Boltzmann weighting factors

We now briefly consider finite size effects at first-order phase transitions. The easiest case is transitions driven by the field conjugate to the order parameter in systems at T < T. Then Eq. (40) is easily generalized by introducing Boltzmann weight factors for the two states < according to their Zeemann energies... 

Small oscillatory torsions of molecular units within the segment produce a de-localization of the electronic density according to the Debye-Waller effect . In a very distorted structure as in the melt its smooth intensity distribution is not affected to a measurable extent. Rotational isomers, on the other hand, induce remarkable changes of the ru involved. Moreover, it is easily shown that the Boltzmann weighting factor of the molecular conformation ensemble also has a substantia] influence on the ru. It is strictly not possible to calculate the... 

Fig. 13. Mossbauer spectra and computed Mossbauer spectra for reduced spinach ferredoxin at 4.3 °K. (a) 580 G (b) 46 kG. Applied magnetic fields are parallel to gam-maray direction. Velocities are relative to platinum source matrix. Boltzmann weighting factor for electronic states = — 0.26...

A polar molecule such as HC1 possesses a permanent dipole moment // by virtue of the non-uniform electric charge distribution within the neutral molecule. The electrostatic energy between two interacting dipoles //1 and 112 is strongly dependent on their relative orientation. If all relative orientations are equally probable and each orientation carries the Boltzmann weighting factor e-cW r, the following expression is obtained ... 

The canonical probability distribution of potential energy Pnvt( o T) is then given by the product of the density of states n(E) and the Boltzmann weight factor Wb(E T) ... 

In the multicanonical ensemble [11,12], on the other hand, each state is weighted by a non-Boltzmann weight factor Wmn E) (which we refer to as the multicanonical weight factor), so that a uniform potential energy distribution Pmu E) is obtained ... 

The interaction of the carbon dioxide molecule with the sieve includes electrostatic, induction, dispersion, and repulsion contributions. The CO2 molecule was assumed to be capable of free rotation, so that the directional interactions could be averaged over all orientations using a Boltzmann weighting factor (JJ) this causes the electrostatic ion-quadrupole interaction to depend on the temperature. Mean values were used for the polarizability (a), the diamagnetic susceptibility (x), and the equilibrium radius of the CO2 molecule. Using vector summation for the total electric field at the CO2 molecule, the total potential, c(r), at a given position r is given by ... 

For a system of N atoms or molecules, an instantaneous configuration represents a microstate in the phase space of the system (position and momentum), characterized by the set of state variables a (denoted as a for brevity). The probability for a system being in a given microstate is given by its Boltzmann weighting factor... 

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. 

Performing the summation in (8.13) with the Boltzmann weighting factors amounts to evaluating an ensemble average. Using the subscript zero to denote the ensemble average of a matrix element, we then have... 

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