2022/07/02 by Olivier Peltre, Peltre, Olivier
Physics and Astronomy · #Statistical Mechanics and Entropy #Advanced Thermodynamics and Statistical Mechanics #Quantum many-body systems
paper · pdf · doi:10.48550/arxiv.2207.00841
A statistical system is classically defined on a set of microstates E by a global energy function H : E → ℝ, yielding Gibbs probability measures (softmins) ρβ(H) for every inverse temperature β= T-1. Gibbs states are simultaneously characterized by free energy principles and the max-entropy principle, with dual constraints on inverse temperature β and mean energy \cal U(β) = 𝔼ρβ[H] respectively. The Legendre transform relates these diverse variational principles which are unfortunately not tractable in high dimension. The global energy is generally given as a sum H(x) = ∑\rm a ⊂ Ω h\rm a(x|\rm a) of local short-range interactions h\rm a : E\rm a → ℝ indexed by bounded subregions \rm a ⊂ Ω, and this local structure can be used to design good approximation schemes on thermodynamic functionals. We show that the generalized belief propagation (GBP) algorithm solves a collection of local variational principles, by converging to critical points of Bethe-Kikuchi approximations of the free energy F(β), the Shannon entropy S(\cal U), and the variational free energy \cal F(β) = \cal U - β-1 S(\cal U), extending an initial correspondence by Yedidia et al. This local form of Legendre duality yields a possible degenerate relationship between mean energy \cal U and β.