Abstract Theory of Bogoliubov Linearizations with Application to Nonlinear Thermodynamic Formalism

Bogoliubov's 1947 approximation, originally developed in the microscopic theory of superfluidity, laid the foundation for solving previously intractable quantum models and later became part of "quantum mathematics". Regarding mathematically rigorous results, one of its most advanced forms - the only one that handles quantum equilibrium states - was published in the Memoirs of the AMS in 2013. Building on key results from convex analysis, the present work significantly extends it to obtain a general mathematical theory that enables nonlinear variational problems on convex compact spaces to be fully studied via a linearization process, referred to here as the "Bogoliubov linearization". This problem is particularly timely, given the current development of quantum algorithms and computers, which are inherently linear machines. A deep connection with the optimal transport is also proven. As a paradigmatic example of application, the approach proposed here is applied to the nonlinear thermodynamic formalism - an emerging field that can have important impacts on various fields of mathematics, such as ergodic transport, the fractals and multifractal formalism, discrete-time linear dynamics, C*-algebras, etc. Notably, even in the case of finite alphabets the obtained results go beyond the scope of the existing literature in nonlinear thermodynamic formalism.