Invariant Averages for Schemes Using the A, B and O Splitting

The analysis of schemes using the splitting strategy coded with the A, B and O pieces is particularly simple, as we use a new random number for each stochastic update meaning that the random numbers and state variables are not correlated between steps. Hence we have 

We can solve this linear system to find the invariant averages computed by the chosen discretization scheme for the harmonic oscillator with Hamiltonian (7.6). 

In the case of the one-dimensional harmonic oscillator, we have yielding the compact form of one iteration as 

Taking expectations of (7.12) and then taking the limit as n oo, we see immediately that 

Hence we can see that the [ABO scheme gives a first-order error in the variance of q. For significantly small friction (relative to h), the leading order term will be dominated by the term at order h , potentially giving us greater accuracy than we would expect from a first order scheme. 

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