Canonical statistical weight

I will now compute the most probable value of the freeon quantum number, p by means of the canonical partition function. Here it is finally necessary to acknowledge the existence of the conjugate spin space whose dimension (statistical weight) is... 

To construct an ensemble conjugate to the MD ensemble, we have to introduce a second parameter which gives 3 a nonzero average value. This is the generalized canonical ensemble, whose statistical weight function is... 

With regard to the motion of the electrons, we choose as energy origin the fundamental level in the atom, and these electrons are not excited, so that the contribution to the atomic partition function of electrons is reduced to its statistical weight, according to the equation z = go, and therefore the electron contribution to the canonical partition function will be ... 

Integrating this expression over either momenta or coordinates shows that the distribution is also canonical in both configurations (i.e., configurations are sampled with a statistical weight proportional to and momenta (i.e., momenta are sampled with a statistical weight proportional to e ). In Cartesian coordinates, such a... 

Strictly speaking, the proper definition of the coexistence point in the canonical statistical analysis is the temperature, where the weights under the two peaks are equal, see, e.g.. Refs. [319,320]. 

In a grand canonical ensemble, the numbers of molecules of species. vt are fixed only on average, and statistical mechanics shows that the weight... 

Reiteration of such a procedure gives a Markov chain of molecular configurations distributed in the phase space of the system, with the probability density proportional to the Boltzmann weight factor corresponding to the canonical NVT statistical ensemble. 

To preserve Boltzmann statistics in the canonical ensemble, the relationship between the weight fnnctions, (r) = [w (r)] results in the following fluctuation-dissipation relation (Espafiol and Warren 1995) ... 

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