Regression of a fluctuation

The first application of the effective Liouvillians [9,65] models the regression of a fluctuation. We consider one observable A and its flux A. It is convenient to introduce an orthonormal base in the Liouville space of the operators A and A  

AA and AA are fluctuations because the observable A is centered. The matrix representation ot the effective Liouvillian (see (B.7) in Appendix B) in the basis 1), 2) is 

A = and M.22 z) is the matrix element (2 M(z) 2). All the other matrix elements ot the memory operator are zero [63]. The appearance ot the tactor i corresponds to the Hermitian definition ot the Liouvillian which is not universal. Equation (98) is an exact z-dependent expression. This z-dependence can be eliminated by approximating M22(z) by a positive constant r (Markovian approximation). This approximation is justified since the microscopic correlation time is much shorter than the regression time ot the fluctuation. Then Eq. (98) transtorms into the z-independent effective Liouvillian 

The non-hermiticity ot L arises trom the dissipative term —iT. It we are interested in the temporal evolution ot the fluctuation, the dynamics can be projected into the one-dimensional (scalar) frequency-dependent Liouvillian 

The inverse Fourier-Laplace transformation ot the Green tunction in the case ot a weak coupling (A T) leads to 

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The two previous secfions were devoted to modeling quantum resonances by means of effective Hamiltonians. From the mathematical point of view we have used two principal tools projection operators that permit to focus on a few states of interest and analytic continuation that allows to uncover the complex energies. Because the time-dependent Schrodinger equation is formally equivalent to the Liouville equation, it is attractive to try to solve the Liouville equation using the same tools and thus establishing a link between the dynamics and the nonequilibrium thermodynamics. For that purpose we will briefly recall the definition of the correlation functions which are similar to the survival and transition amplitudes of quantum mechanics. Then two models of regression of a fluctuation and of a chemical kinetic equation including a transition state will be presented. 

Figure 1.16 Regression of a fluctuation. Temporal evolution of (A)(t). (a) weak coupling A = 1 (b) intermediate coupling A = 4 (c) strong coupling A = 16 (arbitrary units).

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