Kinetic energy finiteness

The temperature of the system is proportional to the average kinetic energy (eq. (16.12), and therefore determines which parts of the energy surface the particles can exploit. Owing to the finite precision by which the atomic forces are evaluated, and the finite time step used, the total energy in practice is not constant (preservation of the energy to within a given threshold may be used to define the maximum permissible time step). 

The motion of particles of the film and substrate were calculated by standard molecular dynamics techniques. In the simulations discussed here, our purpose is to calculate equilibrium or metastable configurations of the system at zero Kelvin. For this purpose, we have applied random and dissipative forces to the particles. Finite random forces provide the thermal motion which allows the system to explore different configurations, and the dissipation serves to stabilize the system at a fixed temperature. The potential energy minima are populated by reducing the random forces to zero, thus permitting the dissipation to absorb the kinetic energy. 

Often the actions of the radial parts of the kinetic energy (see Section IIIA) on a wave packet are accomplished with fast Fourier transforms (FFTs) in the case of evenly spaced grid representations [24] or with other types of discrete variable representations (DVRs) [26, 27]. Since four-atom and larger reaction dynamics problems are computationally challenging and can sometimes benefit from implementation within parallel computing environments, it is also worthwhile to consider simpler finite difference (FD) approaches [25, 28, 29], which are more amenable to parallelization. The FD approach we describe here is a relatively simple one developed by us [25]. We were motivated by earlier work by Mazziotti [28] and we note that later work by the same author provides alternative FD methods and a different, more general perspective [29]. 

There will always be a position a, where the electron has positive kinetic energy even though it is outside the metal and there will be a finite probability that it leaks through the barrier and leaves the metal permanently, in a process of cold emission. 

For finite values of /n the system moves within a limited width, given by the fictitious electronic kinetic energy, above the Born-Oppenheimer surface. Adiabacity is ensured if the highest frequency of the nuclear motion separated from the lowest frequency associated with the fictitious motion of the electronic degrees of freedom cofm. It can be shown [30] that eo in is proportional to the gap Eg ... 

The critical density is traditionally dehned as that density which separates the closed (finite) universe from the open (infinite) universe in the simplest model available, i.e. in a universe without cosmological constant or quintessence. It corresponds to a universe with zero total energy, where the kinetic energy due to expansion is exactly balanced by gravitational potential energy. The value of the critical density is 10 gcm, which amounts to very httle when compared to a chunk of iron ... 

In the absence of an external field, electrons in the metal are confronted by a semi-infinite potential barrier (upper solid line in Fig. la), so that escape is possible only over the barrier. The process of thermionic emission consists of boiling electrons out of the Fermi sea with kinetic energy > x + M- The presence of a field F volts/cm. at and near the surface modifies the barrier as shown. It follows from elementary electrostatics that the potential V will not be noticed by electrons sufficiently far in the interior of the metal. However, electrons approaching the surface are now confronted by a finite potential barrier, so that tunneling can occur for sufficiently low and thin barriers. 

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