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## IV. BULGAC-KUSNEZOV-NOS ´ E-HOOVER DYNAMICS 

The BKNH Hamiltonian 

<img src='content_image/330749.jpg'>

is simply the BK Hamiltonian augmented by the Nos´e variables ( η, p$_{η}$ ) with mass m$_{η}$ . With the antisymmetric BKNH tensor 

<img src='content_image/330750.jpg'>

we obtain from Eq. (3) equations of motion for the phase space variables x = ( q, ζ, ξ, η, p, p$_{ζ}$, p$_{ξ}$, p$_{η}$ ) = ( x$_{1}$, x$_{2}$, x$_{3}$, x$_{4}$, x$_{5}$, x$_{6}$, x$_{7}$, x$_{8}$ ): 

<img src='content_image/330751.jpg'>

<img src='content_image/330752.jpg'>

<img src='content_image/330753.jpg'>

<img src='content_image/330754.jpg'>

<img src='content_image/330755.jpg'>

<img src='content_image/330756.jpg'>

<img src='content_image/330757.jpg'>

<img src='content_image/330758.jpg'>

Here, a single Nos´e variable is coupled to both of the BK demons ζ and ξ . The associated invariant measure is discussed in Appendix C. 

