Information and complexity measures in the interface of a metal and a superconductor

Fisher information, Shannon information entropy and Statistical Complexity are calculated for the interface of a normal metal and a superconductor, as a function of the temperature for several materials. The order parameter \(\Psi({\bf r})\) derived from the Ginzburg-Landau theory is used as an input together with experimental values of critical transition temperature \(T_c\) and the superconducting coherence length \(\xi_0\). Analytical expressions are obtained for information and complexity measures. Thus \(T_c\) is directly related in a simple way with disorder and complexity. An analytical relation is found of the Fisher Information with the energy profile of superconductivity i.e. the ratio of surface free energy and the bulk free energy. We verify that a simple relation holds between Shannon and Fisher information i.e. a decomposition of a global information quantity (Shannon) in terms of two local ones (Fisher information), previously derived and verified for atoms and molecules by Liu et al. Finally, we find analytical expressions for generalized information measures like the Tsallis entropy and Fisher information. We conclude that the proper value of the non-extensivity parameter \(q\simeq 1\), in agreement with previous work using a different model, where \(q\simeq 1.005\).