the six-vertex model

We observe that theN-state integrable chiral Potts model can be considered as a part of some new algebraic structure related to the six-vertex model. As a result, we obtain a functional equation which is supposed to determine all the eigenvalues of the chiral Potts model transfer matrix.     

Corner transfer matrices of the chiral Potts model. II. The triangular lattice

We consider a two-dimensional edge-interaction model satisfying the star-triangle relations. For the triangular lattice, the corner transfer matrices are functions of three rapidities: we show that they possess various factorization properties and satisfy certain equations. We indicate how these equations can be solved for the Ising model. We then consider the three-state chiral Potts model and obtain low-temperature solutions to the equations. The conjectured formula for the order parameter (the spontaneous magnetization) is verified to one more order in a series expansion.     

Baxter's Solution for the Free Energy of the Chiral Potts Model

In honor of Baxter's sixtieth birthday, we would like to review some of his work on the free energy of the chiral Potts model. In spite of the enormous complexity and difficulty of the problem, Baxter, using functional relations was able to calculate not only the free energy, but also the interfacial tension. We here show that the integral for the free energy simplifies in the superintegrable case and is identical to his earlier results using entirely different approaches. His calculations are extended to include other regions. We also attempt to clarify some of his reasoning as several steps may be mysterious at first glance.     

Three-dimensional vertex model in statistical mechanics from Baxter-Bazhanov model

We find that the Boltzmann weight of the three-dimensional Baxter-Bazhanov model is dependent on four spin variables which are the linear combinations of the spins on the corner sites of the cube, and the Wu-Kadanoff-Wegner duality between the cube- and vertex-type tetrahedron equations is obtained explicitly for the Baxter-Bazhanov model. Then a three-dimensional vertex model is obtained by considering the symmetry property of the weight function, which corresponds to the three-dimensional Baxter-Bazhanov model. The vertex-type weight function is parametrized as the dihedral angles between the rapidity planes connected with the cube. We write down the symmetry relations of the weight functions under the actions of the symmetry groupG of the cube. The six angles with a constraint condition appearing in the tetrahedron equation can be regarded as the six spectra connected, with the six spaces in which the vertextype tetrahedron equation is defined.     

Numerical results for the three-state critical Potts model on finite rectangular lattices

Partition functions for the three-state critical Potts model on finite square lattices and for a variety of boundary conditions are presented. The distribution of their zeros in the complex plane of the spectral variable is examined and is compared to the expected infinite-lattice result. The partition functions are then used to test the finite-size scaling predictions of conformal and modular invariance.     

On a one-dimensional model for the three-dimensional vorticity equation

The one-dimensional model for the three-dimensional vorticity equation proposed by Constantin, Lax, and Majda is discussed. Some unsatisfactory points are examined, especially when the viscosity is introduced. A different model is suggested, which, while less solvable than the previous one, can be more strictly connected with the three-dimensional vorticity behavior. The study is of interest for the numerical treatment of the three-dimensional vorticity equation.     

A numerical study of the phase structure of the q-state generalization of the hard-square model

An investigation is made of how a recently foundq-state generalization of the hard-square model fits into a more general phase diagram. The investigation is done by Monte Carlo and series expansion methods. Evidence is presented that the one-dimensional manifold of parameters along which the model is exactly solvable represents a line of first-order phase transitions.     

A lattice Bhatnagar--Gross--Krook model for a class of the generalized Burgers equations

A new lattice Bhatnagar--Gross--Krook (LBGK) model for a class of the generalized Burgers equations is proposed. It is a general LBGK model for nonlinear Burgers equations with source term in arbitrary dimensional space. The linear stability of the model is also studied. The model is numerically tested for three problems in different dimensional space, and the numerical results are compared with either analytic solutions or numerical results obtained by other methods. Satisfactory results are obtained by the numerical simulations.     

An idealized radiative transfer scheme for use in a mechanistic general circulation model from the surface up to the mesopause region

A new and numerically efficient method to compute radiative flux densities and heating rates in a general atmospheric circulation model is presented. Our method accommodates the fundamental differences between the troposphere and middle atmosphere in the long-wave regime within a single parameterization that extends continuously from the surface up to the mesopause region and takes the deviations from the gray limit and from the local thermodynamic equilibrium into account. For this purpose, frequency-averaged Eddington-type transfer equations are derived for four broad absorber bands. The frequency variation inside each band is parameterized by application of the Elsasser band model extended by a slowly varying envelope function. This yields additional transfer equations for the perturbation amplitudes that are solved numerically along with the mean transfer equations. Deviations from local thermodynamic equilibrium are included in terms of isotropic scattering, calculating the single scattering albedo from the two-level model for each band. Solar radiative flux densities are computed for four energetically defined bands using the simple Beer-Bougert-Lambert relation for absorption within the atmosphere. The new scheme is implemented in a mechanistic general circulation model from the surface up to the mesopause region. A test simulation with prescribed concentrations of the radiatively active constituents shows quite reasonable results. In particular, since we take the full surface energy budget into account by means of a swamp ocean, and since the internal dynamics and turbulent diffusion of the model are formulated in accordance with the conservation laws, an equilibrated climatological radiation budget is obtained both at the top of the atmosphere and at the surface.     

A half-moment model for radiative transfer in a 3D gray medium and its reduction to a moment model for hot, opaque sources

We present in this paper a new 3D half-moment model for radiative transfer in a gray medium, called the model, which uses maximum entropy closure. This model is a generalization to 3D of the 1D version recently proposed in (J. Comp. Phys. 180 (2002) 584). The direction space Ω is divided into two pieces, Ω⁺ and Ω^-, in a dynamical way by the plane perpendicular to the total radiative flux, and the half moments are defined from these subspaces. The model closure and the integrations of the radiative transfer equation performed on the moving Ω^± spaces are detailed. 1D planar results, which have motivated the extension of the model of (J. Comp. Phys. 180 (2002) 584) to multi-dimensions, are shown. These results are very good. The model is thereafter derived for 3D spherically symmetric geometry, where the correctness of the non-trivial border terms can be checked. Two 3D spherically symmetric problems are numerically solved in order to show the accuracy of the closure and the role of the border terms. Once again, compared to the solution obtained with a ray tracing solver, results are very good. From the 3D half-moment model, a new moment model, called , is derived for the particular case of a 3D hot and opaque source radiating into a cold medium, for applications such as simulations of stellar atmospheres and fires. Two-dimensional numerical results are presented and compared to those obtained solving the RTE and with other moment models. They demonstrate the very good accuracy of the model, its good convergence properties, and better prediction compared to all other existing moment models in its domain of applicability.