  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.
