content_image/357614.jpg content_image/357615.jpg content_image/357616.jpg content_image/357617.jpg content_image/357618.jpg content_image/357619.jpg content_image/357620.jpg content_image/357621.jpg content_image/357622.jpg content_image/357623.jpg content_image/357624.jpg content_image/357625.jpg
## A. Algorithm for BKNHC chain dynamics 

Splitting the BKNHC chain Hamiltonian as 

<img src='content_image/357614.jpg'>

<img src='content_image/357615.jpg'>

<img src='content_image/357616.jpg'>

<img src='content_image/357617.jpg'>

<img src='content_image/357618.jpg'>

<img src='content_image/357619.jpg'>

<img src='content_image/357620.jpg'>

<img src='content_image/357621.jpg'>

<img src='content_image/357622.jpg'>

<img src='content_image/357623.jpg'>

we obtain the corresponding measure-preserving splitting of the Liouville operator 

<img src='content_image/357624.jpg'>

At this stage we go directly to Eqs (20). The antisymmetric Nos´e-Hoover-Bulgac- Kusnezov tensor becomes 

<img src='content_image/357625.jpg'>

