Error Analysis due to finite time step in the GLE integration in a simple case

We are interested in a statistically stationary process and so we proceed to evaluate mean average energies and correlation functions as function of . To do so we multiply the equation E.1 for and , respectively, and then take the average. We obtain the following pair of equations:

0 | |||

0 |

Becuase we are interested in the equilibrium distribution, we can assume that and . Thus we have three unknown quantities , and . To get a third relation among this quantities we square the Eq. E.1 to obtain the relation:

After solving this equation system we can evaluate the potential and the kinetic energy:

(E.2) | |||

(E.3) | |||

(E.4) |

It is easy to show that in the limit of small the potential and the kinetic energies converge to:

(E.5) | |||

(E.6) |

This show that at least in this simple model the impulse integrator leads to a quadratic error in in both kinetic and poterntial energy.