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

When we discretize the equation 5.13 we introduce an error due to the finite integration time step . Following the idea of Ref. (111) we can evaluate this error analytically in the case of a simple harmonic oscillator. Consider the equation:

 (E.1)

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.

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Claudio Attaccalite 2005-11-07