Maxwell-Boltzmann, distribution equation

The basic principles are described in many textbooks [24, 26]. They are thus only sketchily presented here. In a conventional classical molecular dynamics calculation, a system of particles is placed within a cell of fixed volume, most frequently cubic in size. A set of velocities is also assigned, usually drawn from a Maxwell-Boltzmann distribution appropriate to the temperature of interest and selected in a way so as to make the net linear momentum zero. The subsequent trajectories of the particles are then calculated using the Newton equations of motion. Employing the finite difference method, this set of differential equations is transformed into a set of algebraic equations, which are solved by computer. The particles are assumed to interact through some prescribed force law. The dispersion, dipole-dipole, and polarization forces are typically included whenever possible, they are taken from the literature. 

Similar convection-diffusion equations to the Navier-Stokes equation can be formulated for enthalpy or species concentration. In all of these formulations there is always a superposition of diffusive and convective transport of a field quantity, supplemented by source terms describing creation or destruction of the transported quantity. There are two fundamental assumptions on which the Navier-Stokes and other convection-diffusion equations are based. The first and most fundamental is the continuum hypothesis it is assumed that the fluid can be described by a scalar or vector field, such as density or velocity. In fact, the field quantities have to be regarded as local averages over a large number of particles contained in a volume element embracing the point of interest. The second hypothesis relates to the local statistical distribution of the particles in phase space the standard convection-diffusion equations rely on the assumption of local thermal equilibrium. For gas flow, this means that a Maxwell-Boltzmann distribution is assumed for the velocity of the particles in the frame-of-reference co-moving with the fluid. Especially the second assumption may break dovm when gas flow at high temperature or low pressure in micro channels is considered, as will be discussed below. 

In summary, Eq. (86) is a general expression for the number of particles in a given quantum state. If t = 1, this result is appropriate to Fenni-rDirac statistics, or to Bose-Einstein statistics, respectively. However, if i is equated torero, the result corresponds to the Maxwell -Boltzmann distribution. In many cases the last is a good approximation to quantum systems, which is furthermore, a correct description of classical ones - those in which the energy levels fotm a continuum. From these results the partition functions can be calculated, leading to expressions for the various thermodynamic functions for a given system. In many cases these values, as obtained from spectroscopic observations, are more accurate than those obtained by direct thermodynamic measurements. 

Equation (30) is the Maxwell-Boltzmann distribution function in rectangular coordinates. Thus, in a system of N total molecules, the fraction of molecules, dN/ N, with velocity components in the ranges x component, vx to vx + dvx y component, vy to Vy + dvy, and z component, vz to vz + dvz is given by... 

Equation (8.55) comes ultimately from the Maxwell - Boltzmann distribution in Equation (1.16). 

At this point, it is worthwhile to return on the theoretical basis of the kinetic method, and make some considerations on the assumptions made, in order to better investigate the validity of the information provided by the method. In particular some words have to been spent on the effective temperamre The use of effective parameters is common in chemistry. This usually implies that one wishes to use the form of an established equation under conditions when it is not strictly valid. The effective parameter is always an empirical value, closely related to and defined by the equation one wishes to approximate. Clearly, is not a thermodynamic quantity reflecting a Maxwell-Boltzmann distribution of energies. Rather, represents only a small fraction of the complexes generated that happen to dissociate during the instrumental time window (which can vary from apparatus to apparatus). 

The standard theories of chemical kinetics are equilibrium theories in which a Maxwell-Boltzmann distribution of reactants is postulated to persist during a reaction.68 The equilibrium theory first passage time is the TV -> oo limit in Eq. (6), Corrections to it then are to be expected when the second term in this equation is no longer negligible, i.e., when N is not much greater than e — e- )-1. The mean first passage time and rate of activation deviate from their equilibrium value by more than 10% when... 

Insertion of equation 3 into equation 1, approximation of the Fermi distribution by a classical Maxwell-Boltzmann distribution, and integration of equation 1 yield the expression for the total number of electrons in the conduction band ... 

Specialized to thermal equilibrium, the velocity distributions for the molecules are the Maxwell-Boltzmann distribution (a special case of the general Boltzmann distribution law). The expression for the rate constant at temperature T, k(T), can be reduced to an integral over the relative speed of the reactants. Also, as a consequence of the time-reversal symmetry of the Schrodinger equation, the ratio of the rate constants for the forward and the reverse reaction is equal to the equilibrium constant (detailed balance). 

That is, the Maxwell-Boltzmann distribution for the two molecules can be written as a product of two terms, where the terms are related to the relative motion and the center-of-mass motion, respectively. After substitution into Eq. (2.18) we can perform the integration over the center-of-mass velocity Vx. This gives the factor y/2iVksTjM (IZo eXP( —ax2)dx = sjnja) and, from the equation above, we obtain the probability distribution for the relative velocity, irrespective of the center-of-mass motion. 

These two different concepts lead to different mathematical expressions which can be tested with the experimental data. The derivation is similar to that of equations (1-5) but with the inclusion of a term, calculated from the Maxwell-Boltzmann distribution, for the fraction of molecules in the activated state. With these formulas it can be shown that when the reciprocal of the velocity constant is plotted against the reciprocal of the initial pressure a straight line is produced, according to Theory I, but a curved line is produced if Theory II is correct. Moreover the extent of the curvature depends on the complexity of the molecule. It is found that simple molecules like nitrous oxide give astraight line, and more complicated molecules, like azomethane, give er curved line. ... 

Equation (1.4) expresses what is called the Maxwell-Boltzmann distribution law. If Eq. (1.4) gives the probability of finding any particular molecule in the fctii state, it is clear lhat it also gives the fraction of all molecules to be found in that state, averaged through the assembly. 

For a gas mixture at rest, the velocity distribution function is given by the Maxwell-Boltzmann distribution function obtained from an equilibrium statistical mechanism. For nonequilibrium systems in the vicinity of equilibrium, the Maxwell-Boltzmann distribution function is multiplied by a correction factor, and the transport equations are represented as a linear function of forces, such as the concentration, velocity, and temperature gradients. Transport equations yield the flows representing the molecular transport of momentum, energy, and mass with the transport coefficients of the kinematic viscosity, v, the thermal diffirsivity, a, and Fick s diffusivity, Dip respectively. 

We ask you to accept that the second fraction on the right-hand side of Equation 4.4 is the probability that an oscillator of frequency v is activated at a given temperature T. Chapter 9 presents the origin of this probability in the famous Maxwell-Boltzmann distribution, but in this chapter we want to use the result to demonstrate some additional consequences of Planck s hypothesis. 

All chemical process involves the motion of atoms within a molecule. Molecular dynamics (MD), in the broadest sense, is concerned with molecules in motion. It combines the energy calculations from molecular mechanics with equations of motion. Generally, an appropriate starting structure is selected (normally an energy minimized structure). Each atom in the system is then assigned a random velocity that is consistent with the Maxwell-Boltzmann distribution for the temperature of interest. The MM formalism is used to calculate the forces on all the atoms. Once the atom positions are known, the forces, velocities at time t, and the position of the atoms at some new time t + 5t can be predicted. More details about the method can be found in Ref.. ... 

To obtain Eqs (1.203) and (1.206) we need to assume that P vanishes asx - 00 faster than Physically this must be so because a particle that starts at x = 0 cannot reach beyond some finite distance at any finite time if only because its speed cannot exceed the speed of light. Of course, the diffusion equation does not know the restrictions imposed by the Einstein relativity theory (similarly, the Maxwell-Boltzmann distribution assigns finite probabilities to find particles with speeds that exceed the speed of light). The real mathematical reason why P has to vanish faster than jg that in... 

An easy way to find this correction factor is to look at the history of an exit trajectory. This history is followed by starting atx = xb trajectories with velocity sampled from a Maxwell-Boltzmann distribution in the outward direction—these represent the outgoing equilibrium flux, then inverting the velocity (y -> —v) so that the particle is heading into the well, and integrating the equations of motion... 

We have illustrated the calculation of the averages from the Langevin equation for sharp initial conditions. The solution of the Langevin equation subject to a Maxwell-Boltzmann distribution of velocities is called the stationary solution. Clearly for the stationary solution... 

Step 3 involved smearing out of the discrete ions of the ionic atmosphere into a continuous cloud of charge so that the Poisson equation could be used. The problem in this step, however, is to find the actual distribution of the discrete ions of the ionic atmosphere around the central y -ion. Statistical mechanics in the form of the Maxwell-Boltzmann distribution is used. The derivation of the Maxwell-Boltzmann distribution automatically involves an implicit averaging process, but this statistical mechanical averaging is different from that used for the Poisson equation when smearing out has been done. 

Finding a relation between the smeared out pj of Poisson s equation and the n, corresponding to the discrete ions of the Maxwell-Boltzmann distribution. 

The problem is to get some device which would substitute the average distribution of the discrete ions in the ionic atmosphere around the centralj-ion, given by n, in the Maxwell-Boltzmann expression, by a continuous charge density which could be taken to be equivalent to pj in the Poisson equation. This would enable Poisson s equation to be combined with a Maxwell-Boltzmann distribution. 

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