Time Evolution of a Harmonic Chain with Fixed Boundary Conditions

We investigate the time evolution of a coupled harmonic-oscillator chain under two boundary conditions: two ends fixed and one end fixed. The dynamics of the coupled chain and the steady variances of the coordinates are explicitly analyzed by the entire Hamiltonian using a diagonalization approach. Our result shows the desirable symmetry for the case with two ends fixed. In particular, a Langevin simulation technique is proposed to sample the harmonic chain across the entire equilibrium distribution.