Motion of a single particle

17) and (E3.18) determine both the position and the velocity at a given time of each particle initially at radius Ro- In addition, each particle reaches an asymptotic velocity as R approaches infinity. 

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An important property of the time autocorrelation function CaU) is that by taking its Fourier transform, F CA(t) a, one gets a spectral decomposition of all the frequencies that contribute to the motion. For example, consider the motion of a single particle in a hannonic potential (harmonic oscillator). The time series describing the position of the... 

Fig. 6.13 Diagonal motion of a single particle (represented by solid dot) as induced by successively applying BBMCA rule (b) (see figure 6.12) to even (i.e. thicJc-lined) and odd (i.e. thin-lined) partitions of the lattice.

This describes the motion of a single particle having the reduced mass of the two-particle system, whose position is that of particle two with respect to particle one, and which is acted upon by the interparticle force. 

The motion of a particle in the flow field can be described in the Lagrangian coordinate with the origin placed at the center of the moving particle. There are two modes of particle motion, translation and rotation. Interparticle collisions result in both the translational and the rotational movement, while the fluid hydrodynamic forces cause particle translation. Assuming that the force acting on a particle can be determined exclusively from its interaction with the surrounding liquid and gas, the motion of a single particle without collision with another particle can be described by Newton s second law as... 

The phase space for three-dimensional motion of a single particle is defined in terms of three cartesian position coordinates and the three conjugate momentum coordinates. A point in this six-dimensional space defines the instantaneous position and momentum and hence the state of the particle. An elemental hypothetical volume in six-dimensional phase space dpxd Pydpzdqxdqydqz, is called an element, in units of (joule-sec)3. For a system of N such particles, the instantaneous states of all the particles, and hence the state of the system of particles, can be represented by N points in the six-dimensional space. This so-called /r-space, provides a convenient description of a particle system with weak interaction. If the particles of a system are all distinguishable it is possible to construct a 6,/V-dimensional phase space (3N position coordinates and 3N conjugate momenta). This type of phase space is called a E-space. A single point in this space defines the instantaneous state of the system of particles. For / degrees of freedom there are 2/ coordinates in /i-space and 2Nf coordinates in the T space. 

Before beginning a quantitative discussion, let us recall the classical equation of one dimensional motion of a single particle (n) in the crystal... 

The transition from a macroscopic description to the microscopic level is always a complicated mathematical problem (the so-called many-particle problem) having no universal solution. To illustrate this point, we recommend to consider first the motion of a single particle and then the interaction of two particles, etc. The problem is well summarized in the following remark from a book by Mattuck [18] given here in a shortened form. For the Newtonian mechanics of the 18th century the three-body problem was unsolvable. The general theory of relativity and quantum electrodynamics created unsolvable two-body and single-body problems. Finally, for the modem quantum field... 

To illustrate the point, let us consider the collinear reaction AB + C— A + BC. It is known (c.f., Baer [1982]) that motion of the system in the center-of-mass frame is equivalent to motion of a single particle of mass... 

MOTION OF A SINGLE PARTICLE IN CO-AXIAL HORIZONTAL IMPINGING STREAMS... 

Thus, with the subscript x denoting the motion direction being saved for the onedimensional motion of a single particle, Eq. (2.1) can be simplified to... 

Solution This is a one-dimensional problem. The equation of motion of a single particle in the ft-direction is obtained from Eq. (3.96) as... 

To yield a constitutive relation for the mass flux of particles, it is convenient to begin with the equation of motion of a single particle, which is expressed by... 

In quantum mechanics (18,19) the vibration of a diatomic molecule can be treated as a motion of a single particle having mass n whose potential energy is expressed by (1-21). The Schrodinger equation for such a system is written as... 

If fl = 1, every atom in the slider has the same velocity at every instant of time, once steady state (not necessarily smooth sliding) has been reached. Hence the problem is reduced to the motion of a single particle, for which one obtains Fj = 1. This provides an upper bound of Fj for arbitrary a. If the walls are incommensurate or disordered, one can again make use of the argument that the motion of all atoms relative to their preferred positions is the same up to temporal shifts once steady state has been reached. Owing to the incommensurability, the distribution of these temporal shifts with respect to a reference trajectory cannot change with time in the thermodynamic limit, and the instantaneous value of Fk is identical to Fk at all times. This gives a lower bound for Fj for arbitrary a. The static friction for arbitrary commensurability and/or finite systems lies in between the upper and the lower bound. 

Through quantum mechanical considerations [2,7], the vibration of two nuclei in a diatomic molecule can be reduced to the motion of a single particle of mass p, whose displacement q from its equilibrium position is equal to the change of the intemuclear distance. The mass p is called the reduced mass and is represented by... 

As expected, the larger the radius of gyration the larger the relaxation times and the larger the diffusion constant the smaller r. The k = 0 mode represents translational diffusion of the molecular center of resistance and is given by the Langevin equation for the diffusive motion of a single particle (see Section 5.9). 

Gs(r,t) is in general simpler to visualize and calculate theoretically than Gd(/V) since the former describes the motion of a single particle. 

Motion of a Single Particle Relative to a Fixed Axis... 

Without loss of generality, we may consider the motion of a single particle along one direction x and eventually generalize what we learn to many dimensions. From elementary principles of quantum mechanics, we know that Jf(x, x ) has the composition property [66]. That is, Jf(x, x ) can be expressed as... 

Katayama Y., Terauti R., Brownian motion of a single particle under shear flow, Fur. J. Phys., 1966, Vol. 

The use of a wave packet instead of a plane wave provides a description of the motion of a single particle which bears a close similarity to the classical description It should be noted, however, that the width ax of a wave packet in the general case increases with time /46c,47/f which is a consequence of the dispersion of the plane waves in the pactet, having different velocities of propagation (phase velocities) Therefore, a localization of the particle is possible only for relatively short time intervals for which the condition /47/... 

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