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## III. PHASE SPACE FORMULATION OF THE BK THERMOSTAT 

A phase space formulation of the BK thermostat can be achieved upon introducing the Hamiltonian 

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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. 

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Substituting B BK and H BK into Eq. (3), we obtain the energy-conserving equations 

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The associated invariant measure for the BK flow is discussed in Appendix B. 

