Postulating the smoothness of the phase space distribution function, as done by Tuckerman et al. [Phys. Rev. Lett. 78 (1997) 2042], we show that the phase space density is conserved when a proper smooth geometrical factor (the Jacobian) can be defined and is correctly taken into account. As a result, many thermodynamic concepts can be carried over to characterize the non-equilibrium steady state.
