In this paper, we analyze Gaussian processes using statistical mechanics. Although the input variable variable is originally multidimensional, we simplify our model by considering the input variable variable as one-dimensional for statistical mechanical analysis. Furthermore, we employ periodic boundary conditions as an additional modeling approach. By using periodic boundary conditions, we can diagonalize the covariance matrix. The diagonalized covariance matrix is then applied to Gaussian processes. This allows for a statistical-mechanical analysis of Gaussian processes using the derived diagonalized matrix. We indicate that the analytical solutions obtained in this method closely match the results from simulations.
