Path-integral techniques

This approximation has been originally derived and extensively explored in the path-integral techniques (see the review [Leggett et al. 1987]). Most of the results cited in section 2.3 can be obtained from (5.62) and (5.63). Equation (5.62) makes it obvious that only when the integrand/(r) falls off sufficiently fast, can the rate constant be defined, and it equals... 

Basically the perturbative techniques can be grouped into two classes time-local (TL) and time-nonlocal (TNL) techniques, based on the Nakajima-Zwanzig or the Hashitsume-Shibata-Takahashi identity, respectively. Within the TL methods the QME of the relevant system depends only on the actual state of the system, whereas within the TNL methods the QME also depends on the past evolution of the system. This chapter concentrates on the TL formalism but also shows comparisons between TL and TNL QMEs. An important way how to go beyond second-order in perturbation theory is the so-called hierarchical approach by Tanimura, Kubo, Shao, Yan and others [18-26], The hierarchical method originally developed by Tanimura and Kubo [18] (see also the review in Ref. [26]) is based on the path integral technique for treating a reduced system coupled to a thermal bath of harmonic oscillators. Most interestingly, Ishizaki and Tanimura [27] recently showed that for a quadratic potential the second-order TL approximation coincides with the exact result. Numerically a hint in this direction was already visible in simulations for individual and coupled damped harmonic oscillators [28]. 

In all of the preceding contexts, it might prove expedient to employ a different solution technique to the analysis of the situation. The path integral technique of Kardar and co-workers appear promising in this regard [39]. 

The path integral technique was first proposed by Feynmann (Feynmann Hibbs, 1965). The purpose of this technique was to deal with questions in quantum mechanics. It has been applied to the study of the statistical mechanics of polymer systems (Kreed, 1972 Doi Edwards, 1986) and liquid crystalline polymers as well (Jahnig, 1981 Warner et al, 1985 Wang Warner, 1986). The path integrals relate the configurations of a polymer chain to the paths of a particle when the particle is undergoing Brownian or diffusive motion. 

Finally, we note that quantum mechanical tunneling effects may also be incorporated through the use of path integral techniques [14]. These effects are believed to be important in electron transfer processes in condensed phases and proteins [15-16]. 

We have considered the stochastic dynamics of a particle interacting with its environment of two-level systems in the presence of an external potential field. The treatment is based on the canonical quantization procedure. This approach directly yields the dissipative term and the noise operators. It may be pertinent to mention that, although the calculation of dissipative effects is straightforward, the treatment of noise is not simple as far as the path integral techniques are concerned. 

Because the path integral techniques can account for quantum effect directly in the simulations, the methodology has been used mostly in studies of the behavior of quantum solutes, including tunneling, charge transfer between solutes, and hydrated electrons. Simulations of pure water ° investigated quantum corrections to effective potentials. The Feynman-Hibbs effective potential is a computationally simple method for estimating quantum effects and has been used to examine the differences in the properties of H2O and 02 . ... 

Using bosonization procedures and real-time path integral techniques, the quantum conductance of a rigid (i.e., no lattice distortion) ID quantum wire has been computed. Analytical theories predict that, in the presence of even a single defect, the conductance of a strictly ID quantum wire would vanish at 0 K. Bosonization of this system yields an equivalent spin-boson model with an infinite number of tight-binding states, which turns out to be the same model as for CTCs discussed in... 

Another class of methods have been developed to handle the problem of the solvated electron. These usually involve the treatment of the electron as a quantum particle embedded in a classical fluid using Feynman path-integral techniques. 

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