Localization exponential

Remark 2.5. Tikhonov s theorem holds only for bounded time intervals. Under the additional assumption that the slow system (2.1 f) is also locally exponentially stable, a similar result exists for infinite time intervals (Khalil 2002). 

If the system displays local exponential instability, then... 

Thus, the composition of such a chemical system may be obtained, at time t, by solving the set of non-linear differential equations (76). The solution is generally obtained by numerical integration, which must be highly stable ones, as the problems encountered are locally exponential , giving rise to the phenomenon called. .stiffness . Warner gives a critical review of the different numerical methods... 

Baker, R.S., Larsen, E.W., "A Local Exponential Transform Method for Global Variance Reduction in Monte Carlo Transport Problem", Karlsruhe, 1993, Vol.2. p.725. 

Definition 1.17 Standard assumptions on w. Unless specified otherwise, if uj is random it has to be meant IID and locally exponentially integrable. Moreover, without loss of generality, u>i and o>i are centered and E[a j] =... 

Fig. 6.1 When cji is only locally exponentially integrable and the left-tail of its distributions does not decay fast enough, so that the interval Dm is bounded from the left and —2mXo 9Dm for some Ao > 0, it may happen that lim / j(l/2mA) logM(—2mA) is finite, while (l/2mAo) log M(—2mAo) = oo. In this case, and only in this case, /i( )(Ao) (l/2mAo) logM(—2mAo).

The key to this method is thus to act with each operator (exponential of the potential or kinetic tenn) in the representation (coordinate or momentum grid) in which it is local [M,... 

In rare gas crystals [77] and liquids [78], diatomic molecule vibrational and vibronic relaxation have been studied. In crystals, VER occurs by multiphonon emission. Everything else held constant, the VER rate should decrease exponentially with the number of emitted phonons (exponential gap law) [79, 80] The number of emitted phonons scales as, and should be close to, the ratio O/mQ, where is the Debye frequency. A possible complication is the perturbation of the local phonon density of states by the diatomic molecule guest [77]. 

Coupling to these low-frequency modes (at n < 1) results in localization of the particle in one of the wells (symmetry breaking) at T = 0. This case, requiring special care, is of little importance for chemical systems. In the superohmic case at T = 0 the system reveals weakly damped coherent oscillations characterised by the damping coefficient tls (2-42) but with Aq replaced by A ft-If 1 < n < 2, then there is a cross-over from oscillations to exponential decay, in accordance with our weak-coupling predictions. In the subohmic case the system is completely localized in one of the wells at T = 0 and it exhibits exponential relaxation with the rate In k oc - hcoJksTY ". 

Local Thermodynamic Equilibrium (LTE). This LTE model is of historical importance only. The idea was that under ion bombardment a near-surface plasma is generated, in which the sputtered atoms are ionized [3.48]. The plasma should be under local equilibrium, so that the Saha-Eggert equation for determination of the ionization probability can be used. The important condition was the plasma temperature, and this could be determined from a knowledge of the concentration of one of the elements present. The theoretical background of the model is not applicable. The reason why it gives semi-quantitative results is that the exponential term of the Saha-Eggert equation also fits quantum-mechanical expressions. 

In disordered materials such as amorphous silicon, the mobility is so low that it would correspond to a mean free path lower than the distance between atomic sites, which is not physically pertinent. In a classical paper, Anderson [20 has shown that disorder in a solid may result in a localization of the states, in which case the one-electron wave function takes an exponential form... 

A crucial element in MTR is the profile of the localized state density as a function of eneigy, the so-called density of states (DOS). Unfortunately, a direct derivation of the DOS from the variation of the mobility is not straightforward. In two papers published in 1972 and 1976 [116, 117], Spear and Le Comber developed a method based on a simplified description of the accumulation layer, which was assumed to behave like a depletion (Schottky) layer, with a constant density of carrier up to a given thickness L This method has been more recently analyzed by Powell [118], who concluded that is was only able to give a rough estimate of the DOS. Nevertheless, we have used this method to estimate the DOS in 6T and DH6T [115] and found an exponential distribution of the form... 

More recently, a comprehensive model has been developed by Vissenberg and Matters [120 to account for these data. The model is based on a variable range-hopping system with an exponential distribution of localized slates (Eq. (14.71)). The... 

The solidity of gel electrolytes results from chain entanglements. At high temperatures they flow like liquids, but on cooling they show a small increase in the shear modulus at temperatures well above T. This is the liquid-to-rubber transition. The values of shear modulus and viscosity for rubbery solids are considerably lower than those for glass forming liquids at an equivalent structural relaxation time. The local or microscopic viscosity relaxation time of the rubbery material, which is reflected in the 7], obeys a VTF equation with a pre-exponential factor equivalent to that for small-molecule liquids. Above the liquid-to-rubber transition, the VTF equation is also obeyed but the pre-exponential term for viscosity is much larger than is typical for small-molecule liquids and is dependent on the polymer molecular weight. 

In the search for a better approach, investigators realized that the ignition of a combustible material requires the initiation of exothermic chemical reactions such that the rate of heat generation exceeds the rate of energy loss from the ignition reaction zone. Once this condition is achieved, the reaction rates will continue to accelerate because of the exponential dependence of reaction rate on temperature. The basic problem is then one of critical reaction rates which are determined by local reactant concentrations and local temperatures. This approach is essentially an outgrowth of the bulk thermal-explosion theory reported by Fra nk-Kamenetskii (F2). 

Beyond the Boson peak, the reduced DOS reveals for all studied glasses a temperature-independent precisely exponential behavior, g(E)/E exp( / o) with the decay energies Eo correlating with the energies E of the Boson peak. This finding additionally supports the view that the low-energy dynamics of the glasses are indeed delocalized collective motions because local and quasilocal vibrations would be described in terms of a power law or a log-normal behavior [102]. 

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