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AN OBJECTIVISTACCOUNT of PROBABILITIES in STATISTICAL MECHANICS D. A. Lavis

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AN OBJECTIVISTACCOUNT of PROBABILITIES in STATISTICAL MECHANICS D. A. Lavis 3 AN OBJECTIVISTACCOUNT OF PROBABILITIES IN STATISTICAL MECHANICS D. A. Lavis 1 Introduction The foundations of classical (meaning nonquantum) statistical mechanics can be characterized by the triplet (i) classical mechanics ! (ii) probability ! (iii) thermodynamics, and the primary-level classification of theories of probability is between (a) epistemic interpretations, in which probability is a measure of the degree of belief (rational, or otherwise) of an individual or group, and (b) objective interpretations, in which probability is a feature of the material world independent of our knowledge or beliefs about that world. Hoefer (2007, p. 550) offers the opinion that ‘the vast majority of scientists using non-subjective probabilities overtly or covertly in their research feel little need to spell out what they take their objective probabilities to be.’ My experience (as a member of that group) accords with that view; indeed it goes further, in finding that most scientists are not even particularly interested in thinking about whether their view of probability is objective or subjective. The irritation expressed by Margenau (1950, p. 247) with ‘the missionary fervor with which everybody tries to tell the scientist what he ought [his italics] to mean by probability’ would not be uncommon. The relevant question here is, of course, the meaning of probability in statistical mechanics, and the approach most often used in textbooks on statistical mechanics (written for a scientific, as distinct from philosophically inclined, audience) is to either sidestep discussion of interpretations1 or offer remarks of a somewhat ambiguous nature. Compare, for example, the assertion by Tolman (1938, pp. 59–60) that in ‘the case of statistical mechanics the typical situation, in which statistical predictions are desired, arises when our knowledge of the condition of some system of interest [his italics] is not sufficient for a specification of its precise state’ with the remark two pages later that by ‘the 1See, for example, Lavis & Bell 1999, Ch. 2. C. Beisbart & S. Hartmann (eds), Probabilities in Physics, Oxford University Press 2011, 51–81 52 D. A. Lavis probability [his italics] of finding the system in any given condition, we mean the fractional number of times it would actually be found in that condition on repeated trials of the same experiment.’ The intention of this essay is to propose the use of a particular hard-objective, real-world chance, interpretation for probability in statistical mechanics. This approach falls into two parts and is applicable to any system for which the dynamics supports an ergodic decomposition. Following von Plato 1989b we propose the use of the time-average definition of probability within the mem- bers of the ergodic decomposition. That part of the program has already been presented in Lavis 2005 and 2008. It is now augmented by the second part, in which a version of Cartwright’s (1999) nomological machine is used to as- sign probabilities over the members of the decomposition. It is shown that this probability scheme is particularly well adapted to a version of the Boltzmann approach to statistical mechanics, in which the preoccupation with the tem- porally local increase in entropy is replaced by an interest in the temporally global entropy profile, and in which the binary property of being or not being in equilibrium is replaced by a continuous equilibrium-like property which is called ‘commonness.’ The place of the Gibbs approach within this program is also discussed. 2 The dynamic system In this section we introduce the dynamic structure of the system and the idea of ergodic decomposition.2 2.1 Dynamic flow and invariant subsets The states of the dynamic system are represented by points x 2 G, the phase space of the system, and the dynamics is given by a flow which is a semigroup ? ? f ft j t ≥0 g of automorphisms on G, parameterized by time t 2 Z or R . The points of G can also form a continuum or be discrete. The Hamiltonian motion of the particles of a gas in a box is a case where both G and t are continuous, the baker’s gas (see, for example, Lavis 2005) is a case where G is continuous and t is discrete, and the ring model of Kac (1959) (see also Lavis 2008) is a case where both G and t are discrete. In the following we shall, for simplicity, restrict most of the discussion to systems, like that of Ex. 2 below, where G is a suitably compactified Euclidean space; implying, of course, a metric measuring the distance between points and a Lebesgue measure, which we denote by mL. Ergodic decomposition applies to compact metric spaces (see Thm A.2), and also to systems, like Ex. 1 below, which have discrete time and a phase space consisting of a finite number of points; 2For the interested reader the mathematical underpinning of this section is provided in App. A. An Objectivist Account of Probabilities in Statistical Mechanics 53 there the ergodic decomposition is just the partition into cycles. For all systems the dynamics is assumed to be3 (i) forward-deterministic: if the phase point at some time t0 > 0 is x0, then the trajectory of phase points x0 ! ftx0, for all t ≥ 0, is uniquely determined by the flow; (ii) reversible (or invertible): there exists a self-inverse operator I on the points −1 of G, such that ftx0 = x1 =) ftIx1 = Ix0; then f−t := (ft) = IftI, and fftg becomes a group, with t 2 R or Z. A forward-deterministic system defined by an invertible flow is also backward-deterministic and thus simply deterministic; (iii) autonomous (or stationary): if x0 and x1 are related by the condition that, if x0 is the phase point at t0 then x1 is the phase point at t0 +t1, then this relationship between x0 and x1 would also hold if t0 were replaced by any 0 other time t0. Determinism, as defined by (i) and (ii), means that, if two trajectories x(t) and x˜(t) occupy the same state x(t0) = x˜(t0) at some specified time t0, then they coincide for all times. If the system is also autonomous, then the determinism condition is generalized to one in which, if two trajectories x(t) and x˜(t) 0 0 have x(t0) = x˜(t0 +t ), then they coincide with x(t) = x˜(t+t ), for all times t. The time parameter now measures relative time differences rather than absolute times, and for the trajectory x(t) containing x0, which we denote by Lx0 , we can, without loss of generality, let x(0) = x0. Hamiltonian systems, for which the Hamiltonian is nontrivially a function of t, are examples of nonautonomous systems. For our discussion we need to define on G a sigma-algebra S of subsets, which is preserved by the flow, meaning that (a) s 2 S =) fts 2 S; (b) s1, s2 2 S =) ft(s1[ s2) = (fts1) [ (fts2) and ft(s1\ s2) = (fts1) \ (fts2). A set s 2 S is f-invariant iff every trajectory with a phase point in s is completely contained in s, that is, fts = s.A f-invariant s 2 S is Lebesgue-indecomposable iff 4 there do not exist any f-invariant proper subsets of s. We denote by Sf the set of f-invariant members of S and by Sef the subset of Sf consisting of Lebesgue- indecomposable members. It is not difficult to show that Sef is a decomposition (or 3Although we state the following assumptions cumulatively, and (ii) and (iii) are predicated on (i), they could be applied separately. Indeed, most of the discussion in this essay could be adapted to a system which is only forward-deterministic. 4 0 0 0 s is a proper subset of s iff mL(s ) > 0 and mL(sns ) > 0. 54 D. A. Lavis Γ5 Γ4 Γ3 Γ2 Γ1 FIG. 1. The ergodic decomposition of G, where G1,..., G5 are Lebesgue- indecomposable. partition) of G, meaning that almost every5 x 2 G belongs to exactly one member of Sef. We denote the members of Sef by Gl, where l could take integer values or a range of real values. This situation is represented schematically in Fig. 1, where Sef := fG1, ... , G5g and Sf is Sef together with any unions of its members; thus G1 [ G2 belongs to Sf but not to Sef. 2.2 Invariant measures and ergodicity A measure m is said to be f-invariant if it is preserved by the dynamics; that is, m(fts) = m(s), for all s 2 S and t. We denote by Mf the set of f-invariant measures on S such that6 (1) m(G) = 1 (the measure is normalized over phase space); (2) m is absolutely continuous with respect to mL, meaning that for all s 2 S, mL(s) = 0 =) m(s) = 0. A set s is total (or of total measure) if m(Gns) = 0 for all m 2 Mf, and an important consequence of assuming absolute continuity is that we can interpret ‘almost everywhere’ as ‘in a set of total measure,’ which is the condition for the validity of the conclusions reached in this section and App. A. The motivation for 5 ‘Almost every’ means, for all but a set of points of mL-measure zero. ‘Almost everywhere’ is used in the same sense. 6 We shall assume that Mf 6= ?, which is true (Mañé 1987, Thm 1.8.1) if f is continuous. An Objectivist Account of Probabilities in Statistical Mechanics 55 restricting attention to normalized absolutely continuous f-invariant measures is discussed in Sec. 3.1 when we define the Boltzmann entropy.
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