An interation of probability theory and statistical physics produces a new branch of mathematics --- interacting particle systems. A central topic in the field is studying the phase transitions. There are mainly two mathematical tools: the Peierls method and
the first non-trivial eigenvalue. The second approach leads us to reeximine the known mathematics on the spectral theory and conversely, produces some new results even in a very classical area. The talk begins with a short overview about these methods, and then introduce some recent results on the basic estimates of the principle eigenvalues with different boundaries for one-dimensional elliptic operators.