Ballistic motion

Table 10.4 lists the values of trap density and binding energy obtained in the quasi-ballistic model for different hydrocarbon liquids by matching the calculated mobility with experimental determination at one temperature. The experimental data have been taken from Allen (1976) and Tabata et ah, (1991). In all cases, the computed activation energy slightly exceeds the experimental value, and typically for n-hexane, 0/Eac = 0.89. Some other details of calculation will be found in Mozumder (1995a). It is noteworthy that in low-mobility liquids ballistic motion predominates. Its effect on the mobility in n-hexane is 1.74 times greater than that of diffusive trap-controlled motion. As yet, there has been no calculation of the field dependence of electron mobility in the quasi-ballistic model. 

This relation connects heat conduction and diffusion, quantitatively. As expected, normal diffusion (a =1)corresponds to the size-independent (/ = 0) heat conduction obeying the Fourier law. Moreover, a ballistic motion (a = 2) implies that the thermal conductivity is proportional to... 

Figure 5. Comparison of prediction (4) with numerical data. Normal diffusion ( ). The ballistic motion ( ). Superdiffusion ID Ehrenfest gas channel (Li et al, 2005)(v) the rational triangle channel (Li et al, 2003) (empty box) the polygonal billiard channel with (i = (V > — 1)7t/4), and 2 = 7r/3 (Alonso et al, 2002)(A) the triangle-square channel gas(Li et al, 2005) (<>) / values are obtained from system size L e [192, 384] for all channels except Ehrenfest channel (Li et al, 2005). The FPU lattice model at high temperature regime (Li et al, 2005) ( ), and the single walled nanotubes at room temperature ( ). Subdiffusion model from Ref. (Alonso et al, 2002) (solid left triangle). The solid curve is f3 = 2 — 2/a.

Constraints may be introduced either into the classical mechanical equations of motion (i.e., Newton s or Hamilton s equations, or the corresponding inertial Langevin equations), which attempt to resolve the ballistic motion observed over short time scales, or into a theory of Brownian motion, which describes only the diffusive motion observed over longer time scales. We focus here on the latter case, in which constraints are introduced directly into the theory of Brownian motion, as described by either a diffusion equation or an inertialess stochastic differential equation. Although the analysis given here is phrased in quite general terms, it is motivated primarily by the use of constrained mechanical models to describe the dynamics of polymers in solution, for which the slowest internal motions are accurately described by a purely diffusive dynamical model. 

We next consider the effective force balance for all >N variables, while treating the system as an unconstrained system. For simplicity, we consider the case in which the crossover from ballistic motion to diffusion occurs at a timescale much less than any characteristic relaxation time for vibrations of the hard coordinates, so that the vibrations are overdamped, but in which the vibrational relaxation times are much less than any timescale for the diffusion of the soft coordinates. In this case, we may assume local equilibration of all 3N momenta at timescales of order the vibration time. Repeating the analysis of the Section V.A, while now treating all 3N coordinates as unconstrained, yields an effective force balance... 

J.-L. Martin Prof. Fleming, you pointed out the importance of having ballistic motions along the reaction coordinate If the motion was not ballistic through the transition region, the quantum yield could be much less Along this line, what would be the influence of bal-... 

Microwaves can be produced by four types of macroscopic cavity resonators that use the ballistic motion of electrons across a cavity opening the klystron, the magnetron, the traveling-wave tube (TWT), and the gyrotron. They can also be generated by field-effect transistors at low frequencies, by Gunn42 diodes, and by IMP ATT diodes. 

In the Levy walk case, this is true for q < p — 1. For q > p — 1, the slope becomes equal to 1, ballistic motion, according to... 

The frequency T - corresponding to the ballistic motion of electron between two successive scatterings by impurities is here the characteristic one. At oJQ, the usual Drude... 

Thermal conductivity at the nanoscale may significantly deviate from the bulk material value from various reasons, of classical and quantum origin [1], First, size effects are influential if the structure dimension is comparable (or smaller) to the phonon characteristic length. This may result in a ballistic motion and the failure of the Fourier s law, even when phonons are treated as particles. Moreover, if the phases of the phonon waves are fixed, for example when interfaces in the structure are flat, wave effects sustain, leading to phonon interference and diffraction effects. 

MD simulations and experiments clearly show that the single particle motion of water molecules next to a protein surface is different than in the bulk. Here, single particle refers to measures of the average behavior of individual water molecules, as opposed to coherent behavior of collections of water molecules, which will be discussed in more detail below. The perturbation of the translational and rotational mobility of protein hydration water (defined using the 4 A distance criterion) is depicted in Fignre 16.1a and b, respectively. We will discuss the data for the native (N) state first, and snbsequently compare the native and MG states. In bulk water, after an initial rapid ( 2ps) rise corresponding to ballistic motion. 

The Landau-Zener model assumes the ballistic motion along the reaction coordinate with constant velocity in the vicinity of the crossing point [73]. The applicability condition of this approach is... 

Both situations are illustrated in Figure 9.25. It can be shown that perturbation of the ballistic motion by additive noise can cause dramatic changes in the transition probability [164-167]. So the applicability of the... 

The particles travel with speed y and turn with frequency jx. The persistent random walk is characterized by two parameters, in contrast to the ordinary random walk or Brownian motion, which is completely characterized by the diffusion coefficient D. The persistent random walk spans the whole range of dispersal, from ballistic motion, in the limit /r 0, to diffusive motion, in the limit y oo, p. oo, such that lim y 2p = Z) = constant. The total density of the dispersing particles is given by... 

The linear dependence of the mean squared deviation with respect to time is a defining hallmark of Brownian motion. However, this dependence is a result of the averaging over a large number of kicks received by the particle from the surrounding fluid molecules. Since such kicks impart a ballistic motion to the particle, a natural question arises What does the motion of the particle look like at short times and distances after each kick, i.e., during the transition from ballistic to Brownian motion ... 

The relation (X (0) = Kf between mean squared displacement and time presents a range of possibilities for varying a, in which canonical Brownian motion (a = 1) and ballistic motion (a = 2) represent just two possibilities. AU cases apart from a = 1 are clubbed under the label anomalous Brownian motion and encompass a range of interesting phenomena from biology to electronics [9]. The value of a is determined by two quantities - the variance in the step size waiting time /x, between consecutive steps - as discussed below. 

Barati Farimani A, Alum NR. Spatial diffusion of water in carbon nanotubes from Fickian to ballistic motion. J Phys Chem B 2011 115 12145-9. 

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