Quantum-classical Liouville Hamiltonian equation

We now show how the mean field equations can be derived as an approximation to the quantum-classical Liouville equation (8) [9]. The Hamiltonian... 

The basic issue is at a higher level of generality than that of the particular mechanical assumptions (Newtonian, quantum-theoretical, etc.) concerning the system. For simplicity of exposition, we deal with the classical model of N similar molecules in a closed vessel "K, intermolecular forces being conservative, and container forces having a force-function usually involving the time. Such a system is Hamiltonian, and we assume that the potentials are such that its Hamiltonian function is bounded below. The statistics of the system are conveyed by a probability density function 3F defined over the phase space QN of our Hamiltonian system. Its time evolution is completely determined by Liouville s equation... 

Section III is devoted to illustrating the first theoretical tool under discussion in this review, the GME derived from the Liouville equation, classical or quantum, through the contraction over the irrelevant degrees of freedom. In Section III.A we illustrate Zwanzig s projection method. Then, in Section III.B, we show how to use this method to derive a GME from Anderson s tight binding Hamiltonian The second-order approximation yields the Pauli master equation. This proves that the adoption of GME derived from a Hamiltonian picture requires, in principle, an infinite-order treatment. The case of a vanishing diffusion coefficient must be considered as a case of anomalous diffusion, and the second-order treatment is compatible only with the condition of ordinary... 

This contribution deals with the description of molecular systems electronically excited by light or by collisions, in terms of the statistical density operator. The advantage of using the density operator instead of the more usual wavefunction is that with the former it is possible to develop a consistent treatment of a many-atom system in contact with a medium (or bath), and of its dissipative dynamics. A fully classical calculation is usually suitable for a many-atom system in its ground electronic state, but is not acceptable when the system gets electronically excited, so that a quantum treatment must then be introduced initially. The quantum mechanical density operator (DOp) satisfies the Liouville-von Neumann (L-vN) equation [1-3], which involves the Hamiltonian operator of the whole system. When the system of interest, or object, is only part of the whole, the treatment can be based on the reduced density operator (RDOp) of the object, which satisfies a modified L-vN equation including dissipative rates [4-7]. 

To date, there has only been one attempt to develop a dynamic density functional theory for systems in which inertia plays a role [8]. However, it has been shown that the formal proof for the existence of a quantum mechanical dynamical density functional theory by Runge and Gross can be applied to classical systems [9] by starting from the Liouville equation for Hamiltonian systems (instead of the time-dependent Schrodinger equation), which therefore includes inertia terms. However, the proof is not of practical use (see below). 

Both for classical and quantum physics, elimination of interaction means existence of a unitary operator, which we call f/. We have shown that this operator U may be expressed in terms of the kinetic operators that I quoted above. Therefore our dynamics of correlation is equivalent to the various methods to obtain diagonalization of the Hamiltonian. In this case we also have the diagonalization of the Liouville operator for the density matrix. In short, Poincare integrability is equivalent to the integrability of the Liouville equation. But there is a much more interesting case, as there exists a class of non-integrable... 

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