Boltzman probability function

Becke and Roussel (BR) functional, 185 Bell, Evans, Polanyi (BEP) principle, 364 Bell correction, tunnelling, 391 Bending energy, in force fields, 11 BLYP function, 188 Boltzman probability function, 374 Bond critical points, electron density analysis, 225... 

The Boltzman probability distribution function P may be written either in a discrete energy representation or in a continuous phase space formulation. 

Using the fact that we have a well defined energy function (equation 10.9), we know from statistical mechanics that when the system has reached equilibrium, the probability that it is in some state S = Si, S, , Sm) is given by the Boltzman distribution ... 

There are two basic approaches to the computer simulation of liquid crystals, the Monte Carlo method and the method known as molecular dynamics. We will first discuss the basis of the Monte Carlo method. As is the case with both these methods, a small number (of the order hundreds) of molecules is considered and the difficulties introduced by this restriction are, at least in part, removed by the use of artful boundary conditions which will be discussed below. This relatively small assembly of molecules is treated by a method based on the canonical partition function approach. That is to say, the energy which appears in the Boltzman factor is the total energy of the assembly and such factors are assumed summed over an ensemble of assemblies. The summation ranges over all the coordinates and momenta which describe the assemblies. As a classical approach is taken to the problem, the summation is replaced by an integration over all these coordinates though, in the final computation, a return to a summation has to be made. If one wishes to find the probable value of some particular physical quantity, A, which is a function of the coordinates just referred to, then statistical mechanics teaches that this quantity is given by... 

To describe polydispersed multiphase systems the Boltzmann equation can be extended by including the dependency of the internal property coordinates such as the particle size and shape in the definition of the distribution function. In this way a statistical balance formulation can be obtained by means of a distribution function on the form p, r,v, c,t)d dr dv dc, defined as the probable number of particles with internal properties in the range about with a velocity range in property space dv about v, located in the spatial range dr about the position r, with a velocity range dc about c, at time t. The particular Boltzman type of equation is given by ... 

---