IV. BULGAC-KUSNEZOV-NOS ´ E-HOOVER DYNAMICS The BKNH Hamiltonian is simply the BK Hamiltonian augmented by the Nos´e variables ( η, p$_{η}$ ) with mass m$_{η}$ . With the antisymmetric BKNH tensor 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$_{8}$ ): 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. 12 H BKNH = H ( q, p ) + K$_{1}$ ( p$_{ζ}$ ) m$_{ζ}$ + K$_{2}$ ( p$_{ξ}$ ) m$_{ξ}$ + p 2 η 2 m$_{η}$ + k$_{B}$T ( ζ + ξ ) + 2 k$_{B}$T η (28) B BKNH =       0 0 0 0 1 0 − G$_{2}$ 0 0 0 0 0 0 ∂G$_{1}$ ∂p 0 0 0 0 0 0 0 0 ∂G$_{2}$ ∂q 0 0 0 0 0 0 0 0 1 − 1 0 0 0 0 − G$_{1}$ 0 0 0 − ∂G$_{1}$ ∂p 0 0 G$_{1}$ 0 0 − p$_{ζ}$ G$_{2}$ 0 − ∂G$_{2}$ ∂q 0 0 0 0 − p$_{ξ}$ 0 0 0 − 1 0 p$_{ζ}$ p$_{ξ}$ 0       . (29) ˙ q = ∂H ∂p − G$_{2}$ ( q, p ) m$_{ξ}$ ∂K$_{2}$ ∂p$_{ξ}$ (30a) ˙ ζ = 1 m$_{ζ}$ ∂G$_{1}$ ∂p ∂K$_{1}$ ∂p$_{ζ}$ (30b) ˙ ξ = 1 m$_{ξ}$ ∂G$_{2}$ ∂q ∂K$_{2}$ ∂p$_{ξ}$ (30c) ˙ η = p$_{η}$ m$_{η}$ (30d) ˙ p = − ∂H ∂q − G$_{1}$ ( q, p ) m$_{ζ}$ ∂K$_{1}$ ∂p$_{ζ}$ (30e) ˙ p$_{ζ}$ = G$_{1}$ ∂H ∂p − k$_{B}$T ∂G$_{1}$ ∂p − p$_{ζ}$ p$_{η}$ m$_{η}$ (30f) ˙ p$_{ξ}$ = G$_{2}$ ∂H ∂q − k$_{B}$T ∂G$_{2}$ ∂q − p$_{ξ}$ p$_{η}$ m$_{η}$ (30g) ˙ p$_{η}$ = p$_{ζ}$ m$_{ζ}$ ∂K$_{1}$ ∂p$_{ζ}$ + p$_{ξ}$ m$_{ξ}$ ∂K$_{2}$ ∂p$_{ξ}$ − 2 k$_{B}$T . (30h)