# Philosophy of Statistical Mechanics and Thermodynamics

### What Gibbsian Statistical Mechanics Says: Defending Bare Probabilism (2024)

In submission.

I expound and defend the ``bare probabilism'' reading of Gibbsian (i.e. mainstream) statistical mechanics, responding to Frigg and Werndl's recent (\emph{BJPS} 72 (2021), 105-129) plea: ``can somebody please say what Gibbsian statistical mechanics says?''

### The local quantum vacuum as the Past Hypothesis (2023)

In submission.

The 'Past Hypothesis', as advocated by David Albert and Barry Loewer, is the hypothesis that the world came into being in whatever particular low-entropy highly-condensed big-bang sort of macrocondition it is that the normal inferential procedures of cosmology will eventually present to us. I consider some hypotheseses about that that macrocondition is likely to be given what cosmology has already presented to us, and explore the consequences of these hypotheses for the broader ('Mentaculus') project of grounding physics and the special sciences in the Past Hypothesis. My main conclusion is that current cosmology suggests a unique, pure quantum state (the local quantum vacuum, or 'Bunch-Davies vacuum') for the initial state of the Universe, in which case statistical-mechanical probabilities emerge from quantum probabilities without any need for an intervening statistical postulate.

### Thermodynamics with and without irreversibility (2023)

To appear in O. Lombardi and C. Lopez (eds.), *The Arrow of Time: From Local Systems to the Whole Universe* (CUP, forthcoming).

Working inside the control-theoretic framework for understanding thermodynamics, I develop a systematic way to characterize thermodynamic theories via their compatibility with various notions of coarse-graining, which can be thought of as parametrizing an agent's degree of control of a system's degrees of freedom, and explore the features of those theories. Phenomenological thermodynamics is reconstructed via the `equilibration' coarse-graining where a system is coarse-grained to a canonical distribution; finer-grained forms of thermodynamics differ from phenomenological thermodynamics only in that some states of a system possess a free energy that can be extracted by reversibly transforming the system (as close as possible) to a canonical distribution. Exceeding the limits of phenomenological thermodynamics thus requires both finer-grained control of a system and finer-grained information about its state. I consider the status of the Second Law in this framework, and distinguish two versions: the principle that entropy does not decrease, and the Kelvin/Clausius statements about the impossibility of transforming heat to work, or moving heat from a cold body to a hotter body, in a cyclic process. The former should be understood as relative to a coarse-graining, and can be violated given finer control than that coarse-graining permits; the latter is absolute, and binds any thermodynamic theory compatible with the laws of physics, even the entirely reversible limit where no coarse-graining is appealed to at all. I illustrate these points via a discussion of Maxwell's demon.

### On relativistic thermodynamics (2023/2024)

In submission.

‘Relativistic thermodynamics’ should be understood not as a generalization of a non-relativistic theory but as an application of a general thermodynamic framework, neutral as to spacetime setting and allowing arbitrary conserved quantities, to the specific case of relativity. That framework gives an unambiguous result as to the thermodynamics of relativistically moving systems (an answer coinciding with Einstein’s, and Planck’s, original results.) Thermodynamic temperature is unambiguously defined as rate of change of energy with entropy at constant momentum; that said, its operational significance is limited and other measures of energy/entropy covariance, which incorporate momentum transfer, are often more useful.

(The 2024 version is a very substantial revision of a paper originally made public in 2023 under the title 'On Relativistic Temperature' - the content is largely unchanged other than the excision of a section on general relativity, but the structure and organization is drastically different and the paper itself has been largely rewritten. (The original version is still available here.)

### Naturalness and Emergence (2019)

*The Monist* 102 (2019) 499-524.

I develop an account of naturalness (that is, approximately: lack of extreme fine-tuning) in physics which demonstrates that naturalness assumptions are not restricted to narrow cases in high-energy physics but are a ubiquitous part of how interlevel relations are derived in physics. After exploring how and to what extent we might justify such assumptions on methodological grounds or through appeal to speculative future physics, I consider the apparent failure of naturalness in cosmology and in the Standard Model. I argue that any such naturalness failure threatens to undermine the entire structure of our understanding of intertheoretic reduction, and so risks a much larger crisis in physics than is sometimes suggested; I briefly review some currently-popular strategies that might avoid that crisis.

### Spontaneous Symmetry Breaking in Finite Quantum Systems: a decoherent-histories approach (2018)

In submission.

Spontaneous symmetry breaking (SSB) in quantum systems, such as ferromagnets, is normally described as (or as arising from) degeneracy of the ground state; however, it is well established that this degeneracy only occurs in spatially infinite systems, and even better established that ferromagnets are not spatially infinite. I review this well-known paradox, and consider a popular solution where the symmetry is explicitly broken by some external field which goes to zero in the infinite-volume limit; although this is formally satisfactory, I argue that it must be rejected as a physical explanation of SSB since it fails to reproduce some important features of the phenomenology. Motivated by considerations from the analogous classical system, I argue that SSB in finite systems should be understood in terms of the approximate decoupling of the system's state space into dynamically-isolated sectors, related by a symmetry transformation; I use the formalism of decoherent histories to make this more precise and to quantify the effect, showing that it is more than sufficient to explain SSB in realistic systems and that it goes over in a smooth and natural way to the infinite limit.

### The Necessity of Gibbsian Statistical Mechanics (2018)

In V. Allori (ed.), *Statistical Mechanics and Scientific Explanation: Determinism, Indeterminism and Laws of Nature* (World Scientific, 2020).

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.

### Probability and Irreversibility in Modern Statistical Mechanics: Classical and Quantum (2016)

To appear in D. Bedingham, O. Maroney and C. Timpson (eds.), *Quantum Foundations of Statistical Mechanics* (Oxford University Press, forthcoming).

Through extended consideration of two wide classes of case studies --- dilute gases and linear systems --- I explore the ways in which assumptions of probability and irreversibility occur in contemporary statistical mechanics, where the latter is understood as primarily concerned with the derivation of quantitative higher-level equations of motion, and only derivatively with underpinning the equilibrium concept in thermodynamics. I argue that at least in this wide class of examples, (i) irreversibility is introduced through a reasonably well-defined initial-state condition which does not precisely map onto those in the extant philosophical literature; (ii) probability is explicitly required both in the foundations and in the predictions of the theory. I then consider the same examples, as well as the more general context, in the light of quantum mechanics, and demonstrate that while the analysis of irreversiblity is largely unaffected by quantum considerations, the notion of statistical-mechanical probability is entirely reduced to quantum-mechanical probability.

### The Nature of the Past Hypothesis (2016)

In K. Chamcham, J. Silk and J.D. Barrow (eds.), *The Philosophy of Cosmology *(Cambridge University Press, 2017).

This is a lightly-edited transcript of my talk at the conference, focussing on the relation between different levels in physics, and in particular between those levels at which the dynamics are time-reversal invariant and those at which they contain irreversibility. We seem to "derive" theories of the latter kind from those of the former kind; what are the assumptions, explicit and tacit, required for us to make sense of these derivations?

### Thermodynamics as Control Theory (2013)

*Entropy* 16.2 (2014) 699-725.

I explore the reduction of thermodynamics to statistical mechanics by treating the former as a control theory: a theory of which transitions between states can be induced on a system (assumed to obey some known underlying dynamics) by means of operations from a fixed list. I recover the results of standard thermodynamics in this framework on the assumption that the available operations do not include measurements which affect subsequent choices of operations. I then relax this assumption and use the framework to consider the vexed questions of Maxwell's demon and Landauer's principle. Throughout I assume rather than prove the basic irreversibility features of statistical mechanics, taking care to distinguish them from the conceptually distinct assumptions of thermodynamics proper.

### The Quantitative Content of Statistical Mechanics (2013)

*Studies in the History and Philosophy of Modern Physics* 52 (2015) 285-293.

I give a brief account of the way in which thermodynamics and statistical mechanics actually work as contemporary scientific theories, and in particular of what statistical mechanics contributes to thermodynamics over and above any supposed underpinning of the latter's general principles. In doing so, I attempt to illustrate that statistical mechanics should not be thought of wholly or even primarily as itself a foundational project for thermodynamics, and that conceiving of it this way potentially distorts the foundational study of statistical mechanics itself.

(This paper was originally titled ``What statistical mechanics actually does''; the title was changed at a referee's request.)

### Inferential vs. Dynamical Conceptions of Physics (2013)

In E.Lombardi, S.Fortin, F.Holik and C.Lopez (eds.), *What is Quantum Information?*, (CUP, 2017)

I contrast two possible attitudes towards a given branch of physics: as inferential (i.e., as concerned with an agent's ability to make predictions given finite information), and as dynamical (i.e., as concerned with the dynamical equations governing particular degrees of freedom). I contrast these attitudes in classical statistical mechanics, in quantum mechanics, and in quantum statistical mechanics; in this last case, I argue that the quantum-mechanical and statistical-mechanical aspects of the question become inseparable. Along the way various foundational issues in statistical and quantum physics are (hopefully!) illuminated.

### Recurrence Theorems: a Unified Account (2013)

*Journal of Mathematical Physics* 65 (2015) 022105.

I discuss classical and quantum recurrence theorems in a unified manner, treating both as generalisations of the fact that a system with a finite state space only has so many places to go. Along the way I prove versions of the recurrence theorem applicable to dynamics on linear and metric spaces, and make some comments about applications of the classical recurrence theorem in the foundations of statistical mechanics.

- Published version (requires subscription)
- Preprint

### Probability in Physics: Statistical, Stochastic, Quantum (2013)

In A. Wilson (ed.),*Chance and Temporal Asymmmetry* (OUP, 2014)

I review the role of probability in contemporary physics and the origin of probabilistic time asymmetry, beginning with the pre-quantum case (both stochastic mechanics and classical statistical mechanics) but concentrating on quantum theory. I argue that quantum mechanics radically changes the pre-quantum situation and that the philosophical nature of objective probability in physics, and of probabilistic asymmetry in time, is dependent on the correct resolution of the quantum measurement problem.

### The Arrow of Time in Physics (2012)

In A. Bardon and H. Dyke (eds.), A Companion to the Philosophy of Time (Wiley, 2013)

I provide an overview of the various asymmetries in time --- "Arrows of time" --- found in contemporary physics, predominantly but not exclusively in statistical mechanics and thermodynamics.

### The logic of the past hypothesis (2011)

In B. Loewer, E. Winsberg and B. Weslake (ed.), *The Probability Map of the Universe: Essays on David Albert's Time and Chance* (Harvard University Press, 2023).

I attempt to get as clear as possible on the chain of reasoning by which irreversible macrodynamics is derivable from time-reversible microphysics, and in particular to clarify just what kinds of assumptions about the initial state of the universe, and about the nature of the microdynamics, are needed in these derivations. I conclude that while a "Past Hypothesis" about the early Universe does seem necessary to carry out such derivations, that Hypothesis is not correctly understood as a constraint on the early Universe's entropy.

### Gravity, Entropy and Cosmology: In Search of Clarity (2009)

*British Journal for the Philosophy of Science* 61 (2010) 513-540

I discuss the statistical mechanics of gravitating systems and in particular its cosmological implications, and argue that many conventional views on this subject in the foundations of statistical mechanics embody significant confusion; I attempt to provide a clearer and more accurate account. In particular, I observe that (i) the role of gravity *in* entropy calculations must be distinguished from the entropy *of* gravity, that (ii) although gravitational collapse is entropy-increasing, this is not usually because the collapsing matter itself increases in entropy, and that (iii) the Second Law of Thermodynamics does not owe its validity to the statistical mechanics of gravitational collapse.

- Published version (requires subscription)
- Preprint

### Implications of quantum theory in the foundations of statistical mechanics (2001)

Online only; cite as http://philsci-archive.pitt.edu/410.

An investigation is made into how the foundations of statistical mechanics are affected once we treat classical mechanics as an approximation to quantum mechanics in certain domains rather than as a theory in its own right; this is necessary if we are to understand statistical-mechanical systems in our own world. Relevant structural and dynamical differences are identified between classical and quantum mechanics (partly through analysis of technical work on quantum chaos by other authors). These imply that quantum mechanics significantly affects a number of foundational questions, including the nature of statistical probability and the direction of time.

(**Note:** though this has been cited a bit, for various reasons I've never actually got round to publishing it. By now my views have evolved sufficiently that I'm unlikely to publish it without considerable modification.)