Velocity distribution function relative

One may also show that MPC dynamics satisfies an H theorem and that any initial velocity distribution will relax to the Maxwell-Boltzmann distribution [11]. Figure 2 shows simulation results for the velocity distribution function that confirm this result. In the simulation, the particles were initially uniformly distributed in the volume and had the same speed v = 1 but different random directions. After a relatively short transient the distribution function adopts the Maxwell-Boltzmann form shown in the figure. 

Prove the assertion in the text that the relative velocity of two sets of particles having individual Maxwellian velocity distribution functions also has a Maxwellian distribution with the masses replaced by the reduced mass. 

In general, gap depends on the volume fractions of each particle type and on the particle diameters. However, it can also depend on other moments of the velocity distribution function. For example, if the mean particle velocities Uq. and Vp are very different, one could expect that the collision frequency would be higher on the upstream side of the slower particle type. The unit vector Xi2 denotes the relative positions of the particle centers at collision. If we then consider the direction relative to the mean velocity difference, (Uq, - U ) xi2, we can model the dependence of the pair correlation function on the mean velocity difference as °... 

Here is the mass of the light molecule and c g are the velocities of the light molecule before and after the collision, respectively da is the collision cross section element (surface element perpendicular to the relative velocity — c ) /(c ) is the Maxwellian velocity distribution function... 

For a chute flow with relatively massive particles, the velocity distribution function of particles,/ should be nearly Maxwellian. Therefore the volume-averaged drag force based on an element volume of particles can be expressed as... 

If the interval r is large compared with the time for a collision to be completed (but small compared with macroscopic times), then the arguments of the distribution functions are those appropriate to the positions and velocities before and after a binary collision. The integration over r2 may be replaced by one over the relative distance variable r2 — rx as noted in Section 1.7, collisions taking place during the time r occur in the volume g rbdbde, where g is the relative velocity, and (6,e) are the relative collision coordinates. Incomplete collisions, or motions involving periodic orbits take place in a volume independent of r when Avx(r) and Av2(r) refer to motion for which a collision does not take place (or to the force-field free portion of the... 

Two other attempts, without the use of a distribution function, are worth mentioning, as these are operationally related to experiments and serve to give a rough estimate of the thermalization time. Christophorou et al. (1975) note that in the presence of a relatively weak external field E, the rate of energy input to an electron by that field is (0 = eEvd, where vd is the drift velocity in the stationary state. Under equilibrium, it must be equal to the difference between the energy loss and gain rates by an electron s interaction with the medium. The mean electron energy is now approximated as (E) = (3eD )/(2p), where fl = vd /E is the drift mobility and D is the perpendicular diffusion coefficient (this approximation is actually valid for a Maxwellian distribution). Thus, from measurements of fl and D the thermalization time is estimated to be... 

For two particles having masses mi, m2, the distribution function of relative velocity is obtained by substituting for m the reduced mass m mil m + m2) (see Problem 6 at the end of this chapter). 

The reactant molecule BC is specified to be in an initial vibrational v and rotational state 7, which determines p and allows R to be set to the maximum bond extension compatible with total vibrational energy. The initial relative velocity uR may be varied systematically or it may be chosen at random from Boltzmann distribution function. The orientation angle, which specify rotational phase and impact parameter b are selected at random. 

In the turbulent core it has been conventional to employ the velocity deficiency (B2) as a single-valued function of the relative position in the channel in order to correlate the velocity distribution outside the boundary flow. The velocity deficiency is defined by... 

As indicated earlier, the logarithmic velocity distribution, Eq. (7), yields a zero value for the relative viscosity in the center of the channel for all values of Reynolds number. Such behavior is unreasonable in view of available information concerning eddy viscosity and eddy conductivity (PI, P3) at the plane of symmetry. For this reason Eq. (18) should not be used at values of l/l0 less than 0.3. Figure 8 shows the relative viscosity as a function of position in a circular conduit for several different Reynolds numbers. A comparison of total viscosity as established from Eqs. (16) and (18) and as measured directly (C3) is presented in Fig. 9. Reasonable agreement is obtained between the predicted and directly measured total viscosity. 

A sufficiently rarefied gas, or a mixture of gases, consists of a number of neutral molecules of species 1 and 2 (which may or may not be the same). We may assume a distribution of velocities (measured in the laboratory frame), fi ( ) d3u, that may be modeled by a Maxwellian distribution function, with i = 1 or 2, as long as the duration of the average collision is short compared to the time between collisions. For binary collisions, one usually transforms from laboratory coordinates, Vj, to relative ( >12) and center-of-mass (1>cm) velocities,... 

Now, we proceed to the evaluation of the collisional integrals Pc, qc, and y by using Eq. (5.293) as the form of a collisional pair distribution function. Use the coordinates in Fig. 5.12, in which ez is chosen to be parallel to the relative velocity vn. 9 and 0 are the polar angles of k with respect to ez and the plane of ez and ex, respectively. ex, ey, and ez are the three mutually perpendicular unit vectors corresponding to each coordinate in Fig. 5.12. k, as mentioned in the previous section, is the unit normal on the collision point directed from the center of particle 1 to the center of particle 2. Thus, we have... 

To find the influence function rj(s), we shall consider shear deformation of the system at velocity gradient 7y, while two macromolecular coils, separated by a distance dj, move beside each other at velocity 7ijdj. We add to the sum the contributions of every coil, apart from the chosen one, and find the density distribution of the energy dissipation for the chosen coil. The proportionality coefficient depends only on the concentration of the Brownian particles, if an assumption is made that local dissipation is determined by relative velocities of macromolecular coils,... 

Let us now consider the relative motion of two particles of the same radius Rp and mass mp, and denote by W(r, Ci r2, c2)dr dcidr2dc2 the probability of finding the first particle between r and n + drt, with the velocity between c and Ci + dc, and the second particle between r2 and r2 + t/r2, with the velocity between c2 and c2 + r/c2. The distribution function W satisfies the steady-state Fokker-Plank equation... 

To evaluate the volume integrals in (84), the radial distribution function must be known. The pair distribution function affected by the Brownian motion and the relative electrophoretic velocity between a pair of particles is generally nonuniform and nonisotropic. When the particles are sufficiently small so that Brownian motion dominates, one can use a simple distribution function based on hard-sphere potential... 

For some variables, for example, the relative collision velocity, the cumulative distribution function does not have closed form, and then a third Monte Carlo method must be adopted. Here, another random number R is used to provide a value of v, but a decision on whether to accept this value is made on the outcome of a game of chance against a second random number. The probability that a value is accepted is proportional to the probability density in the statistical distribution at that value. The procedure is repeated until the game of chance is won, and the successful value of v is then incorporated into the set of starting parameters. 

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