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Corner Transfer Matrix
In statistical mechanics, the corner transfer matrix describes the effect of adding a quadrant to a lattice. Introduced by Rodney Baxter in 1968 as an extension of the Kramers-Wannier row-to-row transfer matrix, it provides a powerful method of studying Lattice model (physics), lattice models. Calculations with corner transfer matrices led Baxter to the exact solution of the hard hexagon model in 1980. Definition Consider an IRF (interaction-round-a-face) model, i.e. a square lattice model with a Spin (physics), spin σ''i'' assigned to each site ''i'' and interactions limited to spins around a common face. Let the total energy be given by :E=\sum_\epsilon\left(\sigma_,\sigma_,\sigma_,\sigma_\right), where for each face the surrounding sites ''i'', ''j'', ''k'' and ''l'' are arranged as follows: For a lattice with ''N'' sites, the Partition function (statistical mechanics), partition function is :Z_=\sum_\prod_w\left(\sigma_,\sigma_,\sigma_,\sigma_\right), where the sum is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Statistical Mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applications include many problems in a wide variety of fields such as biology, neuroscience, computer science Computer science is the study of computation, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied science, ..., information theory and sociology. Its main purpose is to clarify the properties of matter in aggregate, in terms of physical laws governing atomic motion. Statistical mechanics arose out of the development of classical thermodynamics, a field for which it was successful in explaining macroscopic physical properties—such as temperature, pressure, and heat capacity—in terms of microscop ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Diagonal Form
In mathematics, a diagonal form is an algebraic form (homogeneous polynomial) without cross-terms involving different indeterminates. That is, it is of the form :\sum_^n a_i ^m\ for some degree ''m''. Such forms ''F'', and the hypersurfaces ''F'' = 0 they define in projective space, are very special in geometric terms, with many symmetries. They also include famous cases like the Fermat curves, and other examples well known in the theory of Diophantine equations. A great deal has been worked out about their theory: algebraic geometry, local zeta-functions via Jacobi sums, Hardy-Littlewood circle method. Diagonalization Any degree-2 homogeneous polynomial can be transformed to a diagonal form by variable substitution. Higher-degree homogeneous polynomials can be diagonalized if and only if their catalecticant is non-zero. The process is particularly simple for degree-2 forms (quadratic forms), based on the eigenvalues of the symmetric matrix representing the quadratic f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Lattice Models
Lattice may refer to: Arts and design * Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material * Lattice (music), an organized grid model of pitch ratios * Lattice (pastry), an ornamental pattern of crossing strips of pastry Companies * Lattice Engines, a technology company specializing in business applications for marketing and sales * Lattice Group, a former British gas transmission business * Lattice Semiconductor, a US-based integrated circuit manufacturer Science, technology, and mathematics Mathematics * Lattice (group), a repeating arrangement of points ** Lattice (discrete subgroup), a discrete subgroup of a topological group whose quotient carries an invariant finite Borel measure ** Lattice (module), a module over a ring that is embedded in a vector space over a field ** Lattice graph, a graph that can be drawn within a repeating arrangement of points ** Lattice-based cryptography, encryp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
