Parametrization of open systems with effective quadratic Hamiltonians plus stochastic force

The evolution generated by general dissipative Hamiltonians is analyzed when a stochastic force is included. Using a mapping technique, the equations of motion for the observables of interest can be given easily. A general dissipativity condition is extracted, whose fulfilment guarantees that thermal equilibrium is reached as the final stage of the evolution. Several existing frictional Hamiltonians are examined and it is seen that the correlation of the fluctuating force is essential to the destruction of a constant of motion inherent to pure quantal behaviour.