Hamiltonian equations canonical equilibrium

Thus, the fractional equilibrium state (99) can be considered as a consequence of anomalous transport of phase points in the phase space resulting in the anomalous continuity equation (104). Note that the usual form of the evolution (93) is a direct consequence of the canonical Hamiltonian form of the microscopic equations of motion. Thus, the evolution of (105) implies that the microscopic equations of motion are not canonical. The actual form of these equations has not yet been investigated. However, there are strong indications that dissipative effects on the microscopic level become important. 

While the Berendsen thermostat is efficient for achieving a target temperature within your system, the use of a thermostat that represents a canonical ensemble once the system has reached a thermal equilibrium. The extended system method, which was originally introduced by Nosd [1984a, b] and then further developed by Hoover (1985), introduces additional degrees of freedom into the Hamiltonian that describes the system, from which equations of motion can be determined. 

---