V. BULGAC-KUSNEZOV-NOS ´ E-HOOVER CHAIN For simplicity, we explicitly treat only the case in which the p$_{ζ}$ and p$_{ξ}$ demons are each coupled to a standard NH thermostat (length one). It would be straightforward to couple each of the demons to NH chains [32], and the general case can be easily inferred from what follows. Define the Hamiltonian Upon defining the phase space point x = ( q, ζ, ξ, η, χ, p, p$_{ζ}$, p$_{ξ}$, p$_{η}$, p$_{χ}$ ) = ( x$_{1}$, x$_{2}$, x$_{3}$, x$_{4}$, x$_{5}$, x$_{6}$, x$_{7}$, x$_{8}$, x$_{9}$, x$_{10}$ ) and the antisymmetric BKNHC tensor associated non-Hamiltonian equations of motion are with i = 1 , . . . , 10. 18 H BKNHC = H ( q, p ) + K$_{1}$ ( p$_{ζ}$ ) m$_{ζ}$ + K$_{2}$ ( p$_{ξ}$ ) m$_{ξ}$ + p 2 η 2 m$_{η}$ + p 2 χ 2 m$_{χ}$ + k$_{B}$T ( ζ + ξ + η + χ ) . (44) B BKNHC =        0 0 0 0 0 1 0 − G$_{2}$ 0 0 0 0 0 0 0 0 ∂G$_{1}$ ∂p 0 0 0 0 0 0 0 0 0 0 ∂G$_{2}$ ∂q 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 − 1 0 0 0 0 0 − G$_{1}$ 0 0 0 0 − ∂G$_{1}$ ∂p 0 0 0 G$_{1}$ 0 0 − p$_{ζ}$ 0 G$_{2}$ 0 − ∂G$_{2}$ ∂q 0 0 0 0 0 0 − p$_{ξ}$ 0 0 0 − 1 0 0 p$_{ζ}$ 0 0 0 0 0 0 0 − 1 0 0 p$_{ξ}$ 0 0        , (45) ˙ x$_{i}$ = B BKNHC ij ∂H BKNHC ∂x$_{j}$ (46)