Geometry of basic statistical physics mapping (Q2826712)

A main formula of statistical physics \(F=-kT\ln(\sum_{n}e^{-E_{n}/kT})\) suggests the study of the general mapping defined by \(F=\ln(\sum_{j=1}^me^{f_j(x_1,\dots,x_n)})\) with real-valued functions \(f_j\) of \(n\) real variables, and arbitrary \(n,m\). A main aim of the present paper is to investigate geometrical objects relating to the defined mapping and probabilities \(w_j=\frac{e^{f_j(x)}}{\sum_ke^{f_k(x)}}\), \(j\in\{1,\dots,m\}\). In the paper, the induced metric, the Riemann curvature tensor, the Gauss-Kronecker curvature and the associated entropy are calculated. Some special class of ideal statistical hypersurfaces is analyzed as well. Non-ideal hypersurfaces and their singularities similar to those of the phase transitions are considered too. Moreover, the authors discuss a tropical limit of statistical hypersurfaces and double scaling tropical limits.