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

Abstract

A novel formalism, called H-theory, is applied to the problem of statistical equilibrium of a hierarchical complex system with multiple time and length scales. In this approach, the system is formally treated as being composed of a small subsystem---representing the region where the measurements are made---in contact with a set of `nested heat reservoirs' corresponding to the hierarchical structure of the system. The probability distribution function (pdf) of the fluctuating temperatures at each reservoir, conditioned on the temperature of the reservoir above it, is determined from a maximum entropy principle subject to appropriate constraints that describe the thermal equilibrium properties of the system. The marginal temperature distribution of the innermost reservoir is obtained by integrating over the conditional distributions of all larger scales, and the resulting pdf is written in analytical form in terms of certain special transcendental functions, known as the Fox $H$-functions. The distribution of states of the small subsystem is then computed by averaging the quasi-equilibrium Boltzmann distribution over the temperature of the innermost reservoir. This distribution can also be written in terms of $H$-functions. The general family of distributions reported here recovers, as particular cases, the stationary distributions recently obtained by Macêdo {\it et al.} [Phys.~Rev.~E {\bf 95}, 032315 (2017)] from a stochastic dynamical approach to the problem.

20 pages, 2 figures

References in corpus (10)

• Superstatistics, thermodynamics, and fluctuations
• Statistics of 3-dimensional Lagrangian turbulence
• Turbulence Hierarchy in a Random Fibre Laser
• Superstatistics in high energy physics: Application to cosmic ray energy spectra and e+e- annihilation
• Beyond Boltzmann-Gibbs statistics: Maximum entropy hyperensembles out-of-equilibrium
• Superstatistical distributions from a maximum entropy principle
• Universality Classes of Fluctuation Dynamics in Hierarchical Complex Systems
• Stochastic Dynamical Model of Intermittency in Fully Developed Turbulence
• Maximum entropy approach to power-law distributions in coupled dynamic-stochastic systems
• Multicanonical distribution: Statistical equilibrium of multiscale systems

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• Turbulence Hierarchy and Multifractality in the Integer Quantum Hall Transition
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• Probability Thermodynamics and Probability Quantum Field