A Field Guide to Recent Work on the Foundations of Statistical Mechanics Forthcoming in Dean Rickles (ed.): The Ashgate Companion to Contemporary Philosophy of Physics. London: Ashgate. Roman Frigg London School of Economics arXiv:0804.0399v1 [cond-mat.stat-mech] 2 Apr 2008 [email protected] www.romanfrigg.org March 2008 Contents 1 Introduction 4 1.1 Statistical Mechanics—A Trailer . 4 1.2 AspirationsandLimitations . 6 2 The Boltzmann Approach 8 2.1 TheFramework .......................... 9 2.2 TheCombinatorialArgument . 13 2.3 ProblemsandTasks. .. .. 21 2.3.1 Issue 1: The Connection with Dynamics . 21 2.3.2 Issue 2: Introducing and Interpreting Probabilities ... 22 2.3.3 Issue 3: Loschmidt’s Reversibility Objection . 26 2.3.4 Issue 4: Zermelo’s Recurrence Objection . 27 2.3.5 Issue 5: The Justification of Coarse-Graining . 27 2.3.6 Issue 6: Limitations of the Formalism . 28 2.3.7 Issue7:Reductionism . 29 2.3.8 Plan ............................ 30 2.4 TheErgodicityProgramme . 30 2.4.1 ErgodicTheory . .. .. 30 2.4.2 Promises.......................... 34 2.4.3 Problems ......................... 35 2.5 ThePastHypothesis . .. .. 37 2.5.1 The Past Hypothesis Introduced . 37 2.5.2 Problems and Criticisms . 39 2.6 Micro-Probabilities Revisited . 43 2.6.1 ConditionalisingonPH. 43 2.6.2 Interpreting Micro-Probabilities . 46 2.6.3 Loschmidt’s and Zermelo’s Objections . 46 2.7 Limitations ............................ 47 2.8 Reductionism ........................... 51 3 The Gibbs Approach 55 3.1 TheGibbsFormalism. 56 3.2 ProblemsandTasks. .. .. 59 3.2.1 Issue 1: Ensembles and Systems . 59 3.2.2 Issue 2: The Connection with Dynamics and the Inter- pretationofProbability . 60 2 3.2.3 Issue 3: Why does Gibbs phase averaging work? . 60 3.2.4 Issue 4: The Approach to Equilibrium . 61 3.2.5 Issue5:Reductionism . 62 3.2.6 Plan ............................ 62 3.3 Why Does Gibbs Phase Averaging Work? . 62 3.3.1 Time Averages and Ergodicity . 63 3.3.2 Khinchin’s Programme and the Thermodynamic Limit 68 3.4 Ontic Probabilities in Gibbs’ Theory . 71 3.4.1 Frequentism ........................ 71 3.4.2 TimeAverages ...................... 72 3.5 The Approach to Equilibrium . 74 3.5.1 Coarse-Graining. 74 3.5.2 Interventionism . 83 3.5.3 Changing the notion of Equilibrium . 86 3.5.4 AlternativeApproaches. 87 3.6 TheEpistemicApproach . 89 3.6.1 TheShannonEntropy . 90 3.6.2 MEPandSM ....................... 92 3.6.3 Problems ......................... 93 3.7 Reductionism ........................... 96 4 Conclusion 97 4.1 SinsofOmission ......................... 97 4.2 SummingUp ...........................100 Acknowledgements 101 Appendix 102 A.ClassicalMechanics . .102 B.Thermodynamics . .. .. .106 3 1 Introduction 1.1 Statistical Mechanics—A Trailer Statistical mechanics (SM) is the study of the connection between micro- physics and macro-physics.1 Thermodynamics (TD) correctly describes a large class of phenomena we observe in macroscopic systems. The aim of statistical mechanics is to account for this behaviour in terms of the dynam- ical laws governing the microscopic constituents of macroscopic systems and probabilistic assumptions.2 This project can be divided into two sub-projects, equilibrium SM and non-equilibrium SM.This distinction is best illustrated with an example. Consider a gas initially confined to the left half of a box (see Figure 1.1): [Insert Figure 1.1 (appended at the end of the document).] Figure 1.1 Initial state of a gas, wholly confined to the left compartment of a box separated by a barrier. This gas is in equilibrium as all natural processes of change have come to an end and the observable state of the system is constant in time, meaning that all macroscopic parameters such as local temperature and local pressure assume constant values. Now we remove the barrier separating the two halves of the box. As a result, the gas is no longer in equilibrium and it quickly disperses (see Figures 1.2 and 1.3): [Insert Figures 1.2 and 1.3 (appended at the end of the document).] 1Throughout this chapter I use ‘micro’ and ‘macro’ as shorthands for ‘microscopic’ and ‘macroscopic’ respectively. 2There is a widespread agreement on the broad aim of SM; see for instance Ehrenfest & Ehrenfest (1912, 1), Khinchin (1949, 7), Dougherty (1993, 843), Sklar (1993, 3), Lebowitz (1999, 346), Goldstein (2001, 40), Ridderbos (2002, 66) and Uffink (2007, 923). 4 Figures 1.2 and 1.3: When the barrier is removed the gas is no longer in equilibrium and disperses. This process of dispersion continues until the gas homogeneously fills the entire box, at which point the system will have reached a new equilibrium state (see Figure 1.4): [Insert Figure 1.4 (appended at the end of the document).] Figure 1.4: Gas occupies new equilibrium state. From an SM point of view, equilibrium needs to be characterised in micro- physical terms. What conditions does the motion of the molecules have to satisfy to ensure that the macroscopic parameters remain constant as long as the system is not subjected to perturbations from the outside (such as the removal of barriers)? And how can the values of macroscopic parameters like pressure and temperature be calculated on the basis of such a micro- physical description? Equilibrium SM provides answers to these and related questions. Non-equilibrium SM deals with systems out of equilibrium. How does a system approach equilibrium when left to itself in a non-equilibrium state and why does it do so to begin with? What is it about molecules and their motions that leads them to spread out and assume a new equilibrium state when the shutter is removed? And crucially, what accounts for the fact that the reverse process won’t happen? The gas diffuses and spreads evenly over the entire box; but it won’t, at some later point, spontaneously move back to where it started. And in this the gas is no exception. We see ice cubes melting, coffee getting cold when left alone, and milk mix with tea; but we never observe the opposite happening. Ice cubes don’t suddenly emerge from lukewarm water, cold coffee doesn’t spontaneously heat up, and white tea doesn’t un-mix, leaving a spoonful of milk at the top of a cup otherwise filled with black tea. Change in the world is unidirectional: systems, when left alone, move towards equilibrium but not away from it. Let us introduce a term of art and refer to processes of this kind as ‘irreversible’. The fact 5 that many processes in the world are irreversible is enshrined in the so-called Second Law of thermodynamics, which, roughly, states that transitions from equilibrium to non-equilibrium states cannot occur in isolated systems. What explains this regularity? It is the aim of non-equilibrium SM to give a precise characterisation of irreversibility and to provide a microphysical explanation of why processes in the world are in fact irreversible.3 The issue of irreversibility is particularly perplexing because (as we will see) the laws of micro physics have no asymmetry of this kind built into them. If a system can evolve from state A into state B, the inverse evolution, from state B to state A, is not ruled out by any law governing the microscopic constituents of matter. For instance, there is nothing in the laws governing the motion of molecules that prevents them from gathering again in the left half of the box after having uniformly filled the box for some time. But how is it possible that irreversible behaviour emerges in systems whose components are governed by laws which are not irreversible? One of the central problems of non-equilibrium SM is to reconcile the asymmetric behavior of irreversible thermodynamic processes with the underlying symmetric dynamics. 1.2 Aspirations and Limitations This chapter presents a survey of recent work on the foundations of SM from a systematic perspective. To borrow a metaphor of Gilbert Ryle’s, it tries to map out the logical geography of the field, place the different positions and contributions on this map, and indicate where the lines are blurred and blank spots occur. Classical positions, approaches, and questions are discussed only if they have a bearing on current foundational debates; the presentation of the material does not follow the actual course of the history of SM, nor does it aim at historical accuracy when stating arguments and positions.4 Such a project faces an immediate difficulty. Foundational debates in many other fields can take as their point of departure a generally accepted formalism and a clear understanding of what the theory is. Not so in SM. Unlike quantum mechanics and relativity theory, say, SM has not yet found a 3Different meanings are attached to the term ‘irreversible’ in different contexts, and even within thermodynamics itself (see Denbigh 1989a and Uffink 2001, Section 3). I am not concerned with these in what follows and always use the term in the sense introduced here. 4Those interested in the long and intricate history of SM are referred to Brush (1976), Sklar (1993, Ch. 2), von Plato (1994) and Uffink (2007). 6 generally accepted theoretical framework, let alone a canonical formulation. What we find in SM is a plethora of different approaches and schools, each with its own programme and mathematical apparatus, none of which has a legitimate claim to be more fundamental than its competitors. For this reason a review of foundational work in SM cannot simply begin with a concise statement of the theory’s formalism and its basic principles, and then move on to the different interpretational problems that arise. What, then, is the appropriate way to proceed? An encyclopaedic list of the different schools and their programme would do little to enhance our understanding of the workings of SM.