Applications of tridiagonal matrices in non-equilibrium statistical physics

Abstract

Abstract. One dimensional stochastic problems on a finite lattice that model the time dependence of epidemics, particle deposition and voter influence can easily be cast into a simple form dV/dt = MV, where V is a vector with components representing the average occupation of the i-th cell and M is a matrix with coefficients drawn from the equations that give rates of evolution of a particular cell’s occupation due to its dependence upon other cells. These matrices are often in tridiagonal form (the only non-zero elements are along the main diagonal and the two diagonal rows to its right), or can be transformed via a unitary transformation into this form. In the tridiagonal form, eigenvalues and eigenvectors can be extracted via straightforward techniques, and the inverse of the matrix of eigenvectors can be inverted (in arbitrary finite dimension) so as to enforce the system’s initial conditions. Examples of such models are discussed and related to matrix theory. Key words. Tridiagonal matrices, Determinants, triangular matrices. AMS subject classifications. 15A15, 15F10. 1. Introduction. The