We extend the notion of dissipative particle solutions [5] to case Hamiltonian flow in space probability measures $ \mu \in \mathscr{P}( \mathbb {R}^d \times {R}^d) sense [3]. The is form \begin{equation*} H(\mu) = \int V(q,p) \mu(dqdp) + \frac{1}{2} W(q,p,q',p') \mu(dqdp)\mu(dq'dp'), \end{equation*} with at most quadratic growth, so that a conservative \dot q \nabla_p V W \mu, \quad p - \nabla...