When do Gibbsian phase averages and Boltzmannian equilibrium values agree?. (November 2020)

• Record Type: Journal Article
• Authors: Werndl, Charlotte; Frigg, Roman
• Is Part Of: Studies in history and philosophy of modern physics. Volume 72(2020)
• Page Start: 46
• Page End: 69
• Publication Date: 2020-11
• Subjects: Gibbs -- Equilibrium -- Boltzmann -- Phase averages -- Statistical mechanics; Physics -- Philosophy -- Periodicals; Physics -- History -- Periodicals; 530.01
• DOI: 10.1016/j.shpsb.2020.05.003
• Languages: English
• ISSNs: 1355-2198

Abstract

Abstract: This paper aims to shed light on the relation between Boltzmannian statistical mechanics and Gibbsian statistical mechanics by studying the Mechanical Averaging Principle, which says that, under certain conditions, Boltzmannian equilibrium values and Gibbsian phase averages are approximately equal. What are these conditions? We identify three conditions each of which is individually sufficient (but not necessary) for Boltzmannian equilibrium values to be approximately equal to Gibbsian phase averages: the Khinchin condition, and two conditions that result from two new theorems, the Average Equivalence Theorem and the Cancelling Out Theorem. These conditions are not trivially satisfied, and there are core models of statistical mechanics, the six-vertex model and the Ising model, in which they can fail. Highlights: The paper compares Boltzmannian equilibrium values with Gibbsian phase averages. Boltzmann equilibrium values and Gibbsian phase averages sometimes come apart. Conditions are identified when Boltzmann equilibrium values and phase averages agree.