Relative to the previously used Hamiltonian Monte Carlo sampler , we added an adaptive step size to sample 𝒙 from . The adaptive step size optimizes exploration in Hamiltonian dynamics by monitoring acceptance rates: small steps slow sampling, and large steps introduce inaccuracy. The algorithm adjusts step size to balance efficiency (high acceptance) and exploration (finding diverse states). The step size dynamically adjusts based on the acceptance rate over 50sample intervals. Rates below 90% trigger a step-size decrease for more careful exploration, while rates above 90% lead to an increase of the step size for faster exploration. This ensures efficient sampling within the desired constraints.