Lindblad form

Models for the dissipative dynamics can frequently be based on the assumption of fast decay of memory effects, due to the presence of many degrees of freedom in the s-region. This is the usual Markoff assumption of instantaneous dissipation. Two such models give the Lindblad form of dissipative rates, and rates from dissipative potentials. The Lindblad-type expression was originally derived using semigroup properties of time-evolution operators in dissipative systems. [45, 46] It has been rederived in a variety of ways and implemented in applications. [47, 48] It is given in our notation by... 

Wpla)]. The Lindblad form of the dissipation rate can be constructed from state-to-state transition rates r(ft <— a) in the p-region, induced by its interaction with the s-region, and the operator for the transition k = (/ <— a) is... 

Presently we do our calculations with the Lindblad form of the instantaneous dissipation, to describe instead the relaxation of the adsorbate back... 

We apply to this equation the same remarks as those adopted for the comparison among Eqs. (316)—(318). We note, first of all, that the structure of Eq. (325) is very attractive, because it implies a time convolution with a Lindblad form, thereby yielding the condition of positivity that many quantum GME violate. However, if we identify the memory kernel with the correlation function of the 1/2-spin operator ux, assumed to be identical to the dichotomous fluctuation E, studied in Section XIV, we get a reliable result only if this correlation function is exponential. In the non-Poisson case, this equation has the same weakness as the generalized diffusion equation (133). This structure is... 

In the theory of semi-groups, it is proved that the most general acceptable evolution equation has to be of the form (9), [Lindblad 1976 (b)], which is called the Lindblad form. 

Letting At go to zero and assuming all limits exist, we re-derive a Master Equation of the Lindblad form. The problem is that the assumption about the limiting process requires some additional assumptions. These are usually taken to imply that the system 2, in some way, acts as a huge reservoir. 

Here not all terms are of the Lindblad form, which indicates that there may be trouble emerging. The exact nature of this will be seen in the next section. Here we only want to point out the strangeness of the situation. 

Even if we found that the Master Equation is not of the Lindblad form, we may not actually encounter the trouble. In this section we will show that the trouble cannot be safely ignored. 

However, the solution of the system does also apply in the strong driving limit /teir < 2 /2 a. As the original equation of motion is of the Lindblad form, no unphysical features should emerge. In the case of no damping k —> 0, we obtain the simple result... 

We have found that even in these simple cases things can go wrong. A physically reasonable Master Equation may not be of the Lindblad form. The corresponding Lindblad form may, on the other hand, violate simple rules like the fluctuation-dissipation theorem. 

Another problem is that memory kernels seem to be delicate entities. Erroneous kernels can destroy the physical sense of the time evolution of an initially acceptable density matrix. We do not have a general criterion to help us judge from the Master Equation with memory if the evolution is acceptable. In the Markovian limit, we know that the Lindblad form is certain to preserve the physical interpretation. It is a challenge for the theory of irreversibility in quantum systems to find such a criterion when memory effects are important. 

We have also outlined a treatment of systems showing dissipative dynamies. These ean be treated with a natural extension of the formulation shown here. The density operator equation differs in a signifieant way when dissipation is present, because it eontains terms whieh are not derived from a Hamiltonian, and must be written as rates involving superoperators. We have shown here one such situation, with the Lindblad form of the dissipation rate, and other forms can be treated equally well with the present eombination of a PWT and a quasielassieal approximation in phase spaee. 

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 " ... 

Adventitious surfactants also have a marked effect on the mechanism of coalescence. In studying the coalescence of curved water surfaces, Lindblad (L8) used aged distilled water that was stored for about 30 h in a polyethylene bottle opened to the air through a narrow polyethylene tube inserted in the water. He found that if fresh distilled water (water exposed not longer than 1 h to the air) was used, the delay time in coalescence was approximately half as long. Consequently, he concluded that this difference is due to some form of contamination which settled into the water or onto the water surface. 

The operators can be constructed as combinations of position and momentum operators in the p-region, [45] or from empirical transition rates ka <-a between orthonormal eigenstates and 4>, of Hp. [47] The index L then refers to a given transition a —> a, and the corresponding operator is C(pL) = This simplifies the form of the Lindblad dissipa-... 

Fast dissipation is treated numerically within the Markoff approximation, which leads to differential equations in time, and dissipative rates most commonly written in the Redfield [9,10] or Lindblad [11,12] forms. Several numerical procedures have been introduced for dissipative dynamics within the Markoff approximation. The differential equations have been solved using a pseudospectral method [13], expansions of the Liouville propagator in terms of polynomials, [14-16] and continued fractions. [17]... 

Clearly, type-I migration presents a problem for models of planet formation, both in terms of accreting fully formed planets before they migrate into the Sun and in terms of their survival once fully formed. However, it is likely that a sufficiently massive planet would have cleared a gap in the disk gas. Once this gap extended beyond the Lindblad resonances, type-I migration ceased. At present, there is considerable uncertainty about how massive a planet must be to clear a gap in the disk. This depends sensitively on the way in which waves damped in the nebula, and on the disk viscosity, both of which are poorly constrained. A recent estimate is that a body with a mass of 2-3M would have cleared a gap at 1 AU, while at 5 AU, a body 15M would do the job (Rafikov, 2002). 

The problem is, however, that we need to fix the exact conditions of validity of this approximation, this was attempted already in Ref. [Fano 1954], In particular, it has turned out that introduction of the memory effect is a very sensitive issue [Barnett 2001], Highly reasonable but unprecise approximations may lead to non-physical time evolution. An additional problem is that the procedure does not necessarily lead to Master Equations of the Lindblad type, see above. If this is not its form, we may find well known complications, which have to be avoided if we want to escape unphysical results. 

In the subspace spanned by the preferred coordinates (also denoted "pointer basis"), we assume the relevant density matrix to obey a Lindblad-type equation of the form [Lindblad 1976 (b) Barnett 2001]... 

This form was actually the original Lindblad result. Note that this works as long as does not depend on time. For time-dependent generators of the evolution, a somewhat more elaborate scheme is needed. 

The main shortcoming of the Redfield time evolution is that it does not necessarily conserve the positivity property. In fact, it has been shown by Lindblad that a linear Markovian time evolution that satisfies this condition has to be of the form... 

The final form of the Redfield equation [Eq. (20)] is superficially similar to the equation of motion that arises in the axiomatic semigroup theory of Lindblad, Gorini et al. [48,54-57]. They showed that the most general equation of motion that preserves the positivity of the density matrix must have the general form... 

Each dissipation channel k is described by a so-called Lindblad operator, C/c, which is used to represent the energy exchange via inelastic collision with the electrons of the metal. In this particular case, the operators take the form of a transition between two vibrational states of... 

This form of the Lindblad operators suggests that, provided their computation remains numerically tractable, the eigenstates of the vibrational Hamiltonian would provide the most efficient basis to represent the evolution of the reduced density matrix, eqn (16). Indeed, in such a basis, the equations of motion for the matrix elements take the particularly simple and elegant form... 

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