Errors in Configurational Quantities for the Perturbed Harmonic Oscillator

5 Errors in Configurational Quantities for the Perturbed Harmonic Oscillator 

We can consider applying the analysis of Sect. 7.4 to a more complicated example, however in general for nonlinear systems we cannot adopt the same strategy by taking expectations of the update equations, as we do not end up with a closed system of linear equations to solve (the force term now involves terms of higher orders). However, if the nonlinear terms in the force are preceded by a small constant, we can truncate the equations to provide an approximation to the invariant behavior. We consider sampling canonically the Hamiltonian 

We consider estimating the value of q )h by truncating the equations governing the evolution of the averages in powers of s (to ensure closure of the resulting simultaneous equations). We examine the expected behavior of the average in the case of the four second-order schemes listed in Table 7.1, although we may conduct this analysis for any similar scheme. 

We And that truncating the update equations to 0(s ) is sufficient to explicitly resolve the leading order behavior of the error for these discretization schemes, and we give the differences between observed numerical averages and exact averages below  

It would seem that by ordering the A, B and O update terms favourably in an algorithm we are able to achieve significant sampling improvements on systems that are nearly harmonic , at no extra cost (in terms of force evaluations). 

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