Mapping to Relevant Variables and Reversible Dynamics

For every microstate z of the system, the instantaneous values of the relevant variables are defined by a set of phase space functions II(z). The functions II(z) cannot generally be identified with x they are rather connected with x through x = (II(z)), that is, as averages based on a suitable probability density Qx(z) at the microscopic phase space F. Thus, the coarse-grained energy (x) is obtained from the microscopic Hamiltonian H(z) by straightforward averaging. 

Equations (7.6) and (7.7) define the reversible part of GENERIC (7.5) in terms of a coarse-grained Poisson bracket [174,179]. The additional terms related to dissipation and increase in entropy have to be accounted for by the irreversible contribution to GENERIC (7.5) and are described in the following section. 

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