A general maximum entropy framework for thermodynamic variational principles

• Published in: AIP conference proceedings Vol. 1636; no. 1
• Main Author: Dewar, Roderick C
• Format: Conference Proceeding Journal Article
• Language: English
• Published: Melville American Institute of Physics 05.12.2014
• Subjects: APPROXIMATIONS; CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Divergence; ENTROPY; EQUILIBRIUM; FREE ENERGY; LAGRANGIAN FUNCTION; Maximum entropy; MEAN-FIELD THEORY; MINIMIZATION; POTENTIALS; Principles; PROBABILITY; Statistical analysis; STATISTICAL MECHANICS; Thermodynamic equilibrium; THERMODYNAMICS; VARIATIONAL METHODS; Variational principles
• ISSN: 0094-243X; 1551-7616
• DOI: 10.1063/1.4903723

Minimum free energy principles are familiar in equilibrium thermodynamics, as expressions of the second law. They also appear in statistical mechanics as variational approximation schemes, such as the mean-field and steepest-descent approximations. These well-known minimum free energy principles are here unified and extended to any system analyzable by MaxEnt, including non-equilibrium systems. The MaxEnt Lagrangian associated with a generic MaxEnt distribution p defines a generalized potential Ψ for an arbitrary probability distribution ̂p, such that Ψ is a minimum at ̂p = p. Minimization of Ψ with respect to ̂p thus constitutes a generic variational principle, and is equivalent to minimizing the Kullback-Leibler divergence between ̂p and p. Illustrative examples of min–Ψ are given for equilibrium and non-equilibrium systems. An interpretation of changes in Ψ is given in terms of the second law, although min–Ψ itself is an intrinsic variational property of MaxEnt that is distinct from the second law.