Strange Lagrangian systems and statistical mechanics

We consider the canonical ensemble of $N$ particles admitting a strange Hamiltonian description. Each of the particles obeys a set of Newtonian equation of motion, which can also be described by the standard canonical Hamiltonian mechanics. However, the thermodynamics corresponding to the strange description and canonical description differ drastically from each other. In other words, the strange description and the standard canonical description are inequivalent on the level of thermodynamics.