Configurational Entropy for a System of Hard Particles on Lattices.
Configurational entropy for a system of hard particles on various lattices is investigated. A formalism is described to calculate the entropy per site for two-dimensional square, triangular, union-jack and hexagonal lattices by using transfer matrix method. The transfer matrix for a two -dimensional lattice having m rows is obtained from the allowed states of a system of hard particles on a one-dimensional periodic lattice having m sites. The approach to the thermodynamic limit is realized by increasing the number of rows, which, in turn increases the dimension of the transfer matrix and thus limits our ability to process two-dimensional lattices. Following Bighouse, we then describe the diagonalization of cyclic square and rectangular matrices which enables one to reduce the transfer matrix into several smaller submatrices. The largest submatrix, which is shown to contain the largest eigenvalue of the original transfer matrix, is used to obtain the entropy per site. To achieve partial diagonalization, a systematic procedure is developed to classify the one-dimensional states into pods, each represented by a representative state having a certain period, from which all the members of a pod can be constructed. A method is described to compute directly the elements of the largest submatrix from the representative states and their periods. This procedure is extended to a three-dimensional simple cubic lattice where the transfer matrix is constructed from the allowed states of a corresponding two-dimensional square lattice. To achieve thermodynamic limit, in this case, the number of both rows and columns of the two-dimensional lattice are increased. Consequently, the dimension the transfer matrix increases, but the transfer matrix can be reduced into several submatrices by a double partial diagonalization. The double partial diagonalization is possible because the allowed two-dimensional states can be classified into pods, represented by representative states having row and column periodicities. A method is described to compute directly the largest submatrix, which contains the largest eigenvalue, for a simple cubic lattice. The entropy per site is obtained from the largest submatrix. The numerical results for the entropy per site, reported by Metcalf and Yang in 1978, for different lattices are improved considerably. The entroy per site for a triangular lattice agrees with the exact result of Baxter up to one part in a billion.
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- Physics: General