Time-path integration

To describe nonequilibrium phase transitions, there have been developed many methods such as the closed-time path integral by Schwinger and Keldysh (J. Schwinger et.al., 1961), the Hartree-Fock or mean field method (A. Ringwald, 1987), and the l/lV-expansion method (F. Cooper et.al., 1997 2000). In this talk, we shall employ the so-called Liouville-von Neumann (LvN) method to describe nonequilibrium phase transitions (S.P. Kim et.al., 2000 2002 2001 S.P. Kim et.al., 2003). The LvN method is a canonical method that first finds invariant operators for the quantum LvN equation and then solves exactly the... 

Feynman noted that the quantum mechanical centroid density, Pc(xc), can be defined for the path centroid variable which is the path integral over all paths having their centroids fixed at the point in space Xc. Specifically, the formal imaginary time path integral expression for the centroid density is given by... 

As seen from our discussion in Chapter 3, which dealt with onedimensional problems, in many relevant cases one actually does not need the knowledge of the behavior of the system in real time to find the rate constant. As a matter of fact, the rate constant is expressible solely in terms of the equilibrium partition function imaginary-time path integrals. This approximation is closely related to the key assumptions of TST, and it is not always valid, as mentioned in Section 2.3. The general real-time description of a particle coupled to a heat bath is the Feynman-Vernon... 

So far, one can be much more successful in calculating a rate constant when one knows in advance that it exists, than in answering the question of whether it exists. A considerable breakthrough in this area was the solution of the spin-boson problem, which, however, has only limited relevance to any problem in chemistry because it neglects the effects of intrawell dynamics (vibrational relaxation) and does not describe thermally activated transitions. A number of attempts have been made to go beyond the two-level system approximation, but the basic question of how vibrational relaxation affects the transition from coherent oscillations to exponential decay awaits a quantitative solution. Such a solution might be obtained by numerical computation of real-time path integrals for the density matrix using the influence functional technique. 

MONTE CARLO METHODS FOR REAL-TIME PATH INTEGRATION... 

MONTE CARLO METHODS FOR REAL-TIME PATH INTEGRATION which can be evaluated exactly. Thus we arrive at... 

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