Saddle fluctuations

From Eq. (3.23) it is clear that at weak disorder (g 1) the density of states close to the middle of the pseudogap is strongly suppressed. The reason for this is that a large fluctuation of A(jc) is required in order to create an electron state with energy e < Aq. This makes it possible to apply a saddle-point approach to study the typ-... 

The saddle-point disorder fluctuation [49-51 ] (also called the optimal fluctuation) /(a) is the least suppressed one among the required large fluctuations. It can be found by minimizing. 

Thus, in the saddle-poinl approximation, the absorption coefficient is the product of the averaged density of states (which is essentially the probability to find the necessary disorder fluctuation) and the oscillator strength of the optical transition between the two inlragap levels ... 

Let us consider now processes where intermediate stationary Hamiltonians are mediating the interconversion. In these processes, there is implicit the assumption that direct couplings between the quantum states of the precursor and successor species are forbidden. All the information required to accomplish the reaction is embodied in the quantum states of the corresponding intermediate Hamiltonian. It is in this sense that the transient geometric fluctuation around the saddle point define an invariant property. 

An extremum is a minimum if any fluctuation of the coordinates about this point causes the function to increase. It is a maximum if any fluctuation causes the function to decrease. In any other case, we are dealing with a saddle point. We write the fluctuation Af of f(x,y)... 

The relative lifetimes of the two terraee types at any one saddle point location has been measured[31] to differ by a factor of 6 at 1060C. The change in terrace type occurs by the bridging of the short dimension by step fluctuations. Since the probability of a fluctuation of a particular amplitude depends linearly on the step stiffness[8] the observed lifetime ratio is consistent with measured step stiffnesses[37] and the geometrical picture given above[38]. 

Figure 2. Time evolution of the top ( ) and the saddle point, (o) on a cross-sectional view of two evolving mounds. The saddle point annihilates the top of the small mound, while the maximum of the large mound only fluctuates. Both the height and the lateral position are measured in lattice units.

The quasistationary flux is formed by optimal fluctuations which bring the system from the bottom of the well to the saddle during an optimal time... 

It is clear that all of the escape trajectory from the Lorenz attractor lies on the attractor itself. The role of the fluctuations is, first, to bring the trajectory to a seldom-visited area in the neighborhood of the saddle cycle L, and then to induce a crossing of the cycle L. So we may conclude that the role of the fluctuations is different in this case, and the possibility of applying the Hamiltonian formalism will require a more detailed analysis of the crossing process. 

If the voltage is high enough, the noise of isolated contacts can be considered as white at frequencies at which the distribution function / fluctuates. This allows us to consider the contacts as independent generators of white noise, whose intensity is determined by the instantaneous distribution function of electrons in the cavity. Based on this time-scale separation, we perform a recursive expansion of higher cumulants of current in terms of its lower cumulants. In the low-frequency limit, the expressions for the third and fourth cumulants coincide with those obtained by quantum-mechanical methods for arbitrary ratio of conductances Gl/Gr and transparencies Pl,r [9]. Very recently, the same recursive relations were obtained as a saddle-point expansion of a stochastic path integral [10]. 

The instability arises and evolves owing to thermodynamic fluctua tion (3.29). Such a fluctuation may cause complete system state decay (see, e.g., region V of unstable saddles in Figure 3.4). Flowever, it may also happen that the arising instability creates a new state of the system to be stabilized in time and space. An example is the formation of the limit (restricted) cycle in a system that involves the exceptional point of the unstable focus type. The orbital stability of such a system means exactly the existence of certain time stabilized variations in the thermody namic parameters (for example, the concentrations of reactants) that are... 

In contrast, conventional reaction rate theory replaces the dynamics within the potential well by fluctuations at equilibrium. This replacement is made possible by the assumption of local equilibrium, in which the characteristic time scale of vibrational relaxation is supposed to be much shorter than that of reaction. Furthermore, it is supposed that the phase space within the potential well is uniformly covered by chaotic motions. Thus, only information concerning the saddle regions of the potential is taken into account in considering the reaction dynamics. This approach is called the transition state theory. 

In the Sumi-Marcus model of Figure 17, trajectories on the reactant surface are also important. Since molecular-arrangement fluctuations in solvents (the abscissa X) are assumed to be much slower than intrasolute-vibrational fluctuations (the ordinate q), a reactive trajectory should be the one shown in Figure 17 The reaction system is first taken to an X value by slow diffusive motions along the abscissa X, as represented by a zigzag line. Subsequently, when the reaction system is taken to line C by rapid vibrational motions on the ordinate q at this X value, electron tunneling takes place with an Z-dependent intrinsic rate constant k X), as represented by a vertical straight line. For TST to be justified in this model, the most probable X value should be around that of the saddle point S. 

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