Equilibrium distribution functions

Gorse C and Capitelli M 1996 Non-equilibrium vibrational, electronic and dissociation kinetics in molecular plasmas and their coupling with the electron energy distribution function NATO ASI Series C 482 437-49... 

In tliennal equilibrium at tire temperature T, tire distribution of electrons in tire band is given by tire Fenni-Dirac distribution function f E) = [I where k is tire Boltzmann constant. The function/ E) describes... 

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... 

The probability du> a, r) to find an embryo of size a with its center at point r within the metastable state (treated as the equilibrium oue ) is related to the distribution function /(a,r) as... 

To be more precise, let us assume, as Boltzman first did in 1872 [boltz72], that we have N perfectly elastic billiard balls, or hard-spheres, inside a volume V, and that a complete statistical description of our system (be it a gas or fluid) at, or near, its equilibrium state is contained in the one-particle phase-space distribution function f x,v,t) ... 

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 quantities n, V, and (3 /m) T are thus the first five (velocity) moments of the distribution function. In the above equation, k is the Boltzmann constant the definition of temperature relates the kinetic energy associated with the random motion of the particles to kT for each degree of freedom. If an equation of state is derived using this equilibrium distribution function, by determining the pressure in the gas (see Section 1.11), then this kinetic theory definition of the temperature is seen to be the absolute temperature that appears in the ideal gas law. 

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 results of the three-dimensional random walk, based on the freely-jointed chain, has permitted the derivation of the equilibrium statistical distribution function of the end-to-end vector of the chain (the underscript eq denotes the equilibrium configuration) [24] ... 

For most systems in thermal equilibrium, it is sufficient to regard fB as random forces which follows a Gaussian distribution function with mean value = 0 and standard deviation = 2kBT 8(i — j) 8(t — t ) [44],... 

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 ... 

With this distribution function, velocity factors out of Equation 20 yielding the exact YBG equation, Equation 3, for equilibrium fluids whose Interaction potential Is given by Equation 1. 

The Chapman-Enskog method has been used to solve for steady state tracer diffusion (. ). According to the method the singlet distribution function for the diffusing species 1, present In a trace amount n nj, 1 1) In an otherwise equilibrium fluid. Is approximated by... 

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