Assembly, canonical statistical

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... 

RRKM theory is the well-known and consolidated statistical theory for unimolecular dissociation. It was developed in the late 1920s by Rice and Ramsperger [141, 142] and Kassel [143], who treated a system as an assembly of s identical harmonic oscillators. One oscillator is truncated at the activation energy Eq. The theory disregards any quantum effect and the approximation of having all identical is too cmde, such that the derived equation for micro canonical rate constant, k(E),... 

If the forces and fluxes are properly selected, they are canonically conjugated in the sense of irreversible thermodynamics (Lf/ = L ) (Onsager relation). There are two distinctly different kinds of forces which may induce a flux. The first and more familiar is the electrical force. The second kind of forces has its origin in statistics it is not a force in the usual sense, but normally arises from concentration gradients and results in diffusive fluxes. In all our discussions we shall be concerned with an assembly of electrons, i.e. a Fermi system, for which the chemical potential (ji(T)) at r = 0 K is the Fermi energy. Then it is common practice to introduce the electrochemical potential, and to define an effective electric field E = — V(d> + pje), which is the field that is normally observed. 

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