Molecular Dynamics (MD) simulation of a chain of simple harmonic oscillators - AGAIN
This assignment again uses the MD program you wrote to simulate a chain of particles connected by harmonic interactions.
[The particle at x0 = 0 is fixed.]
1. You know that the average energy of a single simple harmonic oscillator at temperature T is given by
U = = kBT,
so that the specific heat is
c = kB.
Demonstrate analytically that the energy per particle and the specific heat per particle are the same for a harmonic chain and a single SHO without using a Fourier transform of the Hamiltonian.
2. Modify your program to calculate the expected kinetic energy per particle from the initial (randomly chosen) energy. Then use the average kinetic energy to predict an effective temperature.
3. Modify your program to find the contributions to the specific heat from the fluctuations of the potential energy under the assumption that the kinetic degrees of freedom act as a thermal reservoir at the predicted effective temperature. [The contributions to the specific heat from the momenta are kB/2 per particle.]
4. Calculate the effective temperature and the specific heat for various lengths of the harmonic chain. To what extent are the relationships between the energy and the effective temperature and the fluctuations and the specific heat confirmed by your data?