Simplified Matrix Method for Statistical Mechanics of Linear Systems

A matrix method is useful for statistical‐mechanical studies of linear systems of interacting units. For a large class of linear systems, the matrix method presented here leads to matrices whose orders are substantially lower than those obtained by the conventionally used matrix method and thus greatly facilitates mathematical operations. The applications of both the conventional and the simplified methods are illustrated for a DNA molecule; when the molecule has n nucleotide pairs the reduction in the order of the matrix is from $2^{\text{n−1}}$ to $n$ . Next, for a more general type of systems, the order is shown to be reduced from $2^{\text{n−1}}$ to $\text{4n–6}$ . Finally, the general steps are outlined for applying the simplified method to any one‐dimensional system of interacting units.