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A novel formalism, called H-theory, is applied to the problem of statisticalequilibrium of a hierarchical complex system with multiple time and lengthscales. In this approach, the system is formally treated as being composed of asmall subsystem---representing the region where the measurements are made---incontact with a set of `nested heat reservoirs' corresponding to thehierarchical structure of the system. The probability distribution function(pdf) of the fluctuating temperatures at each reservoir, conditioned on thetemperature of the reservoir above it, is determined from a maximum entropyprinciple subject to appropriate constraints that describe the thermalequilibrium properties of the system. The marginal temperature distribution ofthe innermost reservoir is obtained by integrating over the conditionaldistributions of all larger scales, and the resulting pdf is written inanalytical form in terms of certain special transcendental functions, known asthe Fox $H$-functions. The distribution of states of the small subsystem isthen computed by averaging the quasi-equilibrium Boltzmann distribution overthe temperature of the innermost reservoir. This distribution can also bewritten in terms of $H$-functions. The general family of distributions reportedhere recovers, as particular cases, the stationary distributions recentlyobtained by Mac\^edo {\it et al.} [Phys.~Rev.~E {\bf 95}, 032315 (2017)] from astochastic dynamical approach to the problem.

A maximum entropy approach to H-theory: Statistical mechanics of hierarchical systems