content_image/369670.jpg content_image/369671.jpg content_image/369675.jpg content_image/369674.jpg content_image/369673.jpg content_image/369672.jpg content_image/369676.jpg content_image/369677.jpg content_image/369679.jpg content_image/369680.jpg content_image/369681.jpg content_image/369678.jpg
## Appendix C: Invariant Measure of the BKNH phase space flows 

The phase space compressibility of the NH controlled Bulgac-Kusnezov thermostat is 

<img src='content_image/369670.jpg'>

Upon introducing the function 

we have 

so that the invariant measure in phase space is 

<img src='content_image/369671.jpg'>

<img src='content_image/369675.jpg'>

<img src='content_image/369674.jpg'>

<img src='content_image/369673.jpg'>

<img src='content_image/369672.jpg'>

## Appendix D: Invariant Measure of the BKNHC chain phase space flows 

The phase space compressibility of the Nos´e-Hoover-Bulgac-Kusnezov chain is 

<img src='content_image/369676.jpg'>

Upon introducing the function 

we have 

so that the invariant measure in phase space reads 

<img src='content_image/369677.jpg'>

<img src='content_image/369679.jpg'>

<img src='content_image/369680.jpg'>

<img src='content_image/369681.jpg'>

<img src='content_image/369678.jpg'>

