Leggett

FI43S7 Variation of residual stresses in aged components Dr R.H. Leggett The Welding Instrtute... 

Leggett, D. L. Instrumental Simplex Optimization, /. Chem. Educ. 1983, 60, 707-710. 

If all the PES coordinates are split off in this way, the original multidimensional problem reduces to that of one-dimensional tunneling in the effective barrier (1.10) of a particle which is coupled to the heat bath. This problem is known as the dissipative tunneling problem, which has been intensively studied for the past 15 years, primarily in connection with tunneling phenomena in solid state physics [Caldeira and Leggett 1983]. Interaction with the heat bath leads to the friction force that acts on the particle moving in the one-dimensional potential (1.10), and, as a consequence, a> is replaced by the Kramers frequency [Kramers 1940] defined by... 

As first shown by Caldeira and Leggett [1981], in the tunneling regime friction reduces the transition probability by the factor... 

According to (2.29), dissipation reduces the spread of the harmonic oscillator making it smaller than the quantum uncertainty of the position of the undamped oscillator (de Broglie wavelength). Within exponential accuracy (2.27) agrees with the Caldeira-Leggett formula (2.26), and similar expressions may be obtained for more realistic potentials. 

The solution of the spin-boson problem with arbitrary coupling has been discussed in detail by Leggett et al. [1987]. The displacement of the equilibrium positions of the bath oscillators in the transition results in the effective renormalization of the tunneling matrix element by the bath overlap integral... 

We just quote the main results of [Leggett et al. 1987], which cover most of the possible situations. The spectral density is taken in the form... 

Nevertheless, Leggett et al. [1987] have argued, with some provisos, that [with the initial condition (O) = 1] and C(t) may be practically taken the same. If C(t) then obeys the damped oscillator equation (2.41), then the inelastic part of the structure factor has the Lorentzian form with the peaks at [Pg.25]

As long as the system can be described by the rate constant - this rules out the localization as well as the coherent tunneling case - it can with a reasonable accuracy be considered in the imaginary-time framework. For this reason we rely on the Im F approach in the main part of this section. In a separate subsection the TLS real-time dynamics is analyzed, however on a simpler but less rigorous basis of the Heisenberg equations of motion. A systematic and exhaustive discussion of this problem may be found in the review [Leggett et al. 1987]. 

That is, Ci 2(t + P) = Ci,2Wexp( j3e). In fact, when the temperature is less than the separation between the two lowest energy levels of the bath, the latter can be approximated by a set of two-level systems [Caldeira and Leggett 1983 Mermin 1991],... 

Two most appealing features of this model draw so much attention to it. First, although microscopically one has very little information about the parameters entering into (5.24), it is known [Caldeira and Leggett 1983] that when the bath responds linearly to the particle motion, the operators q and p satisfying (5.24) can always be constructed, and the only quantity entering into the various observables obtained from the model (5.24) is the spectral density... 

Substituting (5.34) and (5.35) for (5.8) and dropping in Z the constant partition function of unperturbed harmonic oscillator we get the nonlocal effective action derived by Feynman (see also Caldeira and Leggett [1983]),... 

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... 

Leggett et al. [1987] have set forth a rigorous scheme that reduces a symmetric (or nearly symmetric) double well, coupled linearly to phonons, to the spin-boson problem, if the temperature is low enough. However, in the case of nonlinear coupling (which is necessary to introduce in order to describe the promoting vibrations), no such scheme is known, and the use of the spin-boson Hamiltonian together with (3.67) relies rather on intuition, and is not always Justifiable. 

Gay, D. M., and D. J. Leggett (1993). Enhancing Thermal Hazard Analysis Awareness With Compatibility Charts. Journal of Testing and Evaluation 21, 6, 477-80. 

The editor also acknowledges with pleasure the encouragement and assistance he has received from Mr L. W. Derry (Head of Department of Metallurgy) and Dr D. M. A. Leggett (Principal) of the Battersea College of Technology, and from Dr A. M. Ward (Principal) of Sir John Cass College. 

Murrman, T.F. Jenkins D.C. Leggett, Composition and Mass Spectra of Impurities in Military Grade TNT Vapor , Special Rept 158,... 

Leggetter S, Tildesley DJ (1989) Molec Phys 68 519 Guarnieri F, Still WC (1994) J Comput Chem 15 1302... 

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