Boltzmannian Statistical Mechanics Research Articles

Two Approaches to Reduction: A Case Study from Statistical Mechanics

AbstractI argue that there are two distinct approaches to understanding reduction: the ontology-first approach and the theory-first approach. They concern the relation between ontological reduction and inter-theoretic reduction. Further, I argue for the significance of this distinction by demonstrating that either one or the other approach has been taken as an implicit assumption in, and has in fact shaped, our understanding of what statistical mechanics is. More specifically, I argue that Boltzmannian statistical mechanics assumes and relies on the ontology-first approach, whereas Gibbsian statistical mechanics should assume the theory-first approach.

Reducing thermodynamics to Boltzmannian statistical mechanics: the case of macro values

Thermodynamic macro variables, such as the temperature or volume macro variable, can take on a continuum of allowable values, called thermodynamic macro values. Although referring to the same macro phenomena, the macro variables of Boltzmannian Statistical Mechanics (BSM) differ from thermodynamic macro variables in an important respect: within the framework of BSM the evolution of macro values of systems with finite available phase space is invariably modelled as discontinuous, due to the method of partitioning phase space into macro regions with sharp, fixed boundaries. Conceptually, this is at odds with the continuous evolution of macro values as described by thermodynamics, as well as with the continuous evolution of the micro state assumed in BSM. This discrepancy I call the discontinuity problem. I show how it arises from BSM’s framework and demonstrate its consequences, in particular for the foundational project of reducing thermodynamics to BSM: thermodynamic macro values are shown to not supervene on the corresponding BSM macro values. With supervenience being a conditio sine qua non for the kind of reduction envisaged by the foundational project, the latter is in jeopardy.

Poincaré, Poincaré recurrence and the H-theorem: A continued reassessment of Boltzmannian statistical mechanics

In [C. G. Weaver Found. Phys. 51, 1 (2021)], I showed that Boltzmann’s H-theorem does not face a significant threat from the reversibility paradox. I argue that my defense of the H-theorem against that paradox can be used yet again for the purposes of resolving the recurrence paradox without having to endorse heavy-duty statistical assumptions outside of the hypothesis of molecular chaos. As in [C. G. Weaver Found. Phys. 51, 1 (2021)], lessons from the history and foundations of physics reveal precisely how such resolution is achieved.

A Justification of the Probabilistic Explanation of the Entropy Principle

In many ways, entropy and probability are two concepts that complement each other. But it has been argued that there is no ‘straightforward connection’ between them with a no-go thesis from Kevin Davey. However, this skeptical conclusion rests on counterexamples that fail to do justice to the entropy principle and the equivocality of the notion of probability. Proceeding from the disambiguation of probability, and acknowledging the explanatory goal of the entropy principle, it is argued that the Boltzmannian statistical mechanics account can be vindicated with a justification of the explanandum, the reference class, and the standard uniform probability measure.

When do Gibbsian phase averages and Boltzmannian equilibrium values agree?

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.

Mind the Gap: Boltzmannian versus Gibbsian Equilibrium

There are two main theoretical frameworks in statistical mechanics, one associated with Boltzmann and the other with Gibbs. Despite their well-known differences, there is a prevailing view that equilibrium values calculated in both frameworks coincide. We show that this is wrong. There are important cases in which the Boltzmannian and Gibbsian equilibrium concepts yield different outcomes. Furthermore, the conditions under which equilibriums exists are different for Gibbsian and Boltzmannian statistical mechanics. There are, however, special circumstances under which it is true that the equilibrium values coincide. We prove a new theorem providing sufficient conditions for this to be the case.

Rethinking Boltzmannian Equilibrium

Boltzmannian statistical mechanics partitions the phase space of a system into macroregions, and the largest of these is identified with equilibrium. What justifies this identification? Common answers focus on Boltzmann’s combinatorial argument, the Maxwell-Boltzmann distribution, and maximum entropy considerations. We argue that they fail and present a new answer. We characterize equilibrium as the macrostate in which a system spends most of its time and prove a new theorem establishing that equilibrium thus defined corresponds to the largest macroregion. Our derivation is completely general and does not rely on assumptions about the dynamics or interparticle interactions.

Reconceptualising equilibrium in Boltzmannian statistical mechanics and characterising its existence

In Boltzmannian statistical mechanics macro-states supervene on micro-states. 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.

Justifying typicality measures of Boltzmannian statistical mechanics and dynamical systems

A popular view in contemporary Boltzmannian statistical mechanics is to interpret the measures as typicality measures. In measure-theoretic dynamical systems theory measures can similarly be interpreted as typicality measures. However, a justification why these measures are a good choice of typicality measures is missing, and the paper attempts to fill this gap. The paper first argues that Pitowsky's (2012) justification of typicality measures does not fit the bill. Then a first proposal of how to justify typicality measures is presented. The main premises are that typicality measures are invariant and are related to the initial probability distribution of interest (which are translation-continuous or translation-close). The conclusions are two theorems which show that the standard measures of statistical mechanics and dynamical systems are typicality measures. There may be other typicality measures, but they agree about judgements of typicality. Finally, it is proven that if systems are ergodic or epsilon-ergodic, there are uniqueness results about typicality measures.

The Best Humean System for Statistical Mechanics

Classical statistical mechanics posits probabilities for various events to occur, and these probabilities seem to be objective chances. This does not seem to sit well with the fact that the theory’s time evolution is deterministic. We argue that the tension between the two is only apparent. We present a theory of Humean objective chance and show that chances thus understood are compatible with underlying determinism and provide an interpretation of the probabilities we find in Boltzmannian statistical mechanics.

On Gases in Boxes: A Reply to Davey on the Justification of the Probability Measure in Boltzmannian Statistical Mechanics

Kevin Davey claims that the justification of the second law of thermodynamics as it is conveyed by the “standard story” of statistical mechanics, roughly speaking, that low-entropy microstates tend to evolve to high-entropy microstates, is “unhelpful at best and wrong at worst.” In reply, I demonstrate that Davey’s argument for rejecting the standard story commits him to a form of skepticism that is more radical than the position he claims to be stating and that Davey places unreasonable demands on the notion of justification in the physical sciences.

Typicality and the Approach to Equilibrium in Boltzmannian Statistical Mechanics

An important contemporary version of Boltzmannian statistical mechanics explains the approach to equilibrium in terms of typicality. The problem with this approach is that it comes in different versions, which are, however, not recognized as such and not clearly distinguished. This article identifies three different versions of typicality-based explanations of thermodynamic-like behavior and evaluates their respective successes. The conclusion is that the first two are unsuccessful because they fail to take the system's dynamics into account. The third, however, is promising. I give a precise formulation of the proposal and present an argument in support of its central contention.

Mechanical theory of heat

In Boltzmannian statistical mechanics macro-states supervene on micro-states. 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.