Exponentially small

S. Reich, Dynamical Systems, Numerical Integration, and Exponentially Small Estimates, Habilitationsthesis, Konrad Zuse Center, Free University, Berlin (1997). 

The highest probability paths will make the argument of the exponential small. That will be true for paths that follow Newtonian dynamics where mr = F(r). Olender and Elber [45] demonstrated how large values of the time step ht can be used in a way that projects out high frequency motions of the system and allows for the simulation of long-time molecular dynamics trajectories for macromolecular systems. 

Equation (2.2) defines the statistically averaged flux of particles with energy E = P /2m -f V Q) and P > 0 across the dividing surface with Q =0. The step function 6 E — Vq) is introduced because the classical passage is possible only at > Vq. In classically forbidden regions, E < Vq, the barrier transparency is exponentially small and given by the well known WKB expression (see, e.g., Landau and Lifshitz [1981])... 

A typical situation is when (5i > (5q, so that the tunneling distance Qo is overcome mostly at the expense of intermolecular vibration q. The probability p q) is exponentially small, but it is to be compared with the exponentially small barrier transparency, and reduction of the tunneling distance Qo — qhy promoting vibrations may be very large. 

Hr(p which depend on the radial, nh, and the deformation, cr, quantum numbers. For 0cr AU the levels are grouped so that the gaps between groups are approximately the same and equal to hwi whereas the tunneling splitting which occurs for p > 1 in each group of p levels is exponentially small in the parameter 4AC//fcfi>/ 6U21... 

To summarize, the reactive flux method is a great help but it is predicated on a time scale separation, which results from the fact that the reaction time (1/T) is very long compared to all other times. This time scale separation is valid, only if the reduced barrier height is large. In this limit, the reactive flux method, the population decay method and the lowest nonzero eigenvalue of the Fokker-Planck equation all give the same result up to exponentially small corrections of the order of For small reduced barriers, there may be noticeable differences between the different definitions and as aheady mentioned each case must be handled with care. 

The lower root for small y lies at fi/K slightly greater than unity and ass slightly greater than e tending to these values as y tends to zero. The upper root corresponds to the minimum in the curve and for small y clearly lies at large h/k with ass exponentially small. In the limit of the exponential approximation, y - 0, the minimum goes off to infinity and ass = 0. 

Note that the effective radius (4.2.18) of strong tunnelling takes place under the condition (a(R)rl/D) 1, but not (o r lD) 1, which means that strong tunnelling occurs at the contact R. In this case, the effective radius is independent of R and deviates from the exact value by the exponentially small quantity exp(—2x) only. It should be mentioned that equation (4.2.18) does not necessarily mean that Rf.ff R. All what is said above is valid if the so-called binarity collision condition holds [33]. It states that linear approximation holds provided a number of particles within a reaction volume with the radius R ff is small ... 

The slowness of the nuclear motion results (in the time-dependent picture) in a small change of the electron-nuclear interaction during the electronic time Te = (ACg )- -, where Aee is the separation of electronic states. The probability of a transition from a nondegenerate electronic state to an excited state is exponentially small. Thus, the quantum numbers n and v are essentially integrals of motion. 

If the noise is weak, then the probability P exp —S/D) to escape along the optimal trajectory is exponentially small, but it is exponentially greater than the escape probability along any other trajectory, including along other heteroclinic trajectories of the system (37). 

In this paper we demonstrate that the drag resistivity between weakly coupled wires is dominated by the forward scattering in a wide temperature range. Even for identical wires, which is the most favorable for backscattering case, ro oc T2 wins over r2kF at all T above T ep(lo/hkF )1/(1+7. For different wires, r2kF is exponentially small at T < udn, whereas ro has a power-law low-temperature asymptotics here Sn is the mismatch of the electron densities between the wires and u is the characteristic plasma velocity hereafter we sc I, h 1. 

Fig. 2. Sketch of the temperature dependence of the drag resistivity between identical wires. The small momentum transfer contribution considered in this paper dominates at T > T the ratio T /cf is exponentially small for kpd > 1. ...

Why is such a measurement difficult The probabilities to measure correspond to big deviations of the current from its average value, 11 — (J) (/, and are therefore exponentially small. For instance, in the shot noise regime PT(I) — exp (—(/)(/(//(A)jr/ej, Q I/(I)) 1 being the function to characterize. One has to concentrate on very rare measurement outcomes that occur with probability exp(—(/ r/e) 0. Such measurements can only be carried out with threshold detectors that discriminate these rare events. Let us discuss... 

In recent years, substantial efforts have been made to develop a theoretical framework for understanding the nature of such corrections [93]. In the case of lattice models (i.e., models of strictly localized particles) in the NVT ensemble with periodic boundary conditions (PBCs), it has been established a priori [94] and corroborated in explicit simulation [95] that the corrections are exponentially small in the system size [96],... 

As an immediate corollary, the held values that satisfy the equal-areas criterion will identify the coexistence curve of two such ensembles to within exponentially small corrections... 

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