Velocity distribution positional dependence

In 1872, Boltzmaim introduced the basic equation of transport theory for dilute gases. His equation detemiines the time-dependent position and velocity distribution fiinction for the molecules in a dilute gas, which we have denoted by /(r,v,0- Here we present his derivation and some of its major consequences, particularly the so-called //-tlieorem, which shows the consistency of the Boltzmann equation with the irreversible fomi of the second law of themiodynamics. We also briefly discuss some of the famous debates surrounding the mechanical foundations of this equation. 

Nuclei have many times more mass than electrons. During a very small period of time when the movement of heavy nuclei is negligible, electrons are moving so fast that their distribution is smooth. This leads to the approximation that the electron distribution is dependent only on the fixed positions of nuclei and not on their velocities. This approximation allows two simplifications... 

Another method is to measure the disappearance rate of the excited parent molecules, that is, the intensity changes of the disk-like images at various delay times (therefore, at various photolysis laser positions) along the molecular beam. This is very useful when the dissociation rate is slow and the method described above cannot be applied. This measurement requires a small molecular beam velocity distribution and a large variable distance between the crossing points of the pump and probe laser beams with the molecular beam. The small velocity distribution can be obtained through adiabatic expansion, and the available distances between the pump and probe laser beams depend on the design of the chamber. For variable distances from 0 to 10 cm in our system and AV/V = 10% molecular beam velocity distribution, dissociation rates as slow as 3 x 103 s 1 under collisionless condition can be measured. 

Before we present our idea, we give a short summary of the mean-field theory. The core concept there is the velocity (v)-position (r) probability distribution of stars,/(r, v, t), a positive and integrable function that is a priori time-dependent. The number density p(r, t) is the integral over velocities of/(r, v, t) ... 

However, before embarking on this analysis, a brief excursion is of interest [285]. Let us specifically ignore the spatial dependence of the distribution of the N particles. It could have been calculated from the deterministic Newtonian equations of motion. Now considering, in particular, the motion of particles 1 and 2 as above, average the distribution over all position of the N particles to give the velocity distribution function... 

For a system composed of N particles, the complete velocity distribution function is denoted f(N> (r(N>, p(/V), t). It is a function of 6N variables, that is, the three vector coordinates for each of the N molecules rW) and the three components of the momentum of each molecule p(-V). Of course, for a macroscopic system, where IV is a very large number, on the order of Avogadro s number A, it is impossible to obtain f(N). One usually attempts to find a less complete description of the system by looking at f(h which depends on the positions and momentum of a smaller number of molecules h and integrates over the effects of the remaining N — h molecules. 

To illustrate the effect of thermal gradients and temperature dependent viscosity, we can plot a dimensionless velocity, ux/U0 as a function of dimensionless position, y/h, for various values of thermal imbalance between the surfaces, i. Note that Q, the product between the temperature dependence of the viscosity and the temperature imbalance is also a dimensionless quantity. This gives a fully dimensionless graph that can be used to assess many case scenarios. Figure 6.59 presents dimensionless velocity distributions across the plates for various dimensionless temperature imbalances, ft. 

Tennant (T2) also studied velocity distributions, using a six-blade turbine and two viscous corn syrup solutions as well as water. For impeller speeds in the range of 100-200 r.p.m., his results generally confirm those of Sachs. A 300-fold increase in viscosity reduces the fluid velocity by about 30%. Comparison with Sachs data indicates that increasing the number of turbine blades from four to six increases the radial velocities by roughly 10-50%, depending on impeller speed and on radial position in a manner as yet undefined. 

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 °... 

The governing equation to determine the temperature distribution in the tube is the thermal energy equation, (2-110). To solve this equation, we need to know the form of the velocity distribution in the tube. We have already seen that the steady-state velocity profile for an isothermal fluid, far downstream from the entrance to the tube, is the Poiseuille flow solution given by (3-44). In the present problem, however, the temperature must depend on both r and z, and hence the viscosity (which depends on the temperature) will also depend on position. The dependence on z is due to the fact that heat is added for all z > 0, and thus the temperature must continue to increase with the increase of z. The dependence on r is due to the fact that there must be a nonzero conductive heat flux in the fluid at the tube wall to match the prescribed heat flux through the wall, and thus the temperature must have a nonzero r derivative. It follows that the velocity field will generally differ from Poiseuille flow. 

In previous chapters we considered elementary crystal excitation taking into account only the Coulomb interaction between carriers. From the point of view of quantum electrodynamics (see, for example, (1)) such an interaction is conditioned by an exchange of virtual scalar and longitudinal photons, so that the potential energy, corresponding to this interaction, depends on the carrier positions and not on their velocity distribution. As is well-known, the exchange of virtual transverse photons leads to the so-called retarded interaction between charges. 

Boltzmann equation An equation used in me smdy of a collection of particles in non-equilibrium statistical mechanics, particularly meir transport properties. The Boltzmatm equation describes a quantity called me distribution function, which gives a mamematical description of me state and how it is changing. The distribution function depends on a position vector r, a velocity vector v, and the time fi it mus provides a statistical statement about me positions and velocities of the particles at any time. In me case of one species of particle being present, Boltzmann s equation can be written... 

Molecular dynamics consists of examining the time-dependent characteristics of a molecule, such as vibrational motion or Brownian motion within a classical mechanical description [13]. Molecular dynamics when applied to solvent/solute systems allow the computation of properties such as diftiision coefficients or radial distribution functions for use in statistical mechanical treatments. In this calculation a number of molecules are given some initial position and velocity. New positions are calculated a short time later based on this movement, and the process is iterated for thousands of steps in order to bring the system to an equilibrium. Next the data are Fourier transformed into the frequency domain. A given peak can be chosen and transformed back to the time domain, to see the motion at that frequency. 

In order to see how the boundary layer affects particle detachment, let us turn to Fig. X. 1. Depending on the flow velocity, the boundary layer may be either laminar (Fig. X. 1. a) or turbulent (Fig. X. 1. b). The laminar boundary layer is characterized by a linear velocity distribution in the layer. The adherent particles may be completely submerged in this layer if the particle diameter is smaller than the boundary layer thickness (see Position I in Fig. X.l.a). Position II shows the case in which the diameter of the adherent particle is greater than the boundary layer thickness. 

IV. Spatially dependent velocity distributions. When the spectrum is independent of position, the central problem is the determination of the energy-transfer cross sections. The calculation of the spectrum once these cross sections are known is a straightforward procedure. The cross-section aspect of the problem is both more difficult from a physics point of view, and more time consuming from the point of view of machine computation. This situation is reversed when we come to consider the spatial dependence of the slow neutron spectrum. The cross sections needed are the same ones that already have been computed for the infinite medium spectrum problem. The transport equation must now be solved in at least two variables, and in a form for which the existing approximate techniques are not very well adapted. The focus of the problem therefore shifts to the development of appropriate techniques for solution of the transport equation when the energy and position variables are coupled in such a way that neutrons can both gain and lose energy in a collision. 

For the systems considered the gradients are very small. Thus, in any region of the gas, concentration, temperature, and mean velocity are well-defined experimental quantities. There is no objection, either in principle or practice, to inserting a probe which can measure the local values of these macroscopic parameters (as, for example, one might measure temperature at different points in a room). As the gas is not at equilibrium the velocity distribution is positionally dependent but the mean molecular properties change little over distances comparable with the mean free path (estimated in Section 2.2 to be between 30 and 300 nm for a gas at 300 K and 1 atm). In any region, extending over many mean free paths, the properties of the gas can be characterized by the local values of the macroscopic observables. 

Velocity The term kinematics refers to the quantitative description of fluid motion or deformation. The rate of deformation depends on the distribution of velocity within the fluid. Fluid velocity v is a vector quantity, with three cartesian components i , and v.. The velocity vector is a function of spatial position and time. A steady flow is one in which the velocity is independent of time, while in unsteady flow v varies with time. 

The Born-Oppenheimer approximation is the first of several approximations used to simplify the solution of the Schradinger equation. It simplifies the general molecular problem by separating nuclear and electronic motions. This approximation is reasonable since the mass of a typical nucleus is thousands of times greater than that of an electron. The nuclei move very slowly with respect to the electrons, and the electrons react essentially instantaneously to changes in nuclear position. Thus, the electron distribution within a molecular system depends on the positions of the nuclei, and not on their velocities. Put another way, the nuclei look fixed to the electrons, and electronic motion can be described as occurring in a field of fixed nuclei. 

---