A time-irreversible generalized master equation

L. A. Lugiato

Gang Ren, Jian-ming Du

Arturo López

The soluble model of an oscillator coupled to a scalar field is used as an example of irreversible behavior. By studying the reduced density operator for the oscillator, a generalized Fokker-Planck-Kramers-Chandrasekhar equation for the Wigner distribution function is derived. The diagonal matrix elements of the reduced density operator satisfy a generalized non-Markovian master equation. By application of a time-averaged method, the ordinary Pauli master equation is derived. The time evolution of the occupation probabilities of the oscillator levels has been numerically computed, and compared with the solutions of the Pauli equation.

A. Fuliński

V. P. Belavkin

A characterisation of the stochastic bounded generators of quantum irreversible Master equations is given. This suggests the general form of quantum stochastic evolution with respect to the Poisson (jumps), Wiener (diffusion) or general Quantum Noise. The corresponding irreversible Heisenberg evolution in terms of stochastic completely positive (CP) maps is found and the general form of the stochastic completely dissipative (CD) operator equation is discovered.

Boris Leaf

For a system described in a phase space of generalized coordinates w and momenta J, the generalized master equation gives the time evolution of the reduced-density distribution function ρ(t, J) for the momenta. A generalization of the generalized master equation, having a similar non-Markoffian form, is derived for the full distribution function ρ(t, w, J). This equation is an alternate form of the Liouville equation. The derivation is an extension of a previous derivation of the generalized master equation from the Liouville equation utilizing projection operators in a Hilbert space. The time-evolution equation for the reduced distribution function ρr(t, wr, J), depending on the subset wr of the set of coordinates w, is derived. The approach to a stationary state for t → ∞ is discussed.