Internal energies, canonical distribution

In view of the ergodic hypothesis the average value of an observable property may be regarded as the quantity measured under specified conditions. In this way the internal energy of a system corresponds to the average energy of the canonical distribution ... 

The explanation for this paradox is that a canonical or thermal system maintains the equilibrium distribution of internal energy states through collisions. Thus, Eq. (1.8) does not describe the time behavior of a canonical ensemble. What it docs describe is a... 

It is important to remember, in this context, that the clusters leave the thermalization stage with a canonical distribution of internal energies. Afterwards they do not make any collisions and have to be treated microcanonically. 

If k E) is averaged over a thermal internal energy distribution at temperature T, the equation of transition state theory for the canonical rate constant is recovered. It can be expressed as a function of the activation entropy, AS° and of the activation enthalpy,... 

E = total energy h = Planck constant kg = Boltzmeinn constant k(B) = unimolecular micro canonical rate constant k(7) = canonical rate con stant Nq = initial number of ions formed M E - E< = number of states of the transition state up to - Elj above the critical energy E P E) = distribution of internal energies R(E,tj = rate of dissociation T = temperature Af = activation enthalpy A5 = activation entropy A5j = microcanonical entropy of activation Vr = reaction coordinate frequency p E) = density of states of the parent ion at internal energy E o= degeneracy of reaction path. 

A series of Monte Carlo computer simulation studies of the structure and properties of molecular liquids and solutions have recently been carried out in this Laboratory.The calculations employ the canonical ensemble Monte Carlo-Metropolis method based on analytical pairwise potential functions representative of ab initio quantum mechanical calculations of the intennolecular interactions. A number of thermodynamic properties including internal energies and radial distribution functions were determined and are reported herein. The results are analyzed for the structure of the statistical state of the systems by means of quasicomponent distribution functions for coordination number and binding energy. Significant molecular structures contributing to the statistical state of each system are identified and displayed in stereographic form. 

Since 2 and Zoan are constants with respect to the energy, the canonical microstate probability Pi is an exponentially decaying function of the microstate energy, in contrast to the uniformly distributed microstate probability in the microcanonical ensemble. The internal... 

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