Reconceptualising equilibrium in Boltzmannian statistical mechanics and characterising its existence

Abstract

In Boltzmannian statistical mechanics macro-states supervene on microstates. This leads to a partitioning of the state space of a system into regions of macroscopically indistinguishable micro-states. The largest of these regions is singled out as the equilibrium region of the system. What justifies this association? We review currently available answers to this question and find them wanting both for conceptual and for technical reasons. We propose a new conception of equilibrium and prove a mathematical theorem which establishes in full generality -i.e. without making any assumptions about the system's dynamics or the nature of the interactions between its components -that the equilibrium macro-region is the largest macro-region. We then turn to the question of the approach to equilibrium, of which there exists no satisfactory general answer so far. In our account, this question is replaced by the question when an equilibrium state exists. We prove another -again fully general -theorem providing necessary and sufficient conditions for the existence of an equilibrium state. This theorem changes the way in which the question of the approach to equilibrium should be discussed: rather than launching a search for a crucial factor (such as ergodicity or typicality), the focus should be on finding triplets of macro-variables, dynamical conditions, and effective state spaces that satisfy the conditions of the theorem.

Boltzmannian Equilibrium in Stochastic Systems

“…Hence Q Mα-ε-eq is an α-ε-equilibrium of (X, Σ X , µ X , T t ). It follows from the (deterministic) Dominance Theorem (Frigg and Werndl 2015a) that µ X (Q Mα-ε-eq ) > α(1 − ε), which immediately implies that P {M α-ε-eq } > α(1 − ε).…”

When do Gibbsian phase averages and Boltzmannian equilibrium values agree?

“…However, it is now recognised that combinatorial considerations do not provide a general definition of equilibrium (see Uffink, 2007, andWerndl andFrigg, 2015a, for discussions). We therefore work with the time-average conception of equilibrium, which has recently been proposed by Werndl and Frigg (2015a, 2015b, 2017b. This conception is free of the restriction faced by the combinatorial argument and provides a fully general definition of equilibrium.…”

Of Barrels and Pipes: Representation-as in Art and Science

“…But not all targetless models are remnants of failed scientific projects. Models of three-sex reproduction in population dynamics (Weisberg 2013), the φ 4 -model in quantum field theory (Hartmann 1995, the Lorenz model of the atmosphere {Smith, 2007 #1044), the Kac-ring model in statistical mechanics (Werndl and Frigg 2015), the logistic model of population growth (Hofbauer and Sigmund 1998) and baker's model in chaos theory (Frigg et al 2016) are all models without targets. Crucially, they aren't failures.…”