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Thermodynamics, Statistical Mechanics and the Complexity of Reductions
Published online by Cambridge University Press: 28 February 2022
Extract
That reductions of theories and unifications of science frequently occur by means of identifications is a widely accepted hypothesis of methodology. Usually we are concerned with the micro-reductions of things, in which an entity is identified with a structured aggregate of smaller constituents. And, it is frequently alleged, we must also take into account further identifications in which the attributes of the macro-object, expressed by predicates in the macro-theory, are identified with attributes of the aggregate of micro-entities differently expressed by predicates of the micro- or reducing theory.
I am fundamentally in strong agreement with this view, but intend here to explore one case of inter-theoretic reduction where some important qualifications of this account of reduction are in order. Philosophical scepticism is frequently encountered as to the possibility of ever making coherent sense of the notion of attribute identification. This scepticism often shows itself in the opposition to the identificatory account of the reduction of the theory of mental states to that of neurological states.
- Type
- Symposium: The Unity of Science
- Information
- PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association , Volume 1974 , 1974 , pp. 15 - 32
- Copyright
- Copyright © 1976 by D. Reidel Publishing Company, Dordrecht-Holland
Footnotes
Presented as part of a colloquium on the Unity of Science at the 1974 meetings of the Philosophy of Science Association. I am grateful to the John Simon Guggenheim Memorial Foundation for their generous support during the period in which this paper was written. I am grateful to the following people for their helpful comments and suggestions on an earlier draft of the paper: Evan Jobe, Phillip Quinn, Ken Friedman, and especially P. M. Quay.
References
Notes
1 Gibbs, J., Elementary Principles in Statistical Mechanics, Dover, New York (1960), chap. XIV. The date of original publication of this work is 1902.Google Scholar
2 Tolman, R. , The Principles of Statistical Mechanics, Oxford Univ. Press, Oxford (1938), chap. XIII, ‘Statistical Explanation of the Principles of Thermodynamics’, esp. sec. 122. See p. 536 for the introduction and use of Tolman's special symbol noted in the text of this article.Google Scholar
3 Münster, A., Statistical Thermodynamics, Vol. I, English edition, Springer-Verlag, Heidelberg (1969), sec. 2.11.Google Scholar
4 Münster, A., op. cit., chap. IV. See also Ruelle, D., Statistical Mechanics, Rigorous Results, Benjamin, New York (1969), chap. 5.Google Scholar
5 A. Münster, op. cit., p. 213.
6 A. Münster, op. cit., sees. 2.1-3.2 and 4.1-4.4. See also D. Ruelle, op. cit., chaps. 1-3.
7 See, for a brief outline of some recent results in ergodic theory, Ya. Sinai, ‘Ergodic Theory,’ in Cohen, E. and Thirring, W., The Boltzmam Equation, Springer-Verlag, Wien (1973).CrossRefGoogle ScholarPubMed
8 For a non-technical outline of the ergodic approach to ‘rationalizing’ the use of ensemble averages for equilibrium values and some philosophical commentary thereon see the author's ‘Statistical Explanation and Ergodic Theory’, Philosophy of Science 40 (1973), 194-212. In that article I do not do justice, I now believe, to the first type of rationale mentioned here. I still believe, however, that an additional statistical assumption must be added to the ergodic results to complete their rationalizing program. (An additional assumption over and above the inevitable assumption that ‘sets of measure zero in the natural measure don't count.’) This additional assumption, I believe, is that the equilibrium values shown to be computable from phase averages, given a proof both of the ergodicity and the mixing property of the underlying micro-dynamics, is the equilibrium encountered in ordinary phenomenological experience.
9 See Grad, H. , ‘The Many Faces of Entropy’, Communications on Pure and Applied Mathematics 14 (1961), 323-354, esp. pp. 324-328CrossRefGoogle Scholar. See also Jaynes, E., ‘Gibbs vs. Boltzmann Entropies’, American Journal of Physics, 33 (1965), 391-398, esp. sec. VI, pp. 397-398.CrossRefGoogle Scholar
10 On the failure of the one-particle distribution function to give the correct entropy when inter-particle forces are taken into account see Jaynes, E., op. cit., pp. 391-394.Google Scholar See also his ‘Information Theory and Statistical Mechanics’, in K. Ford (ed.), Brandeis University Summer Institute Lectures in Theoretical Physics, 1962, vol. 3, ‘Statistical Physics’, pp. 181-218, esp. sec. 6, ‘Entropy and Probability’, pp. 212-217. For a discussion of how to move to the entropy defined by means of the two-particle distribution function and the generalization of this process see H. Grad, op. cit., passim.
11 On the definition of the Gibbs entropy and its relation to the Boltzmann see R. Tolman, op. cit., sec. 51, pp. 165-179, esp. (d) on pp. 174-177. There are in fact several different definitions for a Gibbs entropy, all of which “converge in the thermodynamic limit.” See J. Gibbs, op. cit., chap. XIV.
12 See Jaynes, E., ‘Gibbs vs. Boltzmann Entropies’, sec. V, pp. 395-397.Google Scholar
13 On the limits of extending thermodynamics entropy to non-equilibrium cases see Landsberg, P., Thermodynamics, Interscience, New York (1961), sec. 21, pp. 128-142. See also the work of Truesdell cited in note 19, below for important criticism of the “orthodox” view that thermodynamic quantities are “well defined” only in equilibrium situations.Google Scholar
14 For a defense of the thesis that by use of the fine-grained entropy one is perfectly able to establish the statistical mechanical “analogue” of the Second Law, see E. Jaynes, ‘Gibbs vs. Boltzmann Entropies’, sec. IV, ‘The Second Law’, pp. 394-395. See also his ‘Information Theory and Statistical Mechanics’, sec. 6, pp. 212-217.
15 The idea of coarse-graining was initiated by Gibbs. See Gibbs, J., op. cit., chap. XII, pp. 139-149.Google Scholar
16 An introduction to coarse-graining can be found in van Kampen, N., ‘Fundamental Problems in Statistical Mechanics of Irreversible Processes’, in E. Cohen (ed.), Fundamental Problems in Statistical Mechanics, Vol. I, North-Holland, Amsterdam (1961), pp. 173-202.Google Scholar See also Penrose, O., Foundations of Statistical Mechanics, Pergamon, Oxford (1970), Chap. I, ‘Basic Assumptions’, attempts a rationalization of coarse-graining. See esp. chap. I, sec. 3, ‘Observation’, where the “coarseness” of macro-observation is used to justify coarse-graining in statistical mechanics.Google Scholar
17 Not surprisingly, in the light of note 14, Jaynes offers a critique of the relevance of coarse-graining to statistical mechanics. See his ‘Gibbs vs. Boltzmann Entropies’, p. 392.
18 Field, Hartry, ‘Theory Change and the Indeterminacy of Reference’, Journal of Philosophy 70, No. 14, pp. 462-481, esp. p. 466.Google Scholar
19 For a discussion of statistical thermodynamics see Tisza, L. and Quay, P., ‘Statistical Thermodynamics of Equilibrium’, Ann. Phys. (N.Y.) 25 (1963), 48-90. Reprinted in Tisza, L., Generalized Thermodynamics, The M.I.T. Press, Cambridge, Mass., 1966.Google Scholar Citations to the work of Einstein, Mandelbrot and Szilard noted in this paper will be found in the ref00liography to this article. See also Tisza, L., ‘Thermodynamics in a State of Flux. A Search for New Foundations’, in E. Stuart, B. Gal-Or, and A. Brainard, (eds.), A Critical Review of Thermodynamics, Mono Book Corp., Baltimore, 1970.Google Scholar See also Truesdell, C., Rational Thermodynamics, McGraw-Hill, New York, 1969,Google Scholar and Glansdorff, P., and Prigogin, I., Thermodynamic Theory of Structure, Stability and Fluctuations, Wiley-Interscience, London, 1971, for other extensions of the concepts of phenomenological thermodynamics beyond the cases and methods of the traditional theory.Google Scholar
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