Difference between thermodynamic entropy and information entropy
It is noticed in Table 2 that the highest SI value was 2.97 observed for Plot I, followed by 2.86 for Plot III and 1.75 for Plot II. It is understandable for Plot II to have the lowest SI value but it is not fully explicable for Plot I to have a higher SI than Plot III. As a commonly applied biodiversity index, SI is thought to be presented as a function of species richness and evenness (Molles, 2016), namely that the higher the level of species richness and evenness, the higher the level of biodiversity. Compared to Plot I, Plot III had a higher maximum SI m = ln(N ) associated with much lower standard deviation SD values forC TB, C TU, Cov ,s , h and f among the species of the community (Table 2). As a statistical parameter given by Eq. 7, SD can be used as an indicator for species evenness. The trend will be that the lower the SD value, the higher the species evenness in related physical quantities. As seen in Table 2, the SD values forC TB, C TU, Cov ,s , h and f were all in confirmative with the on-site field sensation that Plot III had a higher level of species richness and evenness than Plot I. The unique factor left that actually lowered the SI value of Plot III was its highly significant difference in the number of individual plants m iamong species as being indicated by the largest SD value forM (Table 2).
The significant differences between SI i ands i crossing plant species are illustrated in Fig. 2 using the data obtained from the year 2016. The number on the horizontal axis in Fig 2 corresponds to the rank number of species (ranked in an ascending order of m i orSI i) listed in Appendix 1. The generally observed trend was that a great number of species possessing lowerSI i values had relatively highers i values. The typical case in Plot I depicted in Figs. 2a was that the two transplanted tree species (P. fortunei and K. bipinnata ) had extremely lowSI i values (0.0038; 0.0040) associated with the highest s i values (6.2368; 5.5120 ton/hm2). The species that had the highests i value was P. acinosa (0.5142 ton/hm2) in Plot II (Fig. 2b) and B. papyrifera (3.4985 ton/hm2) in Plot III (Fig. 2c) but their SI i values (0.1306; 0.1105) were relatively low. The two transplanted species were not the strongest competitors under natural conditions in this region. Among the native species in Plot III, B. papyrifera was found to be the one possessing the highest s i value though itsSI i value was comparably low (Fig. 2c).
The inconsistence of the variation in m i andC i among species is the reason for the difference between SI i and s i given, respectively, as (m i/M )ln(M /m i) (Eq. 1) andC iln(C T/C i) (Eq. 3). Since most species possessing high C ivalues in Plot I had relatively low m i values, the levels of the correlations of s , h and f withSI crossing species were extremely low as indicated by their correlation coefficient R values (Table 2). The reason for the slightly higher R values of the correlations between SIand s in Plots II and III was attributed to their lower SDvalues in C i among species.
The present study does not give full support to use of SI as a biodiversity index for describing the investigated plant communities simply because SI is a single function of the individual plant number. Biodiversity of an ecosystem should be in principle a macroscopic property or a state function of the ecosystem describing the state of its composition and structure linked to its matter and energy transformation. The s factor defined by Eq. 3 meets this criterion as a biodiversity index as it is not a probability variable but a state function interrelated with another two thermodynamic factorsh and f . It is noted in Table 2 that though theC T value was much higher in Plot I than in Plot III, the s value of Plot I (34.77 ton/hm2) was still lower than that of Plot III (35.05 ton/hm2). This was in agreement with what was observed in the field that Plot I had a higher level of biomass productivity while Plot III had a higher level of biodiversity.