The Velocity Probability Distribution

If a system obeys classical mechanics, averaging over molecular states means averaging over molecular coordinates and momenta. Coordinates and momenta vary continuously, so we must integrate over their possible values instead of summing over discrete energies as in Eq. (9.2-19). Consider a quantity u that can take on any value between Ml and m2. We define a probability distribution f(u) such that 

If the probability density is normalized, the denominator in this equation equals unity  

We now seek a formula that represents the probability distribution for the molecular states in our model system. Because there are no forces on the molecules inside the box we assert that all positions inside the box are equally probable and focus on the probability distribution for molecular velocities. We begin with a reasonable (but unproved) assumption The probability distribution of each velocity component is independent of the other velocity components. The validity of this assumption must be tested by comparing our results with experiment. Consider a velocity with components v, Vy, and Uj. Let the probability that Vx lies between (a particular value of Vx) and 4- dvx be given by 

Because the effects of gravity are assumed to be negligible there is no physical difference between any two directions. The probability distfibutions for Vy and must be given by the same function as that for Vx. Let and u be particular values of Vy and v. The probability that Vx lies between and + dvx, that Vy is in the range 

9 Gas Kinetic Theory The Moiecuiar Theory of Diiute Gases at Equiiibrium 

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The RTD quantifies the number of fluid particles which spend different durations in a reactor and is dependent upon the distribution of axial velocities and the reactor length [3]. The impact of advection field structures such as vortices on the molecular transit time in a reactor are manifest in the RTD [6, 33], MRM measurement of the propagator of the motion provides the velocity probability distribution over the experimental observation time A. The residence time is a primary means of characterizing the mixing in reactor flow systems and is provided directly by the propagator if the velocity distribution is invariant with respect to the observation time. In this case an exact relationship between the propagator and the RTD, N(t), exists... 

The velocity probability distribution function of Eq. 10.20 is the well-known Maxwell-Boltzmann distribution of velocities. Integrating over vx = —cc — oo shows that P(vx) is normalized. It is also easy to calculate the expectation value for the one-dimensional translational energy of a mole of gas as... 

This is the probability of finding particle 1 with coordinate rx and velocity vx (within drx and dVj), particle 2 with coordinate r2 and velocity v2 (within phase space with velocity rather than momentum for convenience since only one type of particle is being considered, this causes no difficulties in Liouville s equation.) The -particle probability distribution function ( < N) is... 

The velocity gradient leads to an altered distribution of configuration. This distortion is in opposition to the thermal motions of the segments, which cause the configuration of the coil to drift towards the most probable distribution, i.e. the equilibrium s configurational distribution. Rouse derivations confirm that the motions of the macromolecule can be divided into (N-l) different modes, each associated with a characteristic relaxation time, iR p. In this case, a generalised Maxwell model is obtained with a discrete relaxation time distribution. 

Fig. 2.8.13 Velocity probability distributions at different positions across the gap in a 5 mm-9 mm Couette cell at a shear of 0.101 s-1 and following long pre-shearing at high shear rate.

Two template examples based on a capillary geometry are the plug flow ideal reactor and the non-ideal Poiseuille flow reactor [3]. Because in the plug flow reactor there is a single velocity, v0, with a velocity probability distribution P(v) = v0 16 (v - Vo) the residence time distribution for capillary of length L is the normalized delta function RTD(t) = T 1S(t-1), where x = I/v0. The non-ideal reactor with the para-... 

For any r-component stochastic process one may ignore a number of components and the remaining s components again constitute a stochastic process. But, if the r-component process is Markovian, the process formed by the sfirst example above each velocity component is itself Markovian in chemical reactions, however, the future probability distribution of the amount of each chemical component is determined by the present amounts of all components. 

We shall call this a quasilinear Fokker-Planck equation, to indicate that it has the form (1.1) with constant B but nonlinear It is clear that this equation can only be correct if F(X) varies so slowly that it is practically constant over a distance in which the velocity is damped. On the other hand, the Rayleigh equation (4.6) involves only the velocity and cannot accommodate a spatial inhomogeneity. It is therefore necessary, if F does not vary sufficiently slowly for (7.1) to hold, to describe the particle by the joint probability distribution P(X, V, t). We construct the bivariate Fokker-Planck equation for it. 

This equation relates the bimolecular rate constant to the state-to-state rate constant ka(ij l) and ultimately to vap(ij, v l). Note that the rate constant is simply the average value of vcrR(ij,v l). Thus, in a short-hand notation we have ka = (vap(ij,v l)). The average is taken over all the microscopic states including the appropriate probability distributions, which are the velocity distributions /a va) and /b( )(vb) in the experiment and the given distributions over the internal quantum states of the reactants. 

We have shown in Appendix A how these results can be obtained directly from the Langevin equation). Thus the conditional probability distribution of the velocities in the Ornstein-Uhlenbeck [11] process is... 

In gas kinetic theory, the probability density for a component of the molecular velocity is a Gaussian distribution. The normalized probability distribution for Vx, the X component of the velocity, is given by... 

Although the normal mode representation is very useful, the physically observable coordinate is usually the system coordinate q. In the Langevin representation of the dynamics, the trajectory q(t) is characterized by a random force which affects its motion. In the equivalent Fokker-Planck representation of Langevin dynamics, all dynamical information lies in the joint probability distribution function that the particle at time t has position and velocity q, v, given that at time r = 0 its position and velocity were q, v. ... 

Atomistic simulations provide the positions (and in the case of MD, velocities) of aU the atoms in the system consistent with the equilibrium probability distribution... 

Possible optimization of pastes and the according apparatus in process engineering by MRI flow experiments were described. The spatially resolved determination of velocities in suspensions by means of NMR imaging techniques was applied to steady tube flows (with regard to the total flow rate) in different geometries. Three types of suspensions with different solid volumetric concentrations were examined in order to demonstrate the effect of the material-specific flow-behavior and of the geometry of the experimental set-up on the observed flow pattern. The local probability distribution of single velocity components is determined and then both the local mean value and the standard deviation can be derived from the probability distribution. The standard deviation can be interpreted as the local dispersion coefficient of the velocity component. 

Through optical fiber measurement, we now know that the bimodal probability distribution exists over the entire range of sofids volume fraction. For a high-velocity CFB, one peak of the bimodal distribution corresponds to the dense cluster phase and the other to the dilute broth phase (Li and Kwauk, 1994). For a low-velocity, bubbling fluidized bed, this bimodal... 

We have derived this distribution for particles without rotation or vibration, but we now assert that rotation, vibration, and electtonic motion occur independently of translation (the only motion of structureless particles) so that we can use this distribution for the translational motion of any molecules in a dilute gas. The normalized probability distribution is represented in Figure 9.7 for a velocity component of oxygen molecules at 298 K. The most probable value of the velocity component is zero, and most of the oxygen molecules have values of the velocity component between —400ms and 400ms ... 

Figure 9.12 shows this probability distribution of speeds for oxygen molecules at 298 K. The most probable speed, the mean speed, and the root-mean-square speed are labeled on the speed axis. Compare this figure with Figure 9.7. The most probable value of a velocity component is zero, while the most probable speed is nonzero and the probability of zero speed is zero. This difference is due to the fact that the speed probability density is equal to the area of the spherical shell in velocity space (equal to 4nxP-) times the probability density of the velocities lying in the spherical shell. Zero speed is improbable not because the velocity probability density is zero (it is at its maximum value), but because the area of the spherical shell vanishes at n = 0.

Figure 6 Velocity probability distribution P(v v)) as a function of v/(v), where v) is the average velocity, for water flowing in a glass bead pack. For short A (solid line), corresponding to a displacement A v) = 0.08d, where d is the bead diameter, the distribution can be described by an exponential decay, in agreement with the expected Stokes behaviour here d=800pm, and A= 9 ms. For long A (dashed line), corresponding to a displacement A v) = 7.3d, the distribution can be described by a Gaussian law in agreement with the classical models of hydrodynamic dispersion in a porous media here d= 80 pm, and. d = 103 ms. Courtesy of Lebon L, Leblond J and Hulin J-P PMMH, CNRS UMR 7636, ES-PCI, Paris, France.

In three dimensions, the velocity distribution function is proportional to the velocity probability density function ... 

For three-dimensional space equipartition of energy is assumed. This means the onedimensional probability distributions are multiplicative. The variance of the square of the three dimensional velocity about a zero mean velocity is thus defined in terms of the mean of the peculiar velocity ... 

In order to accurately resolve the local flow field around a colloid, methods have been proposed which exclude fluid-particles from the interior of the colloid and mimic slip [19,77] or no-slip [78] boundary conditions. The latter procedure is similar to what is known in molecular dynamics as a thermal wall boundary condition fluid particles which hit the colloid particle are given a new, random velocity drawn from the following probability distributions for the normal velocity component, vn, and the tangential component, vt. 

This method has been devised as an effective numerical teclmique of computational fluid dynamics. The basic variables are the time-dependent probability distributions f x, f) of a velocity class a on a lattice site x. This probability distribution is then updated in discrete time steps using a detenninistic local rule. A carefiil choice of the lattice and the set of velocity vectors minimizes the effects of lattice anisotropy. This scheme has recently been applied to study the fomiation of lamellar phases in amphiphilic systems [92, 93]. 

The Maxwell-Boltzmann velocity distribution function resembles the Gaussian distribution function because molecular and atomic velocities are randomly distributed about their mean. For a hypothetical particle constrained to move on the A -axis, or for the A -component of velocities of a real collection of particles moving freely in 3-space, the peak in the velocity distribution is at the mean, Vj. = 0. This leads to an apparent contradiction. As we know from the kinetic theor y of gases, at T > 0 all molecules are in motion. How can all particles be moving when the most probable velocity is = 0 ... 

If Restart is not checked then the velocities are randomly assigned in a way that leads to a Maxwell-Boltzmann distribution of velocities. That is, a random number generator assigns velocities according to a Gaussian probability distribution. The velocities are then scaled so that the total kinetic energy is exactly 12 kT where T is the specified starting temperature. After a short period of simulation the velocities evolve into a Maxwell-Boltzmann distribution. 

In its more advanced aspects, kinetic theory is based upon a description of the gas in terms of the probability of a particle having certain values of coordinates and velocity, at a given time. Particle interactions are developed by the ordinary laws of mechanics, and the results of these are averaged over the probability distribution. The probability distribution function that is used for a given macroscopic physical situation is determined by means of an equation, the Boltzmann transport equation, which describes the space, velocity, and time changes of the distribution function in terms of collisions between particles. This equation is usually solved to give the distribution function in terms of certain macroscopic functions thus, the macroscopic conditions imposed upon the gas are taken into account in the probability function description of the microscopic situation. 

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