content_image/351942.jpg content_image/351943.jpg content_image/351944.jpg content_image/351946.jpg content_image/351945.jpg content_image/351948.jpg content_image/351949.jpg content_image/351947.jpg
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 

<img src='content_image/351942.jpg'>

which in turn induces a splitting of the Liouville operator associated with the non- Hamiltonian bracket in Eq. (1), 

When the phase space flow has a non-zero compressibility 

<img src='content_image/351943.jpg'>

<img src='content_image/351944.jpg'>

the statistical mechanics must be formulated in terms of a modified phase space measure [12–18] 

where 

<img src='content_image/351946.jpg'>

<img src='content_image/351945.jpg'>

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 

<img src='content_image/351948.jpg'>

<img src='content_image/351949.jpg'>

<img src='content_image/351947.jpg'>

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), 

