# Statistical Mechanics

Statistical mechanics (SM) is a reductionist project: it describes the behaviour of macroscopic systems in terms of laws governing their microscopic constituents and probabilistic assumptions. Systems, when left to themselves, approach equilibrium: gases spread and hot crystals cool down. What does the dynamics of the microscopic constituents have to be like for systems to approach equilibrium? What does it mean to reduce macroscopic behaviour to microscopic properties? And how should we interpret probabilities in a deterministic setting?

A currently influential view is that questions about systems' dynamics should be dispelled rather than answered. According to that view, the typicality account, what explains the approach to equilibrium is that equilibrium states are typical in the system’s state space. Dynamical properties such as ergodicity are irrelevant in explaining a system’s behaviour, and probability measures are replaced by typicality measures.

The typicality approach to SM comes in different versions, which are, however, neither recognised as such nor clearly distinguished. I disentangle the different approaches and evaluate their respective success. My conclusion is that bolder versions fail, while more prudent versions leave essential questions unanswered. This forces us to go back to the drawing board. The positive proposal, which I develop in collaboration with Charlotte Werndl, departs from Ludwig Boltzmann’s idea that the approach to equilibrium should be explained in terms of ergodicity. However, Boltzmann’s original proposal suffers from a number of difficulties. We introduce a generalised notion of ergodicity, epsilon-ergodicity, and show that this notion not only avoids the problems of the original approach; it indeed provides a successful explanation of why systems behave thermodynamically.

The dynamics of SM systems is deterministic, and there is a temptation to interpret probabilities in a deterministic theory merely as reflecting an observer’s lack of knowledge. Such an interpretation ought to be resisted. Taking David Lewis’ Humean objective change as our point of departure, Carl Hoefer and I have developed an account of objective probability that makes such probabilities compatible with determinism, and we argue that probabilities in SM should be interpreted as objective probabilities.

Different statements of the reductive aims of SM emphasise different aspects of reduction, but all tend to agree that a successful reduction amounts to deriving the laws of thermodynamics (or laws very similar to them) from the laws of microphysics. This has a familiar ring to it: deducing the laws of one theory from another, more fundamental one, is precisely what Nagel considered a reduction to be. Indeed, the Nagelian model of reduction seems to be the usually unquestioned and unacknowledged ‘background philosophy’ of SM. This puts us in an awkward situation because Nagelian reduction is generally regarded as outdated and misconceived. Together with Foad Dizadji-Bahmani and Stephan Hartmann I reconsider the Nagelian theory of reduction and argue that it is, after all, the right analysis of intertheoretic reduction.

The relation between the Boltzmannian and the Gibbian approach:

‘When do Gibbsian Phase Averages and Boltzmannian Equilibrium Values Agree?, forthcoming in

*Studies in History and Philosophy of Modern Physics*, with Charlotte Werndl.‘Can Somebody Please Say What Gibbsian Statistical Mechanics Says?’, forthcoming in

*The British Journal for Philosophy of Science*, with Charlotte Werndl.‘Statistical Mechanics: A Tale of Two Theories',

*The Monist*102, 2019, 424–438, with Charlotte Werndl.‘Taming Abundance: on the Relation between Boltzmannian and Gibbsian Statistical Mechanics', forthcoming in Valia Allori (ed.):

*Statistical Mechanics and Scientific Explanation: Determinism, Indeterminism and Laws of Nature*, World Scientific, with Charlotte Werndl.‘Ehrenfest and Ehrenfest-Afanassjewa on Why Boltzmannian and Gibbsian Calculations Agree', forthcoming in Giovanni Valente, Charlotte Werndl, and Lena Zuchowski (eds):

*The Philosophy of Tatiana Ehrenfest-Afanassjewa*, with Charlotte Werndl.‘Mind the Gap: Boltzmannian versus Gibbsian Equilibrium’,

*Philosophy of Science*84(5), 2017, 1289–1302, with Charlotte Werndl.

Rethinking equilibrium:

'Reconceptualising Equilibrium in Boltzmannian Statistical Mechanics and Characterising its Existence',

*Studies in History and Philosophy of Modern Physics*49(1), 2015, 19–31, with Charlotte Werndl.'Rethinking Boltzmannian Equilibrium',

*Philosophy of Science*82(5), 2015, 1224–1235, with Charlotte Werndl.'When Does a Boltzmannian Equilibrium Exist?', to be published in Daniel Bedingham, Owen Maroney and Christopher Timpson (eds.):

*Quantum Foundations of Statistical Mechanics*, Oxford: Oxford University Press, with Charlotte Werndl.'Boltzmannian Equilibrium in Stochastic Systems', in: Michela Massimi and Jan-Willem Romeijn (eds):

*Proceedings of the EPSA15 Conference*. Berlin and New York: Springer 2017, 243-254, with Charlotte Werndl.

The approach to SM based on epsilon-ergodicity:

‘Explaining Thermodynamic-Like Behaviour in Terms of Epsilon-Ergodicity’,

*Philosophy of Science*78(3), 2011, 628–652, with Charlotte Werndl.‘A New Approach to the Approach to Equilibrium’, in Yemima Ben-Menahem and Meir Hemmo (eds.):

*Probability in Physics*. Springer: Berlin, Heidelberg, 2012, 99-113, with Charlotte Werndl.

A discussion of the typicality approach:

‘Demystifying Typicality’,

*Philosophy of Science*79(5), 2012, 917–929, with Charlotte Werndl.‘Why Typicality Does Not Explain the Approach to Equilibrium’, in Mauricio Suárez (ed.) ,

*Probabilities, Causes and Propensities in Physics*. Synthese Library. Dordrecht: Springer, 2010, 77-93.‘Typicality and the Approach to Equilibrium in Boltzmannian Statistical Mechanics’,

*Philosophy of Science*76, 2009, 997–1008.

Probability, determinism, and probabilities in statistical mechanics:

‘The Best Humean System for Statistical Mechanics’,

*Erkenntnis*, available on the Journal’s website as Online First Publication, with Carl Hoefer.‘Probability in Boltzmannian Statistical Mechanics’, in Gerhard Ernst and Andreas Hüttemann (eds.):

*Time, Chance and Reduction. Philosophical Aspects of Statistical Mechanics*. Cambridge: Cambridge University Press, 2010.‘Determinism and Chance from a Humean Perspective’, in Dennis Dieks, Wenceslao Gonzalez, Stephan Hartmann, Marcel Weber, Friedrich Stadler and Thomas Uebel (eds.):

*The Present Situation in the Philosophy of Science*. Berlin and New York: Springer, 2010, 351-372, with Carl Hoefer.‘Humean Chance in Boltzmannian Statistical Mechanics’,

*Philosophy of Science*75, 2008, 670–681.

Nagelian reduction vindicated:

- ‘Who’s Afraid of Nagelian Reduction’,
*Erkenntnis*73, 2010, 393–412, with Foad Dizadji-Bahmani and Stephan Hartmann.

Coming to grips with entropy:

- ‘Entropy – A Guide for the Perplexed’, in Claus Beisbart and Stephan Hartmann (eds.):
*Probability in Physics*, Oxford University Press, 2011, 115-42, with Charlotte Werndl.

Surveys of work on SM and deterministic chance:

‘Chance and Determinism’, in Alan Hájek and Christopher Hitchcock (eds):

*The Oxford Handbook of Probability and Philosophy*. Oxford: Oxford University Press, 2016, 460-474.‘A Field Guide to Recent Work on the Foundations of Statistical Mechanics’, in Dean Rickles (ed.):

*The Ashgate Companion to Contemporary Philosophy of Physics*. London: Ashgate, 2008, 99-196.

Introductions to the foundations of SM:

‘What Is Statistical Mechanics?’, forthcoming in Carlos Galles, Pablo Lorenzano, Eduardo Ortiz, and Hans-Jörg Rheinberger (eds.):

*History and Philosophy of Science and Technology*, Encyclopedia of Life Support Systems Volume 4, Isle of Man: Eolss.‘The Ergodic Hierarchy, in Edward N. Zalta (ed.):

*Stanford Encyclopedia of Philosophy*(first published April 2011; substantive revision May 2016), with Joseph Berkovitz and Fred Kronz.‘Qu’est-ce que c’est la mécanique statistique ?’, in Sozig Le Bihan (ed.):

*Précis de Philosophie de la Physique*. Paris: Vuibert, 2013, 152-176.‘Grundprobleme der Statistischen Mechanik’, in Michael Esfeld (ed.):

*Philosophie der Physik*, Frankfurt am Main: Suhrkamp, 2012, 325-41.