Rectilinear motion

By the introduction of the (x, y) coordinate system, one has reduced the problem to the motion of a particle of mass (i in a two-dimensional rectilinear space (x, y). Thus, the problem of the collision between an atom and a diatomic molecule in a collinear geometry has been converted into a problem of a single particle on the potential energy surface expressed in terms of the coordinates x and y rather than the coordinates rAB and rBc The coordinates x and y which transform the kinetic energy to diagonal form in such way that the kinetic energy contains only one (effective) mass are referred to as mass scaled Jacobi coordinates. 

The probability distribution of Eq. (2.21) was derived assuming rectilinear motion in a harmonic potential. The true potential in a crystal is often more complex, especially in the upper parts of the potential surface, which are of importance at higher temperatures. 

Consider a rigid sphere of radius a, executing rectilinear oscillatory motion relative to remote fluid with its velocity given by" ... 

Fig. 12. Potential energy surface for C2v structures of HjO. Rectilinear displacements of the hydrogen atoms would occur along straight lines on this surface. The dotted line shows a curvilinear motion keeping the OH bond length constant. The contours (- 10, - 9, - 8, - 7 eV) have been drawn from the analytical potential of Murrell and Sorbie U35).

The trajectory of an ion moving in such a potential presents a sequence of rectilinear sections placed between the points of elastic reflections of an ion from the walls of the well. We consider two variants of such a model related to one-dimensional and spatial motion of ion, depicted, respectively, in Figs. 47a and 47b. In the first variant the ion s motion during its lifetime59 presents periodic oscillations on the rectilinear section 2 lc between two reflection points. In the second variant we consider a spherically symmetric potential well, to which a spherical hollow cavity corresponds with the radius lc. 

Figure 47. An ion inside a hydration cell. In the variant (a) an ion oscillates along a rectilinear section and in variant (b) inside a hollow reflecting sphere, (c) Schematic of the motion of an ion inside a spherical sheath. Dashed areas denote the hydration sheath.

It is common practice to treat particle motion as the basic dynamic scale for transport processes. This is readily illustrated for particles in steady rectilinear motion. 

As velocity of flow increases, a condition is eventually reached at which rectilinear laminar flow is no longer stable, and a transition occurs to an alternate mode of motion that always involves complex particle paths. This motion may be of a multidimensional secondary laminar form, or it may be a chaotic eddy motion called turbulence. The nature of the motion is governed by both the rheological nature of the fluid and the geometry of the flow boundaries. 

It was assumed that the motion of the fictitious particle within the above time steps is rectilinear. This simplification, which accelerates the computer calculation of the trajectories of the fictitious particle, has been show n to be justified (9). The Philips slip correction factor for an accommodation coefficient of unity (Eq. [4]) was used in the calculation of the diffusion coefficients of particles. The values of the dimensionless coagulation coefficients % obtained by the computer simulation for different particle sizes, are given in Table I. The statistical errors of the Monte Carlo simulation were estimated by the standard 3 a method (corresponding to a probability of 0.997) (13). The number of particle pairs that must be generated in order to lower the error to a reasonable level depends both on the initial distance of separation between... 

Brownian motion on circle and sphere. The equilibrium density on a circle or a sphere deviates from uniformity by an amount proportional to cos 6, where 6 is the angle between the dipole and the field. For the circular motion, that is, a dipole confined to rotate in a plane, the cosine retains its form just as in the rectilinear motion discussed above, and relaxes in amplitude to give... 

Pure rotary diffusion of rigid dipoles in two or three dimensions, then, gives exponential decay of polarization with a single relaxation time, provided the sites are uniformly distributed and D is constant. The description of the motion in terms of D alone breaks down, as we shall see, for very short times. A three-dimensional rigid body in any case executes a more complex motion. Even an internally uniform model of rectilinear charge-carrier difiurion automatically shows multiple relaxation. More realistic models must take account of the dynamic s of molecular motion. 

As in the last section, some aspects of relaxation behaviour are clear in the bounded rectilinear motion of a charged particle. We conrider first... 

For the particular case of rectilinear motion to which we shall... 

If we assume further that the total energy U of the gas is entirely kinetic energy of the rectilinear motion of the molecules. 

These assumptions were (1) The validity of Charles law. (2) That the energy of the gas U consists entirely of the kinetic energy L of the rectilinear motion of the molecules. As diatomic gases obey Charles law, we conclude that the second assumption must be at fault. The equation U=L does not hold for polyatomic gases, and we must substitute for it the equation U=L- -P,... 

Instead of this we must regard the electrons as not entirely free but as colliding from time to time with the atoms of the lattice. Thus the electrons describe zig-zag paths, and the paths described in the magnetic field will exhibit only slight deviations from these if the motion is approximately rectilinear between two collisions, i.e. if the radius of curvature is large compared with the mean free path of the electrons. As we know, the angular velocity of the electrons is... 

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