Distribution functions local equilibrium

As it was already written above, we would like to study structural changes in the charge distribution between macroscopic objects, that is caused by the image forces, and depends on the wall-to-wall distance. To obtain direct structural information about the system, we will introduce a configurational analogue of the phase-space distribution function. At equilibrium, the definition of an fth order distribution function given by Eq. (12) can be applied to the equilibrium probability density [Eq. (13)], and the integration with respect to impulses can easily be carried out. We write for the rth order local density... 

In order to solve the conservation or transport equations (mass, momentum, energy, and entropy) in terms of the dependent variables n, Vo,U, and , we must further resolve the expressions for the flux vectors— P, q, and s and entropy generation Sg. This resolution is the subject of closure, which will be treated in some detail in the next chapter. However, as a matter of illustration and for future reference, we can resolve the flux vector expression for what is called the local equilibrium approximation, i.e., we assume that the iV-molecule distribution function locally follows the equilibrium form developed in Chap. 4, i.e., we write [cf Eq. (4.34)]... 

If it cannot be guaranteed that the adsorbate remains in local equilibrium during its time evolution, then a set of macroscopic variables is not sufficient and an approach based on nonequihbrium statistical mechanics involving time-dependent distribution functions must be invoked. The kinetic lattice gas model is an example of such a theory [56]. It is derived from a Markovian master equation, but is not totally microscopic in that it is based on a phenomenological Hamiltonian. We demonstrate this approach... 

A gas is not in equilibrium when its distribution function differs from the Maxwell-Boltzman distribution. On the other hand, it can also be shown that if a system possesses a slight spatial nonuniformity and is not in equilibrium, then the distribution function will monotonically relax in velocity space to a local Maxwell-Boltzman distribution, or to a distribution where p = N/V, v and temperature T all show a spatial dependence [bal75]. 

Assuming that our LG is in a local equilibrium, it is reasonable to expect that the one-particle distribution functions should depend only on the macroscopic parameters u x,t) and p x,t) and their derivatives [wolf86c]. While there is no reason to believe that this dependence should be a particularly simple one, it is reasonable to expect that both u and p are slowly varying functions of x and t. Moreover, in the subsonic limit, we can assume that li << 1. 

The Burnett Expansion.—The Chapman-Enskog solution of the Boltzmann equation can be most easily developed through an expansion procedure due to Burnett.15 For the distribution function of a system that is close to equilibrium, we may use as a zeroth approximation a local equilibrium distribution function given by the maxwellian form ... 

The Rouse model, as given by the system of Eq, (21), describes the dynamics of a connected body displaying local interactions. In the Zimm model, on the other hand, the interactions among the segments are delocalized due to the inclusion of long range hydrodynamic effects. For this reason, the solution of the system of coupled equations and its transformation into normal mode coordinates are much more laborious than with the Rouse model. In order to uncouple the system of matrix equations, Zimm replaced S2U by its average over the equilibrium distribution function ... 

These hard forces are thus functions of the soft coordinates q alone, and completely independent of the values of c, ..., c. This is because, for this local-equilibrium distribution, the rapid variation of the hard mechanical force —dV/dc along the hard directions is canceled by a compensating variation in the Brownian contribution to... 

The Wigner distribution function for the vibrational ground state of the harmonic oscillator is the product of two Gaussians, one Gaussian in P-space centered at the equilibrium distance Re and one Gaussian in P-space localized at P = 0. 

For a gas at equilibrium, i.e., no mean deformation, there is a spatial homogeneity and thus g(ri,r2) depends only on the separation distance r = ri — r2. Then g = go(r) is termed the radial distribution function, which may be interpreted as the ratio of the local number density at a distance r from the central particle to the bulk number density. For a system of identical spheres, the radial distribution function go O ) at contact (i.e., r = dp) can be expressed in terms of the volume fraction of solids ap as... 

The traditional apparatus of statistical physics employed to construct models of physico-chemical processes is the method of calculating the partition function [17,19,26]. The alternative method of correlation functions or distribution functions [75] is more flexible. It is now the main method in the theory of the condensed state both for solid and liquid phases [76,77]. This method has also found an application for lattice systems [78,79]. A new variant of the method of correlation functions - the cluster approach was treated in the book [80]. The cluster approach provides a procedure for the self-consistent calculation of the complete set of probabilities of particle configurations on a cluster being considered. This makes it possible to take account of the local inhomogeneities of a lattice in the equilibrium and non-equilibrium states of a system of interacting particles. In this section the kinetic equations for wide atomic-molecular processes within the gas-solid systems were constructed. 

Before the transfer starts, the energy distribution of electrons takes the form of a Fermi-Dirac distribution function. While the number of electrons is decreasing steadily with time, the distribution of electrons keep the form of a Fermi-Dirac distribution function. This constancy of the distribution is due to the fact that the capture rate of free electrons by the localized states is much faster than the loss of free electrons caused by the transfer when the occupation probability of localized states is not approximately one. Therefore, electrons are considered to be in their quasi-thermal equilibrium condition i.e., the energy distribution of electrons is described by quasi-Fermi energy EF. Then the total density t of electrons captured by the localized states per unit volume can be written as... 

The kinetic theory leads to the definitions of the temperature, pressure, internal energy, heat flow density, diffusion flows, entropy flow, and entropy source in terms of definite integrals of the distribution function with respect to the molecular velocities. The classical phenomenological expressions for the entropy flow and entropy source (the product of flows and forces) follow from the approximate solution of the Boltzmann kinetic equation. This corresponds to the linear nonequilibrium thermodynamics approach of irreversible processes, and to Onsager s symmetry relations with the assumption of local equilibrium. 

The distribution function/(v) is Maxwellian at local equilibrium, and is defined by... 

This section is devoted to the calculation of the desorption kinetics from a heterogeneous surface characterized by a desorption energy distribution function (peq(E) given by Eq. (9) with E = q, i.e. of the desorption kinetics from a surface which in equilibrium conditions obeys the Freundlich or Temkin isotherm. The local desorption kinetics will be assumed to be of the first order. 

We assume local equilibrium. This means that the distribution function /(r, c,t) does not vary appreciably during a time interval of the order of the duration of a molecular collision, nor does it vary appreciably over a spatial distance of the order of the range of intermolecular forces. 

Distribution Function of Adsorption Energy. From chromatographic data it is possible to relate the amount of solute adsorbed on a solid to the equilibrium pressure and thus to plot its adsorption isotherm. For a heterogeneous surface, the experimentally measured adsorption isotherm can be described as a sum of local isotherms corresponding to different surface-active sites. The isotherm can then be represented by the following integral equation ... 

In describing thermodynamic and equilibrium statistical-mechanical behaviors of a classical fluid, we often make use of a radial distribution function g r). The latter for a fluid of N particles in volume V expresses a local number density of particles situated at distance r from a fixed particle divided by an average number density p = NjV), when the order of IjN is negligible in comparison with 1. Various thermodynamic quantities are related to g(r). For a single-component monatomic system of particles interacting with a pairwise additive potential 0(r), the relationship connecting the pressure P to g(r) is the virial theorem, ... 

---