Rényi entropy, abundance distribution, and the equivalence of ensembles

Distributions of abundances or frequencies play an important role in many fields of science, from biology to sociology, as does the Rényi entropy, which measures the diversity of a statistical ensemble. We derive a mathematical relation between the abundance distribution and the Rényi entropy, by analogy with the equivalence of ensembles in thermodynamics. The abundance distribution is mapped onto the density of states, and the Rényi entropy to the free energy. The two quantities are related in the thermodynamic limit by a Legendre transform, by virtue of the equivalence between the micro-canonical and canonical ensembles. In this limit, we show how the Rényi entropy can be constructed geometrically from rank-frequency plots. This mapping predicts that non-concave regions of the rank-frequency curve should result in kinks in the Rényi entropy as a function of its order. We illustrate our results on simple examples, and emphasize the limitations of the equivalence of ensembles when a thermodynamic limit is not well defined. Our results help choose reliable diversity measures based on the experimental accuracy of the abundance distributions in particular frequency ranges.

Figure 1

Geometric construction of the Rényi entropy from the density of states. In the classical Legendre construction, the free energy $F_{\beta }$ is obtained as the intersection of the tangent to the micro-canonical entropy curve $S\left(E\right)$ (in red) of slope $\beta$, where $\beta$ is the inverse temperature, and the abscissa. The Rényi entropy of order $\beta , H_{\beta }$, is obtained as the intersection between the tangents of slope 1 and $\beta$, projected onto the ordinate. Inset: the micro-canonical entropy curve is equivalent, up to a ${90}^{\circ }$ rotation, to the rank-frequency curve represented on a logarithmic scale.

Figure 2

Illustration of the Legendre construction of the Rényi entropy on the rank-frequency curve of randomly generated T-cell receptor $\beta$ chains [28]. The construction is identical to that of Fig. 1, with slope $\beta$ replaced by slope $-{\beta }^{-1}$ because of the rotation. The projection onto the rank axis gives an approximation to the diversity index of order $\beta , D_{\beta }=exp\left[H_{\beta }\right]$.