Statistical Mechanics and Scientific Explanation

The following sections are included:

• Contents
• Introduction

Building on the Boltzmann’s approach to statistical mechanics, David Albert proposed a framework for a complete physical theory that entails a probability distribution over all physical possible worlds. Albert and I call this framework “the Mentaculus.” In this paper, I provide reasons to think that the Mentaculus entails probabilistic versions of the laws of thermodynamics and other special science laws. In addition, it is the basis for a scientific account of the arrows of time and an account of counterfactuals that express causal relations. I then argue that the best way to understand the laws and probabilities that occur in the Mentaculus are along the lines of David Lewis’ best system account.

We will discuss the link between scientific explanations and probabilities, specially in relationship with statistical mechanics and the derivation of macroscopic laws from microscopic ones.

A common way of characterizing Boltzmann’s explanation of thermodynamics in term of statistical mechanics is with reference to three ingredients: the dynamics, the past hypothesis, and the statistical postulate. In this paper I focus on the statistical postulate, and I have three aims. First, I wish to argue that regarding the statistical postulate as a probability postulate may be too strong: a postulate about typicality would be enough. Second, I wish to show that there is no need to postulate anything, for the typicality postulate can be suitably derived from the dynamics. Finally, I discuss how the attempts to give preference to certain stochastic quantum theories (such as the spontaneous collapse theory) over deterministic alternatives on the basis that they do not need the statistical postulate fail.

In this chapter, I urge a fresh look at the problem of explaining equilibration. The process of equilibration, I argue, is best seen not as part of the subject matter of thermodynamics, but as a presupposition of thermodynamics. Further, the relevant tension between the macroscopic phenomena of equilibration and the underlying microdynamics lies not in a tension between time-reversal invariance of the microdynamics and the temporal asymmetry of equilibration, but in a tension between preservation of distinguishability of states at the level of microphysics and the continual effacing of the past at the macroscopic level. This suggests an open systems approach, where the puzzling question is not the erasure of the past, but the question of how reliable prediction, given only macroscopic data, is ever possible at all. I suggest that the answer lies in an approach that has not been afforded sufficient attention in the philosophical literature, namely, one based on the temporal asymmetry of causal explanation.

Typicality exerts a steady fascination on many experts of statistical mechanics. It is simple, it sounds right, and it falls naturally into the great tradition of Boltzmann’s groundbreaking work. But typicality is a tricky business. How and what exactly typicality is supposed to explain is far from clear and, for several physicists and philosophers, it is too simple to be right. In particular, typicality seems to sideline dynamical considerations, which is much too high a price to pay. In this paper, I try to make order in the confused debate on typicality. I construe a general epistemological framework in which typicality explanations can be discuss and introduce some basic distinctions. My main point is that typicality is a pluralistic concepts that calls for a pluralistic attitude. From this perspective, the confusion of the debate on typicality starts to make sense.

The notion of typicality appears in scientific theories, philosophical arguments, mathematical inquiry, and everyday reasoning. Typicality is invoked in statistical mechanics to explain the behavior of gases. It is also invoked in quantum mechanics to explain the appearance of quantum probabilities. Typicality plays an implicit role in non-rigorous mathematical inquiry, as when a mathematician forms a conjecture based on personal experience of what seems typical in a given situation. Less formally, the language of typicality is a staple of the common parlance: we often claim that certain things are, or are not, typical. But despite the prominence of typicality in science, philosophy, mathematics, and everyday discourse, no formal logics for typicality have been proposed. In this paper, we propose two formal systems for reasoning about typicality. One system is based on propositional logic: it can be understood as formalizing objective facts about what is and is not typical. The other system is based on the logic of intuitionistic type theory: it can be understood as formalizing subjective judgments about typicality.

It is easy to suspect that the appearance of probability measures — or anything close to probability measures — in a physical theory must ultimately trace back to the supposition of some probability measure in the foundations of the theory. This latter could either arise when specifying the dynamics of the theory (if the dynamics is fundamentally stochastic) or else as a probability measure over the space of initial states of the theory. The latter sort of postulate is immediately puzzling: where should a probability measure over initial conditions come from, and what are the truth conditions for ascribing it to a system? This whole problematic is a conceptual illusion. One can derive something close to a probability measure from a deterministic theory without putting such a measure over the space of initial states. The additional conceptual resource one needs is the notion of typicality. I will formally exposit what the concept of typicality is, why it is much less detailed than a probability measure. I will explain how to derive just what one wants from it: the notion of typical empirical frequencies according to the theory. The theory can then be tested by comparing the observed empirical frequencies to the typical ones.

Can the second law of thermodynamics explain our mental experience of the direction of time? By this experience, we mean our psychological feel of distinction between the past and the future; we call this experience the ‘psychological arrow of time’. An influential approach (for example, by Albert and Loewer) argues that the second law gives rise to the psychological temporal arrow. On this approach, accounting for the temporal directionality of the second law in terms of fundamental physics (which is time-symmetric) requires introducing the past hypothesis of low entropy. The idea is that the directionality of the universal increase of entropy explains how the psychological arrow comes about. We argue that there are two necessary conditions on the workings of the brain that any account of the psychological arrow of time must satisfy, and we show that the past hypothesis of a universal low entropy state does not satisfy these conditions. We propose a new reductive (physical) account of the psychological arrow compatible with time-symmetric physics, according to which our two necessary conditions are sufficient. Our proposal has some radical implications, for example, that the psychological arrow is fundamental, whereas the temporal direction of entropy increase in the past hypothesis is derived from it, rather than the other way around.

In this paper, we apply the popular Best System Account of laws to typical eternal recurrence worlds — both classical worlds and worlds of the kind posited by popular contemporary cosmological theories. Our first thesis is that, according to the Best System Account, such worlds will have no laws that meaningfully constrain boundary conditions.

It is generally thought that lawful constraints on boundary conditions are required to avoid skeptical arguments. Thus, the lack of such laws given the Best System Account may seem like a severe problem for the view. Our second thesis is that at eternal recurrence worlds, lawful constraints on boundary conditions do little to help fend off worries. So with respect to handling these worries, the proponent of the Best System Account is no worse off than their competitors.

The paper discusses recent proposals by Carroll and Chen, as well as Barbour, Koslowski, and Mercati to explain the (thermodynamic) arrow of time without a Past Hypothesis, that is, the assumption of a special (low-entropy) initial state of the universe. After discussing the role of the Past Hypothesis and the controversy about its status, we explain why Carroll’s model — which establishes an arrow of time as typical — can ground sensible predictions and retrodictions without assuming something akin to a Past Hypothesis. We then propose a definition of a Boltzmann entropy for a classical N-particle system with gravity, suggesting that a Newtonian gravitating universe might provide a relevant example of Carroll’s entropy model. This invites comparison with the work of Barbour, Koslowski, and Mercati that identifies typical arrows of time in a relational formulation of classical gravity on shape space. We clarify the difference between this gravitational arrow in terms of shape complexity and the entropic arrow in absolute spacetime, and work out the key advantages of the relationalist theory. We end by pointing out why the entropy concept relies on absolute scales and is thus not relational.

This contribution inquires into Clausius’ proposal that “the entropy of the world tends to a maximum.” The question is raised whether the entropy of ‘the world’ actually does have a maximum; and if the answer is “Yes!,” what such states of maximum entropy look like, and if the answer is “No!,” what this could entail for the fate of the universe. Following R. Penrose, ‘the world’ is modelled by a closed Friedman–Lemaître type universe, in which a three-dimensional spherical ‘space’ is filled with ‘matter’ consisting of N point particles, their large-scale distribution being influenced by their own gravity. As ‘entropy of matter’, the Boltzmann entropy for a (semi-)classical macrostate, and Boltzmann’s ergodic ensemble formulation of it for an isolated thermal equilibrium state, are studied. Since the notion of a Boltzmann entropy is not restricted to classical non-relativistic physics, the inquiry will take into account quantum theory as well as relativity theory; we also consider black hole entropy. Model universes having a maximum entropy state and those which don’t will be encountered. It is found that the answer to our maximum entropy question is not at all straightforward at the general-relativistic level. In particular, it is shown that the increase in Bekenstein–Hawking entropy of general-relativistic black holes does not always compensate for the Boltzmann entropy of a piece of matter swallowed by a black hole.

We compare and contrast two distinct approaches to understanding the Born rule in de Broglie-Bohm pilot-wave theory, one based on dynamical relaxation over time (advocated by this author and collaborators) and the other based on typicality of initial conditions (advocated by the ‘Bohmian mechanics’ school). It is argued that the latter approach is inherently circular and physically misguided. The typicality approach has engendered a deep-seated confusion between contingent and law-like features, leading to misleading claims not only about the Born rule but also about the nature of the wave function. By artificially restricting the theory to equilibrium, the typicality approach has led to further misunderstandings concerning the status of the uncertainty principle, the role of quantum measurement theory, and the kinematics of the theory (including the status of Galilean and Lorentz invariance). The restriction to equilibrium has also made an erroneously-constructed stochastic model of particle creation appear more plausible than it actually is. To avoid needless controversy, we advocate a modest ‘empirical approach’ to the foundations of statistical mechanics. We argue that the existence or otherwise of quantum nonequilibrium in our world is an empirical question to be settled by experiment.

In a quantum universe with a strong arrow of time, it is standard to postulate that the initial wave function started in a particular macrostate — the special low-entropy macrostate selected by the Past Hypothesis. Moreover, there is an additional postulate about statistical mechanical probabilities according to which the initial wave function is a “typical” choice in the macrostate (the Statistical Postulate). Together, they support a probabilistic version of the Second Law of Thermodynamics: typical initial wave functions will increase in entropy. Hence, there are two sources of randomness in such a universe: the quantum-mechanical probabilities of the Born rule and the statistical mechanical probabilities of the Statistical Postulate.

I propose a new way to understand time’s arrow in a quantum universe. It is based on what I call the Thermodynamic Theories of Quantum Mechanics. According to this perspective, there is a natural choice for the initial quantum state of the universe, which is given not by a wave function but by a density matrix. Importantly, that density matrix plays two roles in my theories. First, it plays a microscopic role by appearing in the fundamental dynamical equations. Second, it plays a macroscopic/thermodynamical role by being exactly the normalized projection onto the Past Hypothesis subspace (inside the Hilbert space of the universe). This also entails that, given an initial subspace, we obtain a unique choice of the initial density matrix. I call this property the conditional uniqueness of the initial quantum state. The conditional uniqueness provides a new and general strategy to eliminate statistical mechanical probabilities in the fundamental physical theories, by which we can reduce the two sources of randomness to only the quantum mechanical one. The dual role of the initial density matrix also enables us to explore the possibility of an absolutely unique initial quantum state, in a way that might realize Penrose’s idea (1989) of a strongly deterministic universe.

The Gibbs entropy of a macroscopic classical system is a function of a probability distribution over phase space, i.e., of an ensemble. In contrast, the Boltzmann entropy is a function on phase space, and is thus defined for an individual system. Our aim is to discuss and compare these two notions of entropy, along with the associated ensemblist and individualist views of thermal equilibrium. Using the Gibbsian ensembles for the computation of the Gibbs entropy, the two notions yield the same (leading order) values for the entropy of a macroscopic system in thermal equilibrium. The two approaches do not, however, necessarily agree for non-equilibrium systems. For those, we argue that the Boltzmann entropy is the one that corresponds to thermodynamic entropy, in particular, in connection with the second law of thermodynamics. Moreover, we describe the quantum analog of the Boltzmann entropy, and we argue that the individualist (Boltzmannian) concept of equilibrium is supported by the recent works on thermalization of closed quantum systems.

In discussions of the foundations of statistical mechanics, it is widely held that (a) the Gibbsian and Boltzmannian approaches are incompatible but empirically equivalent; (b) the Gibbsian approach may be calculationally preferable but only the Boltzmannian approach is conceptually satisfactory. I argue against both assumptions. Gibbsian statistical mechanics is applicable to a wide variety of problems and systems, such as the calculation of transport coefficients and the statistical mechanics and thermodynamics of mesoscopic systems, in which the Boltzmannian approach is inapplicable. And the supposed conceptual problems with the Gibbsian approach are either misconceived, or apply only to certain versions of the Gibbsian approach, or apply with equal force to both approaches. I conclude that Boltzmannian statistical mechanics is best seen as a special case of, and not an alternative to, Gibbsian statistical mechanics.

Theoreticians working in statistical mechanics seem to be spoilt for choice. The theory offers two different theoretical approaches, one associated with Boltzmann and the other with Gibbs. These approaches are neither theoretically equivalent nor in any obvious way inter-translatable. This raises the question about the relation between them. We argue that Boltzmannian statistical mechanics (BSM) is a fundamental theory while the Gibbisan approach is an effective theory, meaning that the former provides a true description of the systems within its scope while the latter offers an algorithm to calculate the values of physical quantities defined by the fundamental theory. This algorithm is often easier to handle than the fundamental theory and provides results where the fundamental theory is intractable. Being an effective theory, the Gibbsian approach works only within a certain domain of application. We provide a characterisation of the limits of the approach and argue that BSM provides correct results in cases in which the two theories disagree.

The following section is included:

• Index