Understand
We discuss how the integrators used for the Hybrid Monte Carlo (HMC) algorithm not only approximately conserve some Hamiltonian $H$ but exactly conserve a nearby shadow Hamiltonian (\tilde H), and how the difference $\Delta H \equiv \tilde H - H $ may be expressed as an expansion in Poisson brackets.
- By measuring average values of these Poisson brackets over the equilibrium distribution $\propto e^{-H}$ generated by HMC we can find the optimal integrator parameters from a single simulation.
- We show that a good way of doing this in practice is to minimize the variance of $\Delta H$ rather than its magnitude, as has been previously suggested.
- Some details of how to compute Poisson brackets for gauge and fermion fields, and for nested and force gradient integrators are also presented.
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