The sampling of Eq. 3 can be achieved in multiple ways, such as in Monte Carlo, often referred to as the Single-Chain-In-Mean-Field method, or by molecular dynamics with various formulations. We adopt our recently developed Hamiltonian approach, which is the only implementation demonstrated to achieve energy-conserving and alias-free dynamics. In brief, this approach builds on standard particle-mesh operations by assigning particle number densities onto a regular grid via an assignment function P and subsequently performing a convolution with a filter function G that defines the density spread associated with particle species t as