which in turn induces a splitting of the Liouville operator associated with the non- Hamiltonian bracket in Eq. (1), The structure of Eq. (3) can be taken as the starting point for derivation of efficient time-reversible integration algorithms that also preserve the appropriate measure in phase space [19]. Measure-preserving algorithms can be derived upon introducing a splitting of the Hamiltonian When the phase space flow has a non-zero compressibility the statistical mechanics must be formulated in terms of a modified phase space measure [12–18] is the standard phase space volume element (volume form [35]) and the statistical weight w ( x ) is defined by It has been shown that, provided the condition is satisfied, then so that the volume element ω is invariant under each of the L$_{α}$ [19]. The condition (11) is satisfied for all the cases considered below, so that, exploiting the decomposition in Eq. (6), H = n$_{s}$ ∑ α =1 H$_{α}$ (5) L$_{α}$x$_{i}$ = { x$_{i}$, H$_{α}$ } = 2 N ∑ j =1 B$_{ij}$ ∂H$_{α}$ ∂x$_{j}$ . (6) κ = 2 N ∑ i,j =1 ∂ B$_{ij}$ ∂x$_{i}$ ∂H ∂x$_{j}$ (7) ω = e − w ( x $^{)}$ω (8) ω = dx 1 ∧ dx 2 ∧ . . . ∧ dx 2 N (9) dw dt = κ ( x ) . (10) ∂ ∂x$_{j}$ [ e − w ( x $^{)}$B$_{ij}$ ] = 0 , i = 1 , . . . 2 N (11) L$_{α}$ω = 0 for every α , (12) 5 where