The Boltzmann factor

As discussed in Section 3.10, Boltzmann recognized that there was a close link between the entropy of a system and the number of microstates that comprise it which he expressed 

As we have seen in the previous section the entropy change can also be expressed in terms of the number of microstates that make up the system. The small change in entropy, dS, that results from moving a molecule from the Oth to the ith state in a very large system is given by 

We call e ei/feT the Boltzmann factor of the ith energy state and the equation the Boltzmann distribution. It is one of the most important relations in physical science and provides great insight into the systems we deal with in physical chemistry. 

The Boltzmann factor tells us that at low temperatures all molecules will tend to lie in the lowest possible state whereas at very high temperatures the distribution over the energy states tends to be uniform (Fig. 9.5). The energy of a system tends to zero at the absolute zero of temperature and, as all particles then lie in the lowest state, the number of complexions will be unity and S = Jfcln W = 0. This is the basis for the Third Law of Thermodynamics (Section 5.7). We can write the Boltzmann distribution in terms of the probability of an energy state being occupied. If the total number of molecules is iV then the probability of any particular molecule occupying the 

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The idea may be illustrated by considering first a method for increasing the acceptance rate of moves (but at the expense of trying, and discarding, several other possible moves). Having picked an atom to move, calculate the new trial interaction energy for a range of trial positions t = 1.. . k. Pick the actual attempted move from this set, with a probability proportional to the Boltzmann factor. This biases the move selection. 

From the potential energy, calculate the Boltzmann factor, exp(—iC(r )/cBT). 

Add the Boltzmann factor to the accumulated sum of Boltzmann factors and the potent energy contribution to its accumulated sum and return to step 1. 

The density of states increases rapidly with energy but the Boltzmann factor decrease exponentially, meaning that Pcanon(T, E) is bell-shaped, with values that can vary by mar orders of magnitude as the energy changes. In the multicanonical method the simulatic... 

The effect of temperature in Monte Carlo simulations is primarily to modulate the strength of intermolecular interactions, since temperature enters the simulation only through the Boltzmann factor exp(-AE/kT), where AErepresents a difference in potential... 

In this expression, cos 0 is the average value of cos 0 the weighting factor used to evaluate the average is given by the Boltzmann factor exp(-V /RT), where R is the gas constant in the units of and T is in degrees Kelvin. Note that the correction factor introduced by these considerations reduces to unity if... 

The statistical problem. The relative probability associated with the placement of the two coils such that d = °° is unity The solution is so dilute that we can place the polymer molecules anywhere. It is at smaller d s that the placement probability drops off because of a generally unfavorable AG associated with the overlap. We assume that the decrease in probability is described by the Boltzmann factor and write... 

If the spectmm is observed in absorption, as it usually is, and at normal temperatures the intensities of the transitions decrease rapidly as v" increases, since the population of the uth vibrational level is related to Nq by the Boltzmann factor... 

There is considerable literature on material imperfections and their relation to the failure process. Typically, these theories are material dependent flaws are idealized as penny-shaped cracks, spherical pores, or other regular geometries, and their distribution in size, orientation, and spatial extent is specified. The tensile stress at which fracture initiates at a flaw depends on material properties and geometry of the flaw, and scales with the size of the flaw (Carroll and Holt, 1972a, b Curran et al., 1977 Davison et al., 1977). In thermally activated fracture processes, one or more specific mechanisms are considered, and the fracture activation rate at a specified tensile-stress level follows from the stress dependence of the Boltzmann factor (Zlatin and Ioffe, 1973). 

This means that particle configurations where at least two particles overlap, i.e., have a distance r smaller than the diameter cr, are forbidden. They are forbidden because the Boltzmann factor contains a term, exp(—oo) 0, that leads to a vanishing statistical weight. Hence we have an ensemble of... 

We assume that exploring all possible forms for the fields corresponds to exploring the overall usual phase space. To determine the partition function Z the contributions from all the p+ r) and P- r) distributions are summed up with a statistical weight, dependent on p+ r) and p (r), put in the form analogous to the Boltzmann factor exp[—p (F)]], where the effective Hamiltonian p (F)] is a functional of the fields. The... 

Since the equilibrium probability Ed.s, t) contains the Boltzmann factor with an energy Tid.s, ), the condition (12) leads to the ratio of transition probabilities of the forward and backward processes as... 

In the PPF, the first factor Pi describes the statistical average of non-correlated spin fiip events over entire lattice points, and the second factor P2 is the conventional thermal activation factor. Hence, the product of P and P2 corresponds to the Boltzmann factor in the free energy and gives the probability that on<= of the paths specified by a set of path variables occurs. The third factor P3 characterizes the PPM. One may see the similarity with the configurational entropy term of the CVM (see eq.(5)), which gives the multiplicity, i.e. the number of equivalent states. In a similar sense, P can be viewed as the number of equivalent paths, i.e. the degrees of freedom of the microscopic evolution from one state to another. As was pointed out in the Introduction section, mathematical representation of P3 depends on the mechanism of elementary kinetics. It is noted that eqs.(8)-(10) are valid only for a spin kinetics. 

Here the relative intensities of the components of each branch are determined by the Boltzmann factor Correlation function K (t, J), corresponding to Gq(a>, J), is obviously the correlation function of a transition matrix element in Heisenberg representation... 

Clearly, if a situation were achieved such that exceeded Np, the excess energy could be absorbed by the rf field and this would appear as an emission signal in the n.m.r. spectrum. On the other hand, if Np could be made to exceed by more than the Boltzmann factor, then enhanced absorption would be observed. N.m.r. spectra showing such effects are referred to as polarized spectra because they arise from polarization of nuclear spins. The effects are transient because, once the perturbing influence which gives rise to the non-Boltzmann distribution (and which can be either physical or chemical) ceases, the thermal equilibrium distribution of nuclear spin states is re-established within a few seconds. 

Because Pdb(oP,o, Pe,oi) is a higher-order polynomial of the Boltzmann factor, Equation 14.2 cannot have the same structure as Equation 14.1 that is, rewriting Equation 14.2 as... 

Fig. 9.34 Monitoring of inelastic excitations by nuclear resonant scattering. The sidebands of the excitation probability densities for phonon creation, S(E), and for annihilation, S —E), are related by the Boltzmann factor, i.e., S(—E) = S E) tTvp —Elk T). This imbalance, known as detailed balance, is an intrinsic feature of each NIS spectrum and allows the determination of the temperature T at which the spectrum was recorded...

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