Particle momentum collision

How might the interaction between two discrete particles be described by a finite-information based physics Unlike classical mechanics, in which a collision redistributes the particles momentum, or quantum mechanics, which effectively distributes their probability amplitudes, finite physics presumably distributes the two particles information content. How can we make sense of the process A scatters J5, if B s momentum information is dispersed halfway across the galaxy [minsky82]. Minsky s answer is that the universe must do some careful bookkeeping, ... 

B-particle momentum, multiparticle collision dynamics, single-particle friction and diffusion, 115-118... 

The simplest theory of impact, known as stereomechanics, deals with the impact between rigid bodies using the impulse-momentum law. This approach yields a quick estimation of the velocity after collision and the corresponding kinetic energy loss. However, it does not yield transient stresses, collisional forces, impact duration, or collisional deformation of the colliding objects. Because of its simplicity, the stereomechanical impact theory has been extensively used in the treatment of collisional contributions in the particle momentum equations and in the particle velocity boundary conditions in connection with the computation of gas-solid flows. 

Consider a cubic unit volume containing n particles in a Cartesian coordinate system. On average, about n/6 particles move in the +y-direction, and the same number of particles move in the other five directions. Each particle stream has the same averaged velocity (v). Since particle collision is responsible for the momentum transport, the averaged x-component of the particle momentum transported in the y-direction may be reasonably estimated by... 

Consider two molecules, one in the ith cell, one in the jth, of molecular phase space. If these cells happen to correspond to the same value of the coordinates, though to different values of the momenta, there is a chance that the molecules may collide. In the process of collision, the representative points of the molecules will suddenly shift to two other cells, say the kth and Zth, having practically the same coordinates but entirely different momenta. The momenta will be related to the initial values for the collision will satisfy the conditions of conservation of energy and conservation of momentum. These relations give four equations relating the final momenta to the initial momenta, but since there are six components of the final momenta for the two particles, the four equations (conservation of energy and conservation of three components of momentum) will still leave two quantities undetermined. For instance, we may consider that the direction of one of the particles after collision is undetermined, the other quantities being fixed by the conditions of conservation. 

A necessary condition for the two-term expansion of the distribution function of equation (2) to be valid is that the electron collision frequency for momentum transfer must be larger than the total electron collision frequency for excitation for all values of electron energy. Under these conditions electron-heavy particle momentum-transfer collisions are of major importance in reducing the asymmetry in the distribution function. In many cases as pointed out by Phelps in ref. 34, this condition is not met in the analysis of N2, CO, and C02 transport data primarily because of large vibrational excitation cross sections. The effect on the accuracy of the determination of distribution functions as a result is a factor still remaining to be assessed. 

With these specifications, and with the appropriate neutral particle-plasma collision terms put into the combined set of neutral and plasma equations, internal consistency within the system of equations is achieved. Overall particle, momentum and energy conservation properties in the combined model result from the symmetry properties of the transition probabilities W indices of pre-collision states may be permuted, as well as indices of postcollision states. For elastic collisions even pre- and post collision states may be exchanged in W. 

The quantity mu is the momentum of the particle (momentum is the product of mass and velocity), and the expression F = A(mu)/At means that force is the change in momentum per unit of time. When a particle hits a wall perpendicular to the x axis, as shown in Fig. 5.13, an elastic collision occurs, resulting in an exact reversal of the x component of velocity. That is, the sign, or direction, of ux reverses when the particle collides with one of the walls perpendicular to the x axis. Thus the final momentum is the negative, or opposite, of the initial momentum. Remember that an elastic collision means tjiat there is no change in the magnitude of the velocity. The change in momentum in the x direction is ... 

System Collision Energy (eV) Oi Percent of Particle Momentum TVansferred Momentum TVansfer Angle, 0 Hard Cube Model (Mass = 1.5 X surface atom) Binary Collision Model... 

The last relationship is true because the macroscopic velocity is not changed in a microscopic particle collision as the microscopic particle momentum is conserved (i.e., (w ) — -cj = 0). 

Equations (80a)-(80c) are called conservation equations, since their form is a direct consequence of the conservation of number of particles, momentum, and energy in the binary collisions taking place in the gas. In the phenomenological theories of fluid dynamics, equations in the form of Eqs. (80a)-(80c) are derived from the fact that mass, momentum and energy are conserved in the fluid, but in these theories one does not express the heat flow vector Jj- and the pressure tensor P in terms of a microscopic quantity, like the distribution function f(r, v, t). Instead, one relates Jr and P to n, u, and T by means of the so-called linear laws. ° For a one-component fluid with no internal structure, these linear laws are Fourier s law of heat conduction... 

Then (i) there are five eigenfunctions corresponding to A = 0 these are 1, V, as follows from the definition of L given by (72) and the conservation of number of particles, momentum, and energy in a binary collision and (ii) all other eigenvalues of L are strictly negative. This follows from the fact that one can show, using Eq. (46), that... 

To describe the solids phase rheology, the widely used KTGF is adopted in this framework in addition to the mass and momentum conservation equations the granular temperature 0, accounting for frictional stresses due to particle—particle and particle—wall collisions, needs to be solved by ... 

The equation p = hjX also helps us understand the effect of a transfer of momentum in a collision of a photon with another particle, such as an electron. If a photon transfers some of its momentum to another particle, then the momentum, p, of the photon decreases and, as a consequence, its wavelength. A, increases. The change in wavelength that occurs when light is scattered by electrons in atoms in a crystal (the Compton effect) was first observed in 1923. The Compton effect provides additional confirmation that light consists of particle-like entities that can transfer momentum to other particles through collisions. 

Consider a frontal collision of particles moving translationally (without rotation) (Figure 1.36) along the jc-axis. Momentums of particles are directed along this line, and therefore we can change the vector writings to a scalar form (in projections on the axis x) (the sign x for simplicity is deleted). The velocities of the particles after collision are denoted by u. 

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