# Biophotons: A Hard Problem
## 3 Statistical Analysis of the Experimental Data
### The Probability Distribution Function Approach  
In a semiclassical picture of the optical detection process, a phototube converts the continuous cycle-averaged classical intensity \(\overline{I}(t)\) into a series of discrete photocounts. Thus, the photocount m obtained in an integration time \(T\) is proportional to the intensity of the light that arrives on the detector [32]. From the experimental data, it possible to obtain the distribution function \(P_{n}(T)\), which counts how many times a number of photons n has been detected by the phototube in a given acquisition time \(T\). Therefore, after an appropriate normalization process, it gives the probability of obtaining m counts in the chosen acquisition time window. This function is analyzed by determining the mean, variance, and other moments of higher order to highlight some statistical properties of emitted light, considering that, at least in some particular cases, there is a direct correspondence between the statistical properties of the light, the functional form of the \(P_{n}(T)\), and some characteristics of the physical process behind the production of the measured light. We do not repeat here the entire theoretical derivation that leads to the determination of \(P_{n}(T)\), but we want to remember that there are only some cases that lead to an analytical form of this function. Details can be found in the literature [27; 33]. The simplest case is a stable classic light wave, where the cycle-averaged intensity has a fixed value independent of the time [33]. In this case, the distribution has a Poissonian form:  
\[P_{n}(T)=\frac{\left\langle n\right\rangle^{n}}{n!}e^{(-\left\langle n\right\rangle)} \tag{5}\]  
where \(\left\langle n\right\rangle=\alpha T\). Factor \(\alpha\) is a constant that depends on the construction characteristics of the phototube. For a Poisson distribution, the variance is equal to the average \(\sigma^{2}=\left\langle n\right\rangle\). It is convenient to define the so-called Fano factor [32]\(F\) through the relationship \(\sigma^{2}=F\left\langle n\right\rangle\) to quantify some type of departure from the Poisson distribution; this could be an indication of a nonclassical nature of the light. A Poisson distribution is a sign of a system in coherent states; in this case, the quantum states correspond to classical electromagnetic waves [27; 32; 33], but this distribution also occurs for experiments where the integration time \(T\) is much longer than the characteristic time of the intensity fluctuations of the light beam. The photocount distribution can be also derived for complete thermal chaotic light [28; 32; 34]. In this case, the distribution takes the form  
\[P_{n}(T,M)=\frac{(n+M-1)!}{n!(M-1)!}\left(1+\frac{M}{\left\langle n\right\rangle }\right)^{-n}\left(1+\frac{\left\langle n\right\rangle}{M}\right)^{-M} \tag{6}\]  
where \(\left\langle n\right\rangle\) is the average number of photons, and \(M\) is the number of field modes. Thermal states are classical, and there is the following relationship between average and variance:  
\[\sigma^{2}=\left\langle n\right\rangle+\frac{\left\langle n\right\rangle^{2}} {M} \tag{7}\]  
In general, the \(M\) coefficient could be huge, in this case, the variance becomes almost equal to the average value, recovering the equation valid for the Poisson distribution. Therefore, for large \(M\) values, the thermal photocount distribution approaches the Poisson distribution. This implies that it is difficult to discriminate between coherent and thermal states when many modes are present, in agreement with the discussion of in Ref. [34]. In references [17; 27], we have already presented a detailed analysis of the experimental data relating to lentils and a single bean. For brevity, we present here only the comparison between the \(P_{n}(T)\) obtained from the experimental data of lentil seeds related to the time window between 20 and 30 h (original time scale in Figure 1) and two fits performed using Equations (5) and (6).