Modelling with Boltzmann Distribution

As mentioned early, the experimental distributions are described by the sum of two functions:

f(kt |ma)= f1(kt)+f2(kt), (3.27)

subject to the normalisation condition:

/ft |ma) dkt = 1. (3.28)


The f1 and f2 functions are obtained from the Boltzmann statistic:

This function yields unimodal symmetrical curves around kt0i, where the function reaches its maximum. A; determines the function height and X; is related to the width of the distribution function. This can be integrated to:

and this can be analytically inverted:

These characteristics allow the generation of synthetic data of instantaneous val­ues from the CDFs by methods of inferential statistics. Also it is possible to obtain explicitly the kt coefficients.

The coefficients A1 and A2 must satisfy the normalisation condition:

When modelling this dependence for the data of Armilla (Granada, Spain), we found for the maxima distributions, kt01 and kt02, the following expressions:

kt01 = 0.763 -0.0152ma -0.012m2, withR2 = 0.996, (3.33)

kt02 = 0.469 -0.0954ma + 0.01m2, withR2 = 0.992, (3.34)

where R2 is understood as the proportion of response variation “explained” by the parameters in the model.

The position of the principal maximum, kt01, shifts towards lower values as the optical air mass increases. The same trend occurs for the value of kt02, corresponding to the second maximum of the distribution. However, the shift is smaller than that associated with the principal maximum kt01, as it can be concluded by comparing the coefficients of the optical air mass terms in each equation. This implies that, when the optical air mass increases, the two maxima tend to be closer.

The values of the width parameters, X1 and X2, can also be expressed in terms of the optical air mass:

X1 = 91.375 -40.092ma + 6.489m2, withR2 = 0.999, (3.35)

X2 = 6.737 + 1.248m + 0.4246m2, withR2 = 0.975. (3.36)

The coefficient A1 has been fitted using the following expression:

A1 = 0.699 + 0.1217m-21416, withR2 = 0.994. (3.37)

Considering the A1 and A2 dependence (A1 + A2 = 1, because of the normalisa­tion condition), it is obvious that while A1 decreases with air mass, A2 shows the opposite trend. The ratio between the intensity of the two peaks depends on ma. This ratio decreases when ma increases, that is a decrease in ma implies an enhancement of the first maximum relative to the second one.

Figure 3.14 shows the fitting curves using both, the Suehrcke and McCormick’s model and the Tovar’s model based on the Boltzmann statistics. Figure 3.14a shows the case of the best adjustment provided by the Suehrcke-McCormick model for data collected in Armilla (Granada), adapting conveniently the parameters to fit the maxima of the distribution. Figure 3.14b shows the Tovar model adjustment for the same set. Figure 3.14c shows the results by applying the Tovar model to the

data collected in COrdoba. It can be observed that the Boltzmann model provides a reasonable adjustment (Tovar et al. 1998a; Varo et al. 2006). Nevertheless, the maxima of the bimodal distribution depend on the location and its climatic features. Therefore, the fitting parameters inEqs. (3.33-3.37) will also depend on the location and its climatic features. However, there are some common characteristics for all the functions used to fit the distributions.

Updated: July 31, 2015 — 10:19 pm