Effective Liouvillian

This means that the Fokker-Planck equation can be obtained under the basic assumption alone that the time scale of the variable of interest v is much larger than that of its thermal bath, thereby recovering the well-known result of ref. 35. The AEP can thus shed light on the intimate relationship between an effective Liouvillian such as that of Eq. (2.20) and a rigorous one such as that of Eq. (2.7). 

This section is devoted to illustrating the AEP of Section III at work in the case of effective Liouville operators. As pointed out in Section IV, as well as in Chapter I, these effective Liouvillians are constructed for the purpose of simulating real systems. 

To follow the scale of complexity, the review is divided into three parts. The first two parts deal with the key concept of effective Hamiltonians which describe the dynamical and spectroscopic properties of interfering resonances (Section 2) and resonant scattering (Section 3). The third part. Section 4, is devoted to the resolution of the Liouville equation and to the introduction of the concept of effective Liouvillian which generalizes the concept of effective Hamiltonian. The link between the theory of quantum resonances and statistical physics and thermodynamics is thus established. Throughout this work we have tried to keep a balance between the theory and the examples based on simple solvable models. 

We anticipate that the results derived from wavefuncfions and effective Hamiltonians will be generalized in Section 4 to density matrices and effective Liouvillians if the time evolution starts out of a mixed equilibrium state. 

The complex variable z (Im z < 0) is homogenetic to a frequency. The resolvent l/(z — L) is the Fourier-Laplace transform of the evolution operator (see Appendix A). Expression (93) shows that the dynamics is reduced to the determination of the matrix element of the resolvent between two observables. Therefore only a reduced dynamics has to be investigated. For that purpose we shall define more precisely the observables and the operators of interest. The theory is formulated in the framework of the Liouville space of the operators and based on hierarchies of effective Liouvillians which are especially convenient to study reduced dynamics at various macroscopic and microscopic timescales (see Appendix B). 

Figure 1.15 Matrix representation of the Liouvillian. (a) In x In matrix representation of the Liouvillian in the basis of n observables and of their fluxes, (b) n x n matrix representation of the z-dependent effective Liouvillian derived from (a) in the basis of the n observables.

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 ... 

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 roots of irreversibility come from the couplings and Ag between the chemical species and the transition state characterized by its lifetime t i. The dynamics can be obtained exactly by diagonalizing the matrix (108). However, to obtain concentration, which are the observable quantities, it is useful to project the dynamics on the space of fhe operafors representing the chemical concentrations [A] and [B]. One finds fhe exacf 2-dependent two-dimensional effective Liouvillian... 

Assuming a weak coupling between the chemical species and the transition state which plays the role of an effective continuum, we get the frequency-independent effective Liouvillian... 

As a result of the partitioning accomplished above, an effective Liouvillian can be written,... 

It is a commonly encountered situation to find that the solvent molecules in a neighborhood of the solute are strongly perturbed by it. The subsystem of interest is therefore defined so as to include sufficient solvent molecules into the simulation. In this way, the correlations between this redefined subsystem and the reminder may be weak enough to neglect them. The effective Liouvillian describes this latter type of subsystem of interest. 

The moment problem has been almost exclusively studied in the literature having (implicitly) in mind Hermitian operators (classical moment problem). With the progress of the modem projective methods of statistical mechanics and the description of relaxation phenomena via effective non-Hermitian Hamiltonians or Liouvillians, it is important to consider the moment problem also in its generalized form. In this section we consider some specific aspects of the classical moment problem, and in Section V.C we focus on peculiar aspects of the relaxation moment problem. 

We note here that in the case of a linear microscopic interaction, P jiRa) does not depend on R, and this potential simply reduces to a linear one. The effective potential, in this case, turns out to be evirtual potential of the well-known linear itinerant oscillator (see Section II). In the more general case, the virtual potential is given by Eq. (4.9), and the Liouvillian reads... 

When r is a rigorous Liouvillian, time inversion symmetry implies that the odd s parameters vanish. When n is even, the right-hand side of Eq. (1.10) can then be identified with the momoits defined by Eq. (1.1). Note that in this case Eqs. (1.S), (1.7), and (1.10) allow us to recover Eq. (1.2). When r is an effective operator of the type provided by RMT, even the odd moments s can be different from zero. 

A more reahstic and more general treatment would presumably lead to a set of equations like Eqs. (52), with the potential V(x) fluctuating as a consequence of couplings with nonreactive modes (see Section III). For the sake of simplicity, we study separately the two different aspects. While Section III was devoted to pointing out the role of multiphcative fluctuations (derived from nonlinear microscopic Liouvillians) in the presence of additive white noise, this subsection is focused on the effects of a non-Markovian fluctuation-dissipation process (with a time convolution term provided by a rigorous derivation from a hypothetical microscopic Liouvillian) in the presence of a time-independent external potential. 

We have presented a unified formulation of dissipative dynamics based on the quantum theory of resonances. The reversible and dissipative confri-bufions fo fhe dynamics are gafhered in small-dimensional non-Hermifian effective Hamiltonians and effecfive Liouvillians wifh well-defined fheoref-ical sfafus. The formulation fends fo fill the gap between the dynamics and the thermodynamics. It has many advantages ... 

Although it is quite possible to discuss other transformations such as rotations,these would take us too far afield here. To proceed it is important to observe the effects of these transformations on various quantities that come into the evaluation of time correlation function in Eq. (12). For example, the transformation might have an effect on the volume element d F. This would be given by the Jacobian J of the transformation from F to F. In addition, the Hamiltonian, Liouvillian, and equilibrium distribution function might change under the various transformations. In Table 1 we summarize how these quantities are transformed. The primes on the headings of the columns indicate the values of the transformed quantities, whereas the unprimed quantities in the body of the table indicate the untransformed values. 

Note that the term open system refers here to exchange of energy and phase with the environment, as the number of particles is conserved throughout. The reduced density matrix p t) evolves coherently under the influence of the nuclear Hamiltonian, Hnuc, and the non-adiahatic effects enter the equation via the dissipative Liouvillian superoperator jSfn- The latter is also termed memory kernel , as it contains information about the entire history of the environmental evolution and its interaction with the system. The definition of the memory kernel is by no means unique nor straightforward. One possible solution is to start from the microscopic Hamiltonian of the total system, eqn (1). Using the projector formalism, it is possible to separate the evolution of the system, i.e., the... 

For metallic environments, non-adiabatic effects in the form of electron-hole pair coupling is known to dominate the dissipative dy-namics. The bath of electron-hole pairs typically thermalizes within a few femtoseconds, which allows for a great simplification of the memoiy kernel. In view of the adsorbate dynamics, the bath thus remains in its equilibrium state and the memoiy kernel can be treated in the second-order Born-Markov approximation. For Markovian dynamics, the dissipative Liouvillian is often written in its diagonal Lindblad form " ... 

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