III. PHASE SPACE FORMULATION OF THE BK THERMOSTAT A phase space formulation of the BK thermostat can be achieved upon introducing the Hamiltonian where ( q, p ) are the physical degrees of freedom (coordinates and momenta), with mass m , to be simulated at constant temperature T , while ζ and ξ are the BK ‘demons’, with corresponding inertial parameters m$_{ζ}$ and m$_{ξ}$ , and associated momenta ( p$_{ζ}$, p$_{ξ}$ ) [22–24]. K$_{1}$ and K$_{2}$ provide the kinetic energy of demon variables, and for the moment are left arbitrary. Upon defining the phase space point as x = ( q, ζ, ξ, p, p$_{ζ}$, p$_{ξ}$ ) = ( x$_{1}$, x$_{2}$, x$_{3}$, x$_{4}$, x$_{5}$, x$_{6}$ ), one can introduce an antisymmetric BK tensor field as where G$_{1}$ and G$_{2}$ are functions of system variables ( p, q ) only. Substituting B BK and H BK into Eq. (3), we obtain the energy-conserving equations The associated invariant measure for the BK flow is discussed in Appendix B. 7 H BK = p 2 2 m + V ( q ) + K$_{1}$ ( p$_{ζ}$ ) m$_{ζ}$ + K$_{2}$ ( p$_{ξ}$ ) m$_{ξ}$ + k$_{B}$T ( ζ + ξ ) (14a) = H ( q, p ) + K$_{1}$ ( p$_{ζ}$ ) m$_{ζ}$ + K$_{2}$ ( p$_{ξ}$ ) m$_{ξ}$ + k$_{B}$T ( ζ + ξ ) , (14b) B BK =      0 0 0 1 0 − G$_{2}$ 0 0 0 0 ∂G$_{1}$ ∂p 0 0 0 0 0 0 ∂G$_{2}$ ∂q − 1 0 0 0 − G$_{1}$ 0 0 − ∂G$_{1}$ ∂p 0 G$_{1}$ 0 0 G$_{2}$ 0 − ∂G$_{2}$ ∂q 0 0 0      (15) ˙ q = ∂H ∂p − G$_{2}$ ( q, p ) m$_{ξ}$ ∂K$_{2}$ ∂p$_{ξ}$ (16a) ˙ ζ = 1 m$_{ζ}$ ∂G$_{1}$ ∂p ∂K$_{1}$ ∂p$_{ζ}$ (16b) ˙ ξ = 1 m$_{ξ}$ ∂G$_{2}$ ∂q ∂K$_{2}$ ∂p$_{ξ}$ (16c) ˙ p = − ∂H ∂q − G$_{1}$ ( q, p ) m$_{ζ}$ ∂K$_{1}$ ∂p$_{ζ}$ (16d) ˙ p$_{ζ}$ = G$_{1}$ ∂H ∂p − k$_{B}$T ∂G$_{1}$ ∂p (16e) ˙ p$_{ξ}$ = G$_{2}$ ∂H ∂q − k$_{B}$T ∂G$_{2}$ ∂q . (16f)